group theory in quantum mechanics lecture 4 …...2015/01/22 · group theory in quantum mechanics...
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![Page 1: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/1.jpg)
Group Theory in Quantum MechanicsLecture 4 (1.22.15)
Matrix Eigensolutions and Spectral Decompositions (Quantum Theory for Computer Age - Ch. 3 of Unit 1 )
(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 1 )
Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping (and I’m Ba-aaack!) Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and completenessSpectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Diagonalizing Transformations (D-Ttran) from projectors Eigensolutions for active analyzers
4 13 2
⎛⎝⎜
⎞⎠⎟
1Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors)
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
2Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TTUnitary operators and matrices that change state vectors
Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TTUnitary operators and matrices that change state vectors...
Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
...and eigenstates (“ownstates) that are mostly immune to T...
T|ej〉=εj|ej〉
|ej〉
analyzer
Tanalyzer
Teigenstate |ej〉 in
|ej〉
eigenstate |ej〉 out(multiplied by εj )
TFig. 3.1.2 Effect of analyzer
on eigenket | εj 〉 is only to multiply by
eigenvalue εj ( T| εj 〉 = εj | εj 〉 ).
For Unitary operators T=U, the eigenvalues must be phase factors εk=eiαk
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping (and I’m Ba-aaack!) Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors)
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
5Tuesday, January 20, 2015
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Study a real symmetric matrix T by applying it to a circular array of unit vectors c.
A matrix T= maps the circular array into an elliptical one.
1 1/ 21/ 2 1
⎛
⎝⎜⎞
⎠⎟
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
T
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping
6Tuesday, January 20, 2015
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Study a real symmetric matrix T by applying it to a circular array of unit vectors c.
A matrix T= maps the circular array into an elliptical one. Two vectors in the upper half plane survive T without changing direction. These lucky vectors are the eigenvectors of matrix T.
1 1/ 21/ 2 1
⎛
⎝⎜⎞
⎠⎟
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
ε1 = 1
1⎛
⎝⎜⎞
⎠⎟/ 2 , ε2 = −1
1⎛
⎝⎜⎞
⎠⎟/ 2
T
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping
7Tuesday, January 20, 2015
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They transform as follows:to only suffer length change given by eigenvalues ε1 = 1.5 and ε2 = 0.5
Study a real symmetric matrix T by applying it to a circular array of unit vectors c.
A matrix T= maps the circular array into an elliptical one. Two vectors in the upper half plane survive T without changing direction. These lucky vectors are the eigenvectors of matrix T.
1 1/ 21/ 2 1
⎛
⎝⎜⎞
⎠⎟
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
ε1 = 1
1⎛
⎝⎜⎞
⎠⎟/ 2 , ε2 = −1
1⎛
⎝⎜⎞
⎠⎟/ 2
T ε1 = ε1 ε1 = 1.5 ε1 , and T ε2 = ε2 ε2 = 0.5 ε2
T
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping
8Tuesday, January 20, 2015
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They transform as follows:to only suffer length change given by eigenvalues ε1 = 1.5 and ε2 = 0.5
Study a real symmetric matrix T by applying it to a circular array of unit vectors c.
A matrix T= maps the circular array into an elliptical one. Two vectors in the upper half plane survive T without changing direction. These lucky vectors are the eigenvectors of matrix T.
1 1/ 21/ 2 1
⎛
⎝⎜⎞
⎠⎟
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
ε1 = 1
1⎛
⎝⎜⎞
⎠⎟/ 2 , ε2 = −1
1⎛
⎝⎜⎞
⎠⎟/ 2
T ε1 = ε1 ε1 = 1.5 ε1 , and T ε2 = ε2 ε2 = 0.5 ε2
Normalization (〈c|c〉 = 1) is a condition separate from eigen-relations
T
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping
T εk = εk εk
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping (and I’m Ba-aaack!) Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors)
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
10Tuesday, January 20, 2015
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Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
T
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping (and I’m Ba-aaack!)
T-1T-1
Each vector |r〉 on left ellipse maps back to vector |c〉=T-1 |r〉 on right unit circle. Each |c〉 has unit length: 〈c|c〉 = 1 = 〈r|T-1 T-1 |r〉 = 〈r|T-2|r〉. (T is real-symmetric: T†=T=TT.)
c • c = 1= r •T−2 • r = x y( ) Txx Txy
Tyx Ty
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
xy
⎛
⎝⎜
⎞
⎠⎟
c • c = 1= r •T−2 • r = x y( ) Txx Txy
Tyx Ty
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
xy
⎛
⎝⎜
⎞
⎠⎟
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Each vector |r〉 on left ellipse maps back to vector |c〉=T-1 |r〉 on right unit circle. Each |c〉 has unit length: 〈c|c〉 = 1 = 〈r|T-1 T-1 |r〉 = 〈r|T-2|r〉. (T is real-symmetric: T†=T=TT.)
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
T
c • c = 1= r •T−2 • r = x y( ) Txx Txy
Tyx Ty
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
xy
⎛
⎝⎜
⎞
⎠⎟
ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 0
0 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
, and ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
=ε1−2 0
0 ε2−2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
This simplifies if rewritten in a coordinate system (x1,x2) of eigenvectors |ε1〉 and |ε2〉 where T-2|ε1〉 = ε1-2|ε1〉 and T-2|ε2〉 = ε2-2|ε2〉, that is, T, T-1, and T-2 are each diagonal.
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping (and I’m Ba-aaack!)
Geometric visualization of real symmetric matrices and eigenvectors
T-1T-1
c • c = 1= r •T−2 • r = x y( ) Txx Txy
Tyx Ty
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
xy
⎛
⎝⎜
⎞
⎠⎟
ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 0
0 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
, and ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
=ε1−2 0
0 ε2−2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
This simplifies if rewritten in a coordinate system (x1,x2) of eigenvectors |ε1〉 and |ε2〉 where T-2|ε1〉 = ε1-2|ε1〉 and T-2|ε2〉 = ε2-2|ε2〉, that is, T, T-1, and T-2 are each diagonal.
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Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
T
c • c = 1= r •T−2 • r = x y( ) Txx Txy
Tyx Ty
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
xy
⎛
⎝⎜
⎞
⎠⎟
Each vector |r〉 on left ellipse maps back to vector |c〉=T-1 |r〉 on right unit circle. Each |c〉 has unit length: 〈c|c〉 = 1 = 〈r|T-1 T-1 |r〉 = 〈r|T-2|r〉. (T is real-symmetric: T†=T=TT.)
ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 0
0 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
, and ε1 T ε1 ε1 T ε2
ε2 T ε1 ε2 T ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
=ε1−2 0
0 ε2−2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
This simplifies if rewritten in a coordinate system (x1,x2) of eigenvectors |ε1〉 and |ε2〉 where T-2|ε1〉 = ε1-2|ε1〉 and T-2|ε2〉 = ε2-2|ε2〉, that is, T, T-1, and T-2 are each diagonal.
c• c = 1= x1 x2( ) ε1−2 0
0 ε2−2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
x1
x2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x1ε1
⎛
⎝⎜⎞
⎠⎟
2
+x2ε2
⎛
⎝⎜⎞
⎠⎟
2Matrix equation simplifies to an elementary ellipse equation of the form (x/a)2+(y/b)2=1.
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping (and I’m Ba-aaack!)
Geometric visualization of real symmetric matrices and eigenvectors
T-1T-1
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors)
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
14Tuesday, January 20, 2015
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉 Eigenvector
|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
15Tuesday, January 20, 2015
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉
Now M maps vector |q〉 from a quadratic form 1=〈q|M|q〉 to vector |p〉=M|q〉 on surface 1=〈p|M-1|p〉. 1 = 〈q|M|q〉 = 〈q|p〉= 〈p|M-1|p〉
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
M
√ε2 1/√ε1√ε1 1/√ε2
〈q|M|q〉=1〈p|M-1|p〉=1
|q〉|p〉 M maps |q〉 into |p〉=M|q〉
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉
Now M maps vector |q〉 from a quadratic form 1=〈q|M|q〉 to vector |p〉=M|q〉 on surface 1=〈p|M-1|p〉. 1 = 〈q|M|q〉 = 〈q|p〉= 〈p|M-1|p〉
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
M
√ε2 1/√ε1√ε1 1/√ε2
〈q|M|q〉=1〈p|M-1|p〉=1
|q〉|p〉 M maps |q〉 into |p〉=M|q〉
Radii of |p〉 ellipse are square roots of eigenvalues √ε1 and √ε2
Radii of |q〉 ellipse axes are inverse eigenvalue roots 1/√ε1 and 1/√ε2.
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉
Now M maps vector |q〉 from a quadratic form 1=〈q|M|q〉 to vector |p〉=M|q〉 on surface 1=〈p|M-1|p〉. 1 = 〈q|M|q〉 = 〈q|p〉= 〈p|M-1|p〉
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
M
√ε2 1/√ε1√ε1 1/√ε2
〈q|M|q〉=1〈p|M-1|p〉=1
|q〉|p〉 M maps |q〉 into |p〉=M|q〉
Tangent-normal geometry of mapping is found by using gradient ∇of quadratic curve 1=〈q|M|q〉 . ∇(〈q|M|q〉)=〈q|M + M|q〉 = 2 M|q〉 = 2 |p〉
Radii of |p〉 ellipse are square roots of eigenvalues √ε1 and √ε2
Radii of |q〉 ellipse axes are inverse eigenvalue roots 1/√ε1 and 1/√ε2.
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
18Tuesday, January 20, 2015
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉
Now M maps vector |q〉 from a quadratic form 1=〈q|M|q〉 to vector |p〉=M|q〉 on surface 1=〈p|M-1|p〉. 1 = 〈q|M|q〉 = 〈q|p〉= 〈p|M-1|p〉
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
M
√ε2 1/√ε1√ε1 1/√ε2
〈q|M|q〉=1〈p|M-1|p〉=1
|q〉|p〉 M maps |q〉 into |p〉=M|q〉
Tangent-normal geometry of mapping is found by using gradient ∇of quadratic curve 1=〈q|M|q〉 . ∇(〈q|M|q〉)=〈q|M + M|q〉 = 2 M|q〉 = 2 |p〉
M-1
〈q|M|q〉=1〈p|M-1|p〉=1
∇∇〈q|M|q〉/2=M|q〉=|p〉|q〉|p〉
90°90°|q〉 |p〉
M-1 maps |p〉 into |q〉=M-1|p〉
Radii of |p〉 ellipse are square roots of eigenvalues √ε1 and √ε2
Radii of |q〉 ellipse axes are inverse eigenvalue roots 1/√ε1 and 1/√ε2.
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
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(Previous pages) Matrix T maps vector |c〉 from a unit circle 〈c|c〉=1 to T|c〉=|r〉 on an ellipse 1=〈r|T-2|r〉
Now M maps vector |q〉 from a quadratic form 1=〈q|M|q〉 to vector |p〉=M|q〉 on surface 1=〈p|M-1|p〉. 1 = 〈q|M|q〉 = 〈q|p〉= 〈p|M-1|p〉
Eigenvector|ε1〉
ε2|ε2 〉
ε1|ε1 〉
1.0 0.50.5 1.0T =( )
Eigenvector|ε2〉
Eigenvector|ε2〉
Eigenvector|ε1〉
TT
M
√ε2 1/√ε1√ε1 1/√ε2
〈q|M|q〉=1〈p|M-1|p〉=1
|q〉|p〉 M maps |q〉 into |p〉=M|q〉
Tangent-normal geometry of mapping is found by using gradient ∇of quadratic curve 1=〈q|M|q〉 . ∇(〈q|M|q〉)=〈q|M + M|q〉 = 2 M|q〉 = 2 |p〉
M-1
〈q|M|q〉=1〈p|M-1|p〉=1
∇∇〈q|M|q〉/2=M|q〉=|p〉|q〉|p〉
90°90°|q〉 |p〉
M-1 maps |p〉 into |q〉=M-1|p〉
Radii of |p〉 ellipse are square roots of eigenvalues √ε1 and √ε2
Radii of |q〉 ellipse axes are inverse eigenvalue roots 1/√ε1 and 1/√ε2.
Mapped vector |p〉 lies on gradient ∇(〈q|M|q〉) that is normal to tangent to original curve at |q〉.
Original vector |q〉 lies on gradient ∇(〈p|M-1|p〉) that is normal to tangent to mapped curve at |p〉.
Ellipse-to-ellipse mapping (Normal vs. tangent space)
Geometric visualization of real symmetric matrices and eigenvectors
20Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors)
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
21Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
22Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
23Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
24Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
Lagrange says such points have gradient vectors ∇QL and ∇C proportional to each other.
∇QL = λ ∇C,
25Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
Lagrange says such points have gradient vectors ∇QL and ∇C proportional to each other.
∇QL = λ ∇C, Proportionality constant λ iscalled a Lagrange Multiplier.
26Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
Lagrange says such points have gradient vectors ∇QL and ∇C proportional to each other.
∇QL = λ ∇C, Proportionality constant λ iscalled a Lagrange Multiplier.
27Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
Lagrange says such points have gradient vectors ∇QL and ∇C proportional to each other.
∇QL = λ ∇C, Proportionality constant λ iscalled a Lagrange Multiplier.
At eigen-directions the Lagrange multiplier equals quadratic form: λ=QL(r)=〈r|L|r〉
QL(r)=〈εk|L|εk〉= εk at |r〉=|εk〉
28Tuesday, January 20, 2015
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Constraint curve〈r|r〉=C=1
Eigenvector|r〉=|ε2〉
where∇∇QL=λ∇∇Cwithλ=ε2
Quadratic curves〈r|L|r〉=QL=const.
.
QL=ε2
QL=ε1Eigenvector|r〉=|ε1〉
where∇∇QL=λ∇∇Cwithλ=ε1
Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Geometric visualization of real symmetric matrices and eigenvectors
Eigenvalues λ of a matrix L can be viewed as stationary-values of its quadratic form QL=L(r)=〈r|L|r〉
Q: What are min-max values of the function Q(r) subject to the constraint of unit norm: C(r)=〈r|r〉=1.
A: At those values of QL and vector r for which the QL(r) curve just touches the constraint curve C(r).
Lagrange says such points have gradient vectors ∇QL and ∇C proportional to each other.
∇QL = λ ∇C, Proportionality constant λ iscalled a Lagrange Multiplier.
At eigen-directions the Lagrange multiplier equals quadratic form: λ=QL(r)=〈r|L|r〉
QL(r)=〈εk|L|εk〉= εk at |r〉=|εk〉
〈r|L|r〉 is called a quantum expectation value of operator L at r. Eigenvalues are extreme expectation values.
29Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and CompletenessSpectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
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Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with each eigenvector direction.
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε k
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An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with each eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
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ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with each eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε k
33Tuesday, January 20, 2015
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An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 5
First step in finding eigenvalues: Solve secular equation
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An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
a1 = −TraceM,, ak = −1( )k diagonal k-by-k minors of ∑ M,, an = −1( )n det M0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M)
First step in finding eigenvalues: Solve secular equation
where:
35Tuesday, January 20, 2015
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First step in finding eigenvalues: Solve secular equation
where:
Secular equation has n-factors, one for each eigenvalue.
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
a1 = −TraceM,, ak = −1( )k diagonal k-by-k minors of ∑ M,, an = −1( )n det M
detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( ) ε − εn( )
0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5
0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5
36Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and CompletenessSpectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
37Tuesday, January 20, 2015
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First step in finding eigenvalues: Solve secular equation
where:
Secular equation has n-factors, one for each eigenvalue.
Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.
Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
a1 = −TraceM,, ak = −1( )k diagonal k-by-k minors of ∑ M,, an = −1( )n det M
detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( ) ε − εn( )
0 = M − ε11( ) M − ε21( ) M − εn1( )
0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5
0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)
0 00 0
⎛⎝⎜
⎞⎠⎟= 4 1
3 2⎛⎝⎜
⎞⎠⎟
2
− 6 4 13 2
⎛⎝⎜
⎞⎠⎟+ 5 1 0
0 1⎛⎝⎜
⎞⎠⎟
0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5
38Tuesday, January 20, 2015
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1st step in finding eigenvalues: Solve secular equation
where:
Secular equation has n-factors, one for each eigenvalue.
Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.
Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
a1 = −TraceM,, ak = −1( )k diagonal k-by-k minors of ∑ M,, an = −1( )n det M
detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( ) ε − εn( )
0 = M − ε11( ) M − ε21( ) M − εn1( )
0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5
0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)
0 00 0
⎛⎝⎜
⎞⎠⎟= 4 1
3 2⎛⎝⎜
⎞⎠⎟
2
− 6 4 13 2
⎛⎝⎜
⎞⎠⎟+ 5 1 0
0 1⎛⎝⎜
⎞⎠⎟
0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5
Replace jth HC-factor by (1) to make projection operators . pk =j≠k∏ M − ε j1( ) p1 = (1)(M − 5⋅1) = 4 − 5 1
3 2 − 5⎛⎝⎜
⎞⎠⎟= −1 1
3 −3⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1)(1) = 4 −1 13 2 −1
⎛⎝⎜
⎞⎠⎟= 3 1
3 1⎛⎝⎜
⎞⎠⎟
ε j ≠ εk ≠ ...
p1 = 1 ( ) M − ε21( ) M − εn1( )p2 = M − ε11( ) 1 ( ) M − εn1( ) pn = M − ε11( ) M − ε21( ) 1 ( )
(Assume distinct e-values here: Non-degeneracy clause)
39Tuesday, January 20, 2015
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1st step in finding eigenvalues: Solve secular equation
where:
Secular equation has n-factors, one for each eigenvalue.
Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.
Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)
An eigenvector of M is in a direction that is left unchanged by M.
εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives
M ε k = ε k ε k , or: M − ε k1( ) ε k = 0
ε k
ε1 , ε2 , εn{ }
ε1 M ε1 ε1 M ε2 ε1 M εnε2 M ε1 ε2 M ε2 ε2 M εn
εn M ε1 εn M ε2 εn M εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
ε1 0 00 ε2 0 0 0 εn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ε k
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Only possible non-zero {x,y} if denominator is zero, too!
0=det M − ε ⋅1=det 4 13 2
⎛⎝⎜
⎞⎠⎟−ε 1 0
0 1⎛⎝⎜
⎞⎠⎟=det 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟
M ε = 4 13 2
⎛⎝⎜
⎞⎠⎟
xy
⎛
⎝⎜
⎞
⎠⎟ = ε x
y⎛
⎝⎜
⎞
⎠⎟ or: 4 − ε 1
3 2 − ε⎛
⎝⎜⎞
⎠⎟xy
⎛
⎝⎜
⎞
⎠⎟ =
00
⎛⎝⎜
⎞⎠⎟
Trying to solve by Kramer's inversion:
x =det 0 1
0 2 − ε⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
and y =det 4 − ε 0
3 0⎛⎝⎜
⎞⎠⎟
det 4 − ε 13 2 − ε
⎛
⎝⎜⎞
⎠⎟
detM − ε1 = 0 = −1( )n ε n + a1ε
n−1 + a2εn−2 +…+ an−1ε + an( )
a1 = −TraceM,, ak = −1( )k diagonal k-by-k minors of ∑ M,, an = −1( )n det M
detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( ) ε − εn( )
0 = M − ε11( ) M − ε21( ) M − εn1( )
0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5
0 =M2 − 6M + 5M = (M −1⋅1)(M − 5⋅1)
0 00 0
⎛⎝⎜
⎞⎠⎟= 4 1
3 2⎛⎝⎜
⎞⎠⎟
2
− 6 4 13 2
⎛⎝⎜
⎞⎠⎟+ 5 1 0
0 1⎛⎝⎜
⎞⎠⎟
0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5
ε j ≠ εk ≠ ...
Replace jth HC-factor by (1) to make projection operators .
p1 = 1 ( ) M − ε21( ) M − εn1( )p2 = M − ε11( ) 1 ( ) M − εn1( ) pn = M − ε11( ) M − ε21( ) 1 ( )
(Assume distinct e-values here: Non-degeneracy clause)
pk =j≠k∏ M − ε j1( ) p1 = (1)(M − 5⋅1) = 4 − 5 1
3 2 − 5⎛⎝⎜
⎞⎠⎟= −1 1
3 −3⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1)(1) = 4 −1 13 2 −1
⎛⎝⎜
⎞⎠⎟= 3 1
3 1⎛⎝⎜
⎞⎠⎟
Each pk contains eigen-bra-kets since: (M-εk1)pk=0 or: Mpk=εkpk=pkM . Mp1 =
4 13 2
⎛⎝⎜
⎞⎠⎟⋅ −1 1
3 −3⎛⎝⎜
⎞⎠⎟= 1· −1 1
3 −3⎛⎝⎜
⎞⎠⎟= 1·p1
Mp2 =4 13 2
⎛⎝⎜
⎞⎠⎟⋅ 3 13 1
⎛⎝⎜
⎞⎠⎟= 5· 3 1
3 1⎛⎝⎜
⎞⎠⎟= 5·p2
40Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and CompletenessSpectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Idempotent means: P·P=P
41Tuesday, January 20, 2015
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Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
Multiplication properties of pj :
Mpk =ε kpk = pkM
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Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
P1 =(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟
P2 =(M −1⋅1)(5 −1)
= 14
3 13 1
⎛⎝⎜
⎞⎠⎟
43Tuesday, January 20, 2015
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Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟
P2 =(M −1⋅1)(5 −1)
= 14
3 13 1
⎛⎝⎜
⎞⎠⎟
44Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
45Tuesday, January 20, 2015
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Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
“Gauge” scale factors that only affect plots
46Tuesday, January 20, 2015
![Page 47: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/47.jpg)
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
⏐y〉or〈y⏐
⏐x〉 or 〈x⏐
“Gauge” scale factors that only affect plots
47Tuesday, January 20, 2015
![Page 48: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/48.jpg)
Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
48Tuesday, January 20, 2015
![Page 49: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/49.jpg)
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= 1 00 1
⎛⎝⎜
⎞⎠⎟
⏐y〉or〈y⏐
⏐x〉 or 〈x⏐
“Gauge” scale factors that only affect plots
49Tuesday, January 20, 2015
![Page 50: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/50.jpg)
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
⏐y〉or〈y⏐
⏐x〉 or 〈x⏐
“Gauge” scale factors that only affect plots
50Tuesday, January 20, 2015
![Page 51: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/51.jpg)
Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
51Tuesday, January 20, 2015
![Page 52: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/52.jpg)
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
⏐y〉
⏐x〉 or
Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn
MPk =ε kPk
“Gauge” scale factors that only affect plots
52Tuesday, January 20, 2015
![Page 53: Group Theory in Quantum Mechanics Lecture 4 …...2015/01/22 · Group Theory in Quantum Mechanics Lecture 4 (1.22.15) Matrix Eigensolutions and Spectral Decompositions (Quantum Theory](https://reader036.vdocuments.mx/reader036/viewer/2022070911/5fa756a4681f776ac5661fa4/html5/thumbnails/53.jpg)
Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn
MPk =ε kPk
M = 4 13 2
⎛⎝⎜
⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+ 5 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟
53Tuesday, January 20, 2015
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Matrix and operator Spectral Decompositons M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn
MPk =ε kPk
M = 4 13 2
⎛⎝⎜
⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+ 5 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn
54Tuesday, January 20, 2015
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Matrix and operator Spectral Decompositons M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn
MPk =ε kPk
M = 4 13 2
⎛⎝⎜
⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+ 5 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn
Example:
M50= 4 13 2
⎛⎝⎜
⎞⎠⎟=150 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+550 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟=41 1+3·550 550−1
3·550−3 550+3
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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M50= 4 13 2
⎛⎝⎜
⎞⎠⎟=150 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+550 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟=41 1+3·550 550−1
3·550−3 550+3
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Matrix and operator Spectral Decompositons M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn
MPk =ε kPk
M = 4 13 2
⎛⎝⎜
⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟+ 5 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn
Examples:
M = 4 13 2
⎛⎝⎜
⎞⎠⎟= ± 1 4
1 −41
−43
43
⎛
⎝⎜⎜
⎞
⎠⎟⎟± 5 4
341
43
41
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
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M = 4 13 2
⎛⎝⎜
⎞⎠⎟
Last step: make Idempotent Projectors: Pk =
pkε k − εm( )
m≠k∏ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏
Multiplication properties of pj :
p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏
m≠k∏
p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )
m≠k∏ =
0 if : j ≠ k
pk ε k − εm( ) if : j = km≠k∏
⎧⎨⎪
⎩⎪
p1 = (M − 5⋅1) = −1 13 −3
⎛⎝⎜
⎞⎠⎟
p2 = (M −1⋅1) = 3 13 1
⎛⎝⎜
⎞⎠⎟
p1p2 =0 00 0
⎛⎝⎜
⎞⎠⎟
Mpk =ε kpk = pkM
(Idempotent means: P·P=P)
PjPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪
Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM
P1 =(M − 5⋅1)
(1− 5)= 1
41 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Factoring bra-kets into “Ket-Bras:
〈 ε2 |= (3/2 1/2)/k21/21/2
|ε2 〉=k2
1/2-3/2
|ε1 〉=k1
〈 ε1 |= (1/2 -1/2)/k1
| 1 〉 or 〈1 |
| 2 〉or〈2 |
1/4 1/2 3/4 5/41 3/2
-1/2
-1
-3/2
1/4
1/2
3/4Eigen-bra-ketprojectorsof matrix:
M= 4 13 2
The Pj are Mutually Ortho-Normalas are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s
ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn
=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐
P1 + P2 =1 00 1
⎛⎝⎜
⎞⎠⎟
= ε1 ε1 + ε2 ε2
= 1 00 1
⎛⎝⎜
⎞⎠⎟
{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness
{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness
x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .
ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j
⏐y〉or〈y⏐
⏐x〉 or 〈x⏐
Orthonormality vs. Completeness
“Gauge” scale factors that only affect plots
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Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.
PjPk = δ jkPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪1= P1 +P2 +...+Pn
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Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.
PjPk = δ jkPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪1= P1 +P2 +...+Pn
1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk
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{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness
{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness
x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .
ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j
State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.
Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.
PjPk = δ jkPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪1= P1 +P2 +...+Pn
1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk
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{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness
{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness
x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .
ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j
State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.
Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.
PjPk = δ jkPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪1= P1 +P2 +...+Pn
1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk
x y = δ (x, y) = ψ 1(x)ψ *1(y)+ψ 2 (x)ψ *
2 (y)+ ..
However Schrodinger wavefunction notation ψ(x)=〈x⏐ψ〉 shows quite a difference...
Dirac δ-function
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{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness
{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness
x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .
ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j
State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.
Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.
PjPk = δ jkPk =0 if : j ≠ kPk if : j = k
⎧⎨⎪
⎩⎪1= P1 +P2 +...+Pn
1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk
x y = δ (x, y) = ψ 1(x)ψ *1(y)+ψ 2 (x)ψ *
2 (y)+ ..
ε i ε j = δ i, j = ...+ψ *i (x)ψ j (x)+ψ 2 (y)ψ *
2 (y)+ ....→ dx∫ ψ *i (x)ψ j (x)
However Schrodinger wavefunction notation ψ(x)=〈x⏐ψ〉 shows quite a difference… ...particularly in the orthonormality integral.
Dirac δ-function
63Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Spectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
64Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true by Lagrange interpolation)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
65Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1.
66Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) .
67Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
x2= xm
2Pm x( )m=1
N∑
68Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line,
x2= xm
2Pm x( )m=1
N∑
x1
69Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line,
x2= xm
2Pm x( )m=1
N∑
x1 x1 x2
70Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.
x2= xm
2Pm x( )m=1
N∑
x1 x1 x2 x1 x2 x2
71Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.
x2= xm
2Pm x( )m=1
N∑
Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk
72Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.
All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1.
x2= xm
2Pm x( )m=1
N∑
Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk
73Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.
P1 + P2 = j≠1∏ M − ε j1( )j≠1∏ ε1 − ε j( ) + j≠1
∏ M − ε j1( )j≠1∏ ε2 − ε j( ) =
M − ε21( )ε1 − ε2( ) +
M − ε11( )ε2 − ε1( ) =
M − ε21( )− M − ε11( )ε1 − ε2( ) =
−ε21+ ε11ε1 − ε2( ) = 1 (for all ε j )
All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1. Completeness is truer than true as is seen for N=2.
x2= xm
2Pm x( )m=1
N∑
Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk
74Tuesday, January 20, 2015
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A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions
with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.
1=P1+P2 +...+Pn = Pkεk∑ =
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )
M − εm1( )m≠k∏
ε k − εm( )m≠k∏εk
∑
L f (x)( ) = f (xk )·k=1
N∑ Pk (x) where: Pk (x) =
Πj≠k
Nx − x j( )
Πj≠k
Nxk − x j( )
Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.
1= Pm x( )
m=1
N∑
x= xmPm x( )
m=1
N∑
One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.
P1 + P2 = j≠1∏ M − ε j1( )j≠1∏ ε1 − ε j( ) + j≠1
∏ M − ε j1( )j≠1∏ ε2 − ε j( ) =
M − ε21( )ε1 − ε2( ) +
M − ε11( )ε2 − ε1( ) =
M − ε21( )− M − ε11( )ε1 − ε2( ) =
−ε21+ ε11ε1 − ε2( ) = 1 (for all ε j )
All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1. Completeness is truer than true as is seen for N=2.
However, only select values εk work for eigen-forms MΡk= εkΡk or orthonormality ΡjΡk=δjkΡk.
x2= xm
2Pm x( )m=1
N∑
Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk
75Tuesday, January 20, 2015
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Diagonalizing Transformations (D-Ttran) from projectors Eigensolutions for active analyzersSpectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
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Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
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Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.
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ε1 = 21 −2
1( ), ε2 = 23
21( ){ } , ε1 = 2
1
−23
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, ε2 = 21
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
1 −21
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.
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ε1 = 21 −2
1( ), ε2 = 23
21( ){ } , ε1 = 2
1
−23
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, ε2 = 21
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
1 −21
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x K x x K y
y K x y K y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 K ε1 ε1 K ε2
ε2 K ε1 ε2 K ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
21 −2
1
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⋅ 4 13 2
⎛
⎝⎜⎞
⎠⎟ ⋅ 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= 1 00 5
⎛
⎝⎜⎞
⎠⎟
Use Dirac labeling for all components so transformation is OK
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ε1 = 21 −2
1( ), ε2 = 23
21( ){ } , ε1 = 2
1
−23
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, ε2 = 21
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
1 −21
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x K x x K y
y K x y K y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 K ε1 ε1 K ε2
ε2 K ε1 ε2 K ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
21 −2
1
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⋅ 4 13 2
⎛
⎝⎜⎞
⎠⎟ ⋅ 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= 1 00 5
⎛
⎝⎜⎞
⎠⎟
Use Dirac labeling for all components so transformation is OK
ε1 1 ε1 2
ε2 1 ε2 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
1 ε1 1 ε2
2 ε1 2 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 1 ε1 ε1 1 ε2
ε2 1 ε1 ε2 1 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
21 −2
1
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⋅ 21
21
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= 1 00 1
⎛
⎝⎜⎞
⎠⎟
Check inverse-d-tran is really inverse of your d-tran.
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ε1 = 21 −2
1( ), ε2 = 23
21( ){ } , ε1 = 2
1
−23
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, ε2 = 21
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
1 −21
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
Given our eigenvectors and their Projectors.
Diagonalizing Transformations (D-Ttran) from projectors P1 =
(M − 5⋅1)(1− 5)
= 14
1 −1−3 3
⎛⎝⎜
⎞⎠⎟= k1
21
−23
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗
21 −2
1( )k1
= ε1 ε1
P2 =(M −1⋅1)
(5 −1)= 1
43 13 1
⎛⎝⎜
⎞⎠⎟
= k221
21
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⊗ 23
21( )
k2
= ε2 ε2
Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x K x x K y
y K x y K y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 K ε1 ε1 K ε2
ε2 K ε1 ε2 K ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
21 −2
1
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⋅ 4 13 2
⎛
⎝⎜⎞
⎠⎟ ⋅ 2
121
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= 1 00 5
⎛
⎝⎜⎞
⎠⎟
Use Dirac labeling for all components so transformation is OK
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟⋅
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
ε1 1 ε1 ε1 1 ε2
ε2 1 ε1 ε2 1 ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
21 −2
1
23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⋅ 21
21
−23
21
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= 1 00 1
⎛
⎝⎜⎞
⎠⎟
Check inverse-d-tran is really inverse of your d-tran. In standard quantum matrices inverses are “easy”
ε1 x ε1 y
ε2 x ε2 y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
†
=x ε1
* y ε1*
x ε2* y ε2
*
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
x ε1 x ε2
y ε1 y ε2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−1
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Unitary operators and matrices that change state vectors ...and eigenstates (“ownstates) that are mostly immuneGeometric visualization of real symmetric matrices and eigenvectors Circle-to-ellipse mapping Ellipse-to-ellipse mapping (Normal space vs. tangent space) Eigensolutions as stationary extreme-values (Lagrange λ-multipliers) Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness
Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Proof that completeness relation is “Truer-than-true”Diagonalizing Transformations (D-Ttran) from projectors Eigensolutions for active analyzersSpectral Decompositions with degeneracy Functional spectral decomposition
4 13 2
⎛⎝⎜
⎞⎠⎟
Factoring bra-kets into “Ket-Bras:
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Matrix products and eigensolutions for active analyzersConsider a 45° tilted (θ1=β1/2=π/4 or β1=90°) analyzer followed by a untilted (β2=0) analyzer. Active analyzers have both paths open and a phase shift e-iΩ between each path. Here the first analyzer has Ω1=90°. The second has Ω2=180°.
The transfer matrix for each analyzer is a sum of projection operators for each open path multiplied by the phase factor that is active at that path. Apply phase factor e-iΩ1 =e-iπ/2 to top path in the first analyzer and the factor e-iΩ2 =e-iπ to the top path in the second analyzer.
The matrix product T(total)=T(2)T(1) relates input states |ΨIN〉 to output states: |ΨOUT〉 =T(total)|ΨIN〉
We drop the overall phase e-iπ/4 since it is unobservable. T(total) yields two eigenvalues and projectors.
|ΨΙΝ〉|ΨOUT〉|ΨΙΝ〉=|y〉
2Θin =
βin=180°
T 2( ) = e−iπ x x + y y = e−iπ 0
0 1
⎛
⎝⎜
⎞
⎠⎟
T 1( ) = e−iπ / 2 ′x ′x + ′y ′y = e−iπ / 2
12
12
12
12
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
+
12
−12
−12
12
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
=
1− i2
−1− i2
−1− i2
1− i2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
T total( ) = T 2( )T 1( ) = −1 00 1
⎛
⎝⎜⎞
⎠⎟
1− i2
−1− i2
−1− i2
1− i2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
=
−1+ i2
1+ i2
−1− i2
1− i2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
= e−iπ / 4
−12
i2
−i2
12
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
~
−12
i2
−i2
12
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
λ2 − 0λ −1= 0, or: λ=+1, −1, gives projectors P+1 =
−12+1 i
2−i2
12+1
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
1− −1( ) =
−1+ 2 i−i 1+ 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2 2, P−1 =
1+ 2 −ii −1+ 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2 2
|ΨΙΝ〉
=|+1〉|ΨOUT〉=|+1〉
2Θin =
βin=-135°
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