mathematical formulation of quantum mechanics · mathematical formulation of quantum mechanics...
TRANSCRIPT
Mathematical Formulation of Quantum Mechanics
Andreas WackerMathematical Physics
Lund University
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Formulation of Classical Physics
Very successful in everyday lifeand forming our concept of nature
Objects, we observe, are described by points ri in the three-dimensional Euclidean space
Fully determined for given initial position and velocity
Time-evolution according to mi ri=Fi (Newton 1687)
Observation is not a matter of principle relevance
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
The classical concept fails
characteristic frequencies of light (Rydberg 1888 Lund)stability of atom (Rutherford 1911)Magic particle numbers and periodic table of elements
Black-body radiationSpecific heat of solidsPhysics of semiconductors (band gap!)Superconductivity
Features on atomic scale
Macroscopic effects
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
The Stern-Gerlach Experiment (1922)
Force ~ vertical component of the magnetic moment
Two discrete values μz=±μB observed
Drawn by Theresa Knott (wikipedia)
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Formalism to describe quantum effects
Physical systems are described by elements of the ket space (generalizes wave functions)Measurement results become of stochastic natureMeasurement changes the physical system
Well defined formalism, which allows to do calculations and currently agrees with experiment
Interpretation is far from clear and continuously debated
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Ket space as a vector space over the complex numbers
Commutativity of addition |ϕ1>+|ϕ2>=|ϕ2>+|ϕ1>
Associativity of addition (|ϕ1>+|ϕ2>)+|ϕ3>=|ϕ1>+(|ϕ2>+|ϕ3>)
There is a state |null> with | >+|null>=| > for all | >ϕ ϕ ϕEach state | > has an inverse |ϕ ϕ> with | >+|ϕ ϕ>=|null>
(write +|ϕ> as -| >)ϕ1| >=| >ϕ ϕ(α+β)| >=α| >+β| > and α(|ϕ ϕ ϕ ϕ1>+|ϕ2>)=α|ϕ1>+α|ϕ2>
(αβ)| >=α(β| >)ϕ ϕ
kets | > and complex numbers α with two operationsϕAddition of two kets |ϕ1>+|ϕ2> is also a ket |ϕ3>
Multiplication with complex numbers α|ϕ1> is also a ket |ϕ2>
Satisfying:
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
The scalar (or inner) product and bra's
For |ϕ1> and |ϕ2> define inner product <ϕ1|ϕ2>=α with
< | > is real and positive, unless | >=|null>, where <null|null> =0ϕ ϕ ϕ<ϕ1|ϕ2>=<ϕ2|ϕ1>
*
For |ϕ3>=α|ϕ1>+β|ϕ2> we have
< |ϕ ϕ3>=α< |ϕ ϕ1>+β< |ϕ ϕ2>
<ϕ3| >=αϕ *<ϕ1| >+βϕ *<ϕ2| >ϕ
Treat < | as an object on its own, called ϕ bra.
The bras form their own vector space, the dual space
dual correspondence | > ϕ ⇆ < |ϕ
<ϕ1|ϕ2>bra(c)ket
Note |ϕ3>=α|ϕ1>+β|ϕ2> → <ϕ3|=α*<ϕ1|+β*<ϕ2|
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Bases
A basis is a set {|ai>} which allows to construct any | > asϕ | >=ϕ ∑i ci|ai> with unique complex numbers ci
An orthonormal (ON) basis satisfies <ai|aj>=δij
For an ON basis we find
Expansion coefficients ci=<ai| >ϕ
Completeness relation ∑i |ai><ai|=1 ⚠< | >= ϕ ϕ ∑i |ci|
2
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
1. Any physical state is identified by a ket |ψ> with norm 1, i.e. <ψ|ψ>=1.
2. An ideal measurement provides real measurement values αn associated with physical states |an>, which form an ON basis of the ket space.
3. For an arbitrary state |ψ>, the probability to observe the result αn in a measurement is given by Pn= |<an|ψ>|2
Physical States and Measurements (Standard Copenhagen Interpretation)
After measuring the value αn the physical state collapses to the corresponding state |an>.
Fortuitousness is fundamental for quantum mechanics
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Example: Stern-Gerlach Experiment
Force ~ vertical component of the magnetic moment
Two discrete values μz=±μB observed
Drawn by Theresa Knott (wikipedia)
Formulate on board
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Linear Operators
Operator  transforms Â|Φ>=|Φ'>
Linear operator: Â(α|Φ>+β|φ>)=αÂ|Φ>+βÂ|φ>
Product between operators Ĥ=ĈÂ the operator transforming Ĥ|Φ>=Ĉ(Â|Φ>)
Define commutator [Ĉ,Â]=ĈÂ-ÂĈ
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Matrix representation
For ON basis {|ai>} evaluate
State | > ϕ → coefficients ci=<ai| >ϕOperator  → coefficients Aij=<ai|Â|aj>
All calculations become matrix-operations
Note: depend on basis used, only for ON basis
∣φ ' ⟩= A∣φ ⟩⇔(c '1c '2⋮
)=(A11 A22 …
A21 A22 …
⋮ ⋮ ⋱)(
c1
c2
⋮)
ϕ∣φ ⟩=(d1* d2
* …)(c1
c2
⋮)
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
The adjoint operator
Consider <φ|Â|Φ> for arbitrary |Φ>
<φ|Â| is a linear mapping |Φ>→complex numbers
Interpret <φ|Â|=<φ| as a new bra
How is the corresponding ket |φ> related to |φ>?
Relation linear⇒ There is an operator Ĉ with |φ>=Ĉ|φ>
As Ĉ depends on Â, write Ĉ=†, the adjoint operator of Â
†|φ> is dual correspondence of <φ| for all |φ>
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Important relations for the adjoint operator
If †=Â:  is called Hermitian (self-adjoint), Aij=Aji*
If †Â=1:  is called unitary
φ2∣ A∣φ1 ⟩*= φ2∣φ1 ⟩*=φ1∣φ2 ⟩=φ1∣A†∣φ2 ⟩
In matrix representation ai∣A†∣a j ⟩=a j∣A∣ai ⟩
*=A ji*
†|φ> is dual correspondence of <φ| for all |φ>
(∣φ1 ⟩ φ2∣)†=∣φ2 ⟩ φ1∣
( A B)†=B† A† see exercise
( A†)†= A see exercise
(α A)†=α* A† see exercise
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Hermitean Operators
Matrices with Aij=Aji* can be diagonalized
(A11 A12 …
A21 A22 …
⋮ ⋮ ⋱)(c1
(n)
c2(n)
⋮)=λn(
c1(n)
c2(n)
⋮) with real λn
Eigenstates ∣un ⟩=∑ic i
(n)∣ai ⟩ Α∣un ⟩=λn∣un ⟩with
form a new ON basis
Α Hermitean ⇔Α=∑n λn∣un ⟩ ⟨ un∣
If †=Â:  is called Hermitean (self-adjoint), Aij=Aji*
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Common eigenstates of two Hermitean operators  and Ĉ
There exists an ON basis {|an>}, where all |an> are
simultaneously eigenstates of  and Ĉ (possibly with different eigenvalues)
⇕[Â,Ĉ]=0
Note: [Â,Ĉ]=0 does not imply that each eigenstate of  is also eigenstate of Ĉ
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Operators representing observables
Measurement: Real measurement values αn associated with physical states |an> providing ON basis
Expectation value for state |Ψ>:
Physical observable ↔ Hermitean operator
Example: µz and µx of Stern Gerlach
⟨α ⟩=∑n
Pn⏟=∣⟨an∣Ψ ⟩∣2
αn=∑n
⟨ Ψ∣an ⟩αn ⟨an∣Ψ ⟩=⟨ Ψ∣Α∣Ψ ⟩
with Hermitean operator Α=∑n
∣an ⟩αn ⟨an∣
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Phase of a quantum state
The states |Ψ> and |Ψ'>=eiα|Ψ> can not be distinguished by any measurement
The states |Ψ>=|Φ1>+|Φ2> and |Ψ'>=|Φ1>+eiα|Φ2> differ
Examples from Stern Gerlach
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Continuous spectrum and spatial representation
For continuous measurement values, e.g. spatial position x
But continuum of states cannot be counted
Instead: ⟨ x∣x ' ⟩=δ(x−x ' ) and 1=∫dx∣x ⟩ ⟨ x∣
Eigenstates X∣x ⟩=x∣x ⟩ as before Α∣ai ⟩=αi∣ai ⟩
⟨a j∣ai ⟩=δ ji and 1=∑n
∣ai ⟩ ⟨ai∣ not possible
Any state can be written as ∣Ψ ⟩=∫dx∣x ⟩ ⟨ x∣Ψ ⟩
∣⟨ x∣Ψ ⟩∣2Δ x is probability to find position in interval (x , x+Δ x)
Common wave function Ψ(x)=⟨ x∣Ψ ⟩
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Three spatial dimensions
Postulate [ X , Y ]=[ X , Z ]=[Y , Z ]=0
Wave function in 3 dimensions: Ψ(r)=⟨ x , y , z∣Ψ ⟩
⇒ There is a common set of eigenstates ∣x , y , z ⟩
for the operators X , Y , Z
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Function of an operator ITaylor Series
Frequently we write f(Â) – what does it mean?
For f (x)=∑n=0∞
f nn!xn we define f (A)=∑n=0
∞f nn!
An
For f (x)=xn we define f (A)=An≡A A…⏟n times
Exercise: [B , An ]=n [B , A ] An−1 if [[B , A ] , A ]=0
Important implication:
[B , f (A )]=[B , A ] f ' ( A) if [[B , A ] , A ]=0
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Function of an operator IIProjection on eigenstates
Hermitean operators have a complete set of eigenstates
A=∑n
αn |an><an |
f (A)=∑nf (αn)|an><an |Define
Example: ei A is unitary if A is Hermitean
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Change of basis
Old basis {|an>}, new basis {|bn>}
Operator for basis change U=∑n
|bn><an | unitary!
| Ψ >=∑ncn |an>=∑
mc 'm |bm>
c 'm=⟨bm | Ψ⟩=∑n
⟨bm |an⟩⏟=⟨am | U† |an⟩
cn
Matrix for basis change
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Dynamical evolution of ket states:Hamiltonian
Simple systems with mechanical analog and potential V(x,y,z)
with the basic commutation relations
Common notation
Hamilton-operator Ĥ needs to be specified
iℏ ∂∂ t
∣Ψ(t)⟩=H∣Ψ(t) ⟩Define
H=px
2+ p y2 + pz
2
2m+V (r x , r y , rz)
[ p j , pk ]=0, [ r j , rk ]=0 , and [ p j , rk ]=ℏ
iδ jk
p= px ex+ p y e y+ pz e z and r=r x ex+ r y e y+ r z e z
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Eigenstates of Hamiltonian
For time-independent Hamiltonian, search eigenstates
Arbitrary physical state:
provides time-dependence of system for given |Ψ(0)>
Stationary Schrödinger Eq. H∣φn ⟩=En∣φn ⟩
∣Ψ(t)⟩=∑ncn(t)∣φn ⟩ with cn(t)=cn(0)e
−iEn t / ℏ
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Time-dependence of expectation values
ProofExample harmonic oscillator on board
ddt
Ψ∣Α∣Ψ ⟩=iℏ
Ψ∣[H , Α]∣Ψ ⟩+ Ψ∣∂ Α∂ t
∣Ψ ⟩
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Spatial Representation
Consider the states |x> satisfying
In this basis we obtain the matrices
Need to show
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Spatial representation of Schrödinger equation in 3 dimensions
For general operator
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Solving the stationary Schrödinger equation by diagonalization
Search for eigen-energies En and states |φn>
Use matrix representation!
Stationary Schrödinger Eq. H∣φn ⟩=En∣φn ⟩
Lund University / Science Faculty / Mathematical Physics / [email protected] / 2015-01-21
Summary
Any physical state is identified by a ket |ψ> with norm 1
i ℏ∂∣Ψ(t)⟩
∂ t=H ∣Ψ(t) ⟩Time-dependence by Hamilton operator
Observables are represented by Hermitian operators  (†=Â)
The real eigenvalues αn are possible measurement values, which are found with probability Pn= |<an|ψ>|2 , for the corresponding eigenstates |an>
Unitary operators Û (Û†Û=1) describe the basis transformations between ON bases and keep the norm of kets
Average satisfies: ddt
Ψ∣Α∣Ψ ⟩=iℏ
Ψ∣[H , Α]∣Ψ ⟩+ Ψ∣∂ Α∂ t
∣Ψ ⟩