chapter 4 postulates of quantum...

Winter 2013 Chem 350: Introductory Quantum Mechanics

48

Chapter 4 – Postulates of Quantum Mechanics ......................................................................................... 48

MathChapter C ........................................................................................................................................ 48

The Postulates of Quantum Mechanics .................................................................................................. 50

Properties of Hermitian Operators ......................................................................................................... 53

Discussion of Measurement ................................................................................................................... 55

Discussion of Uncertainty Relations ....................................................................................................... 57

The Time-Dependent Schrodinger Equation .......................................................................................... 60

Time-dependence of expectation values ................................................................................................ 62

Chapter 4 – Postulates of Quantum Mechanics

MathChapter C

A brief reminder on vectors:

In 3 dimensions define orthonormal basic vectors

, ,i j k Or 1 2 3, ,e e e

x

u xi yj zk y

z

Or

x

y

z

u

u u

u

x

y x y z

z

v

v v v i v j v k

v

Vector has both direction and a length

Vector Addiction u v

x x

y y

z z

u v

u v u v

u v

Multiplication by a scalar


Chapter 4 – Postulates of Quantum Mechanics 49

x

y

z

u

u u

u

Multiply length by

Inner Product

u v x x y y z zu v u v u v

i j ije e ie i , 2e j , 3e k

, ,

( )i i j j i j i j i j ij i ii j i j i j i

u e v e u v e e u v u v

Length u u v

Angle cosu v u v

2

1

3

u

1

1

2

v

2 2 22 1 3 14u

2 2 21 1 2 6v

( ) 2 1 ( 1) 1 3 ( 2) 5u v

5

cos14 6

....

Examples of use of inner product in physics

Work cosF l F l

cosF l mgh

cosh l

Or: Dipole moment

i ii

u q r Charges iq at position r

2 2

x

r rq q

qr points in the positive direction from – to +

If we apply an electric field E



V E (more later)

Another important product is the vector or cross product

u v : perpendicular to the plane spanned by u and v

Length: sinu v : sign from right hand rule

Mathematical formula

1 2 3

x y z

x y z

e e e

u u u

v v v

(determinant)

1 2 3( ) ( ) ( )y z z y x z z x x y y xu v e u v u v e u v u v e u v u v

Check: Orthogonal to u , v

Famous Physics example:

Lorentz Magnetic Force: ( )F q V B

Apply magnetic field B force acceleration to V , B

charged particle precesses around B

Change direction of velocity, not speed

Also: Angular Momentum

L r p p r

1 3( ) ( ) ( )x y x z z y xyP xP e zP xP e xP yP e

z y xL xP yP Remember

y x zL zP xP Cyclic Permutation

x z yL yP zP

The Postulates of Quantum Mechanics

A) Observables

To any observable, or measurable quantity A , in Quantum Mechanics corresponds a

linear Hermitian operator Â

If one performs a measurement of A , only eigenvalues ia of the operator Â can be obtained

Operators in quantum mechanics are obtained by replacing

xP ix

,

yP iy

, zP i

z



A function of position, eg. ( )V x is interpreted as multiplication by ˆ( ) ( )V x V x

To know possible outcome from experiment:

Solve ˆ ( ) ( )a aA x a x

Eigenvalue

With operator Â we often need to impose boundary conditions

Observable Operator

Position , ,x y z ˆ ˆ ˆ, ,x y z multiply

Momentum , ,x y zP P P ix

, i

y

, i

z

Kinetic energy

2

2

x

x

PT

m

2

2

PT

m

2 2

22m x

2 2 2 2

2 2 22T

m x y z

22

2m

Potential Energy ( )V x ˆ( )V x

: multiply by ˆ( )V r

Total Energy E

ˆ ˆ ˆH T V 2

2 ˆ( )2

V rm

Angular momentum

x z yl yP zP

y x zl zP xP

z y xl xP yP

ˆ

ˆ

ˆ

x

y

z

L i y zz y

L i z xx z

L i x yy x

Spin

ˆ ˆ ˆ, ,x y zS S S

2 2 2 2ˆ ˆ ˆ ˆx y zS S S S

We have discussed linear operators and eigenvalue equations. What does Hermitian mean?

An operator Â is Hermitian if for any pair of functions ( )f x and ( )g x in the domain of the

operator (ie. Satisfying the boundary conditions)



it is true that

*

* ˆ ˆ( ) ( ) ( ) ( )b b

a af x Ag x dx g x Af x dx

In higher dimensions

*

* ˆ ˆ( ) ( ) ( ) ( )domain domain

f Ag d g Af d

The domain of integration is part of the definition of Hermitian operator (Boundary conditions)

In words: act with Â on g and integrate with *f should be equal to act with Â of f , take

complex conjugate, and integrate against ( )g .

We can write the latter expression also as

ˆ ˆ( ) * ( )* *g Af d g Af d

Example of Hermitian operators

( )V x : ( )V x is a real function

*( ) ( ) ( ) ( ) ( ) *( )f x V x g x dx g x V x f x dx

( ) ( ) ( ) *g x V x f x dx (V real!)

xP ix

:

*( ) *( ) ( )g

f x i dx i f x g xx

*( )

P

P

fi g x dx

x

(partial integration)

*

( )f

i g x dxx

( ) *f

g x i dxx

ˆ( )( ( ))*xg x P f x dx

Likewise 2 2

22m x

and other listed operators are Hermitian

Operators in Quantum Mechanics are required to be Hermitian because these operators have very nice

mathematical properties:



Properties of Hermitian Operators

a) The eigenvalues are all real

b) The eigenfunctions can be chosen to be orthonormal

c) Any function ( )x can be expressed as linear combination ( ) ( )n nn

x C x

with *( ) ( )n ndomain

C x x dx

Let us denote eigenvalues of Â to be na with eigenfunctinos ( )n x

Then: Eigenvalues are real

ˆ ˆ*( ) ( )*n n n nA dx A dx

*( ) ( ) * *( ) ( )n n n n n na x x dx a x x dx

*n na a , na is real

2) Eigenfunctions corresponding to different eigenvalues are orthonormal

ˆ ˆ*( ) ( ) ( ( ))*k l l kx A x dx A x dx

( *) *( ) ( ) 0l k k la a x x dx

*k ka a l ka a

*( ) ( ) 0k ldomain

x x dx

k , l are orthogonal if k la a

3) If ( )k x and ( )l x have the same eigenvalue (degenerate) then any linear combination

is also eigenfunction with the same eigenvalue

ˆ ˆ( ) ( ( ) ( ))k k l lA x A C x C x

ˆ ˆ( ) ( )k k l lC A x C A x [Linear]

( ) ( )k k l lC a x C a x [Same Eigenvalue]

( ( ) ( ))k k l la C x C x

( )a x

4) If ( )k x and ( )l x are degenerate (same a ), but are not orthogonal, we can make

them orthonormal

Eg. *( ) ( )k lx x dx c

( ) ( )newl l kx c x k normalized



* ( )( ( ) ( ))k l kx x c x dx

1 0c c normalize newl

This is true in general.

With any Hermitian operator we can associate a set of orthonormal functions ( )k x

ˆ ( ) ( )k k kA x a x

Earlier *( ) ( )k l kldomain

x x dx

1 k l

0 k l

5) Any (suitable) wave function can expressed as a linear combination of the ( )k x

( ) ( )k kk

x C x

Completeness Property

This statement is hard to prove and we assume it to be true

Further consequences

5a) Normalization

*( ) ( )x x dx

* *( ) ( )l l k kl k

c x c x dx

* *( ) ( )l k l kl k

c c x x dx

*l k lkl k

c c

*(0 0 ... 0 ...)l ll

c c

2

*l l ll l

c c c

Normalized Wavefunction: 2

1ll

c

5b) Expectation Values

ˆ*( ) ( )x A x dx

ˆ* *( ) ( )k k l lk l

c x A c x dx

* *( ) ( )k l l k lk l

c c a x x dx



*k l l klk l

c c a

0 0 .... ...k kc a

( * )k k k k kk k

c c a A P a

5c)

22 2 2 2.... l l k l

l l

A c a A P a

From this we deduce that kP the probability to find ka is given by *k kc c if

( ) ( )k kk

x c x

This assumes we know the coefficient. We can calculate it!

*( ) ( ) *( ) ( )k k k l ll

c x x dx x c x dx

*( ) ( )l k l l kll l

c x x dx c

0 0 ... 0 0....kc

kc Indeed!

We can now discuss the predictions of Quantum Mechanics regarding measurements

Discussion of Measurement

If we decide to measure a quantity A , we associate with it a Quantum Mechanical operator Â

Solve for eigenfunctions and eigenvalues

ˆ ( ) ( )k k kA x a x

Type of Measurement Â , { ka ( )k x }

If we measure A for individual quantum system, we get always an eigenvalue (rolling the dice)

Possible Outcomes 1 2 3, , .....a a a



Probability: 1 1 1( )tot

NP a

N , 2 2 2( )

tot

NP a

N

What determines the probabilities? The wavefunction (normalized) ( )x !!

If we do a large number of measurements on identical microscopic systems, collectively described by

the wavefunction we find the value ka with probability kP

2

*k k k kP c c c

*( ) ( )k kc x x dx

How do we prepare an ensemble described by wavefunction ( )x ?

Could be the ground state of a molecule (low T )

0( )x 0 0 0ˆ ( ) ( )H x E x

This is like ‘measuring’ the energy first, then measure Â

We could alternatively measure B̂ and collect all Microsystems with eigenvalue kb

Take those systems that yield eigenvalue kb . They are described by ( )kb x

ˆ ( ) ( )k kb k b

B x b x

Why? If I measure B again I would measure kb with certainty (if it does not change over time)

*( ) ( ) 1k kb b

C x x dx

( ) ( ) ( )k

i

bx x e

A measurement is the standard procedure to prepare an ensemble of microscopic systems in a well-

define wavefunction (takes some effort!)



Discussion of Uncertainty Relations

If I measure Â for an ensemble, can I get a sharp value for Â ?

Yes: use ( ) ( )ax x

Â a , 2 2Â a , 0A

Next question, if I would measure Â and B̂ for an ensemble, can I create a sharp value for both Â and

B̂ ?

If ( )x is eigenstate of Â and eigenstate of B̂

ˆ ( ) ( )A x a x

ˆ ( ) ( )B x b x

,( ) ( )a bx x

When can I create a sharp value for measuring Â and B̂ , for each compatible value of a and b ?

If Â and B̂ have a complete set of common eigenstates

, ,

ˆ ( ) ( )a b a bA x a x

, ,ˆ ( ) ( )a b a bB x b x

Completeness Property:

Any ( ) :x ( )

( ) ( )ab abab

x c x

( ) ( )

ˆ ˆˆ ( ) ( ) ( )ab ab ab abab ab

AB x A c b x c ab x

( ) ( )

ˆˆ ˆ( ) ( ) ( )ab ab ab abab ab

BA x B c a x c ba x

ˆ ˆˆ ˆ( ) ( ) 0AB BA x

Hence, If and only if : ˆ ˆ ˆˆ ˆ ˆ[ , ] 0A B AB BA then

Â and B̂ have complete set of common eigenfunctions

And: We can create samples to measure sharp values of both Â and B̂



If Â and B̂ commute measuring B̂ does not destroy the value measured for Â (compare to my

example of measuring lunchboxes for schoolkids)

After sample preparation: we can obtain completely sharp values 0A B

What if Â and B̂ do not commute?

Mathematical uncertainty Principle

1 ˆ ˆ,2

A B A B

Depends on the state ( ) in general

If Â and B̂ have no common eigenstate then:

One cannot generate an ensemble in which one obtains sharp values for both Â and B̂

Most famous example: Measure position and momentum

ˆˆ[ , ] ( ) ( )xx P x i x i x xx x

( )i x i x i xx x

( )i x

ˆˆ[ , ]xx P i

1 1

2 2x p i



Heisenberg Uncertainty Relation

Measure x and p for ensemble

1

2x p

Not simultaneously measure x , p !! This was discussed originally by Heisenberg, for his famous

microscope, but we can only verify for ensembles as discussed above.

Special Topic: Continuous eigenvalues

Consider ˆxP i

x

(1d)

Eigenfunctions ikxe ˆikx ikx

xP e ke k can have any value

but ( ) ikxx e cannot be normalized : ( )*ikx ikxe e dx

1ikx ikxe e dx dx

!

And orthonormality is ill-defined

1 2 1 2( )ik x ik x i k k x

e e dx e dx

1 2

( )

1 2

1

( )

i k k xe

i k k

Function keeps oscillating, does not vanish at infinity…

Continuous ‘spectra’ require special care

1

( )2

ikx

kc e x dx

Converges if ( )x is normalized

Other example position x̂

ˆ ( ) ( ) ( )a a ax x x x a x

( ) ( ) 0ax a x ( ) 0a x if x a Funny function!!

( ) ?a x if

x a

( ) ( )a x a x a

( ) 1x a



( )x a Dirac delta function

a.k.a. normalization ( ) ( ) ( )x a x dx a

( )P a : the probability to find particle at a

2

( ) ( )P a x dx

2

( ) ( )P x dx x dx

This definition is consistent with the postulates (follows from the postulates).

Continuous eigenvalues require more sophisticated mathematics (distributions).

The Time-Dependent Schrodinger Equation

Time dependence of ( , )x t :

ˆ( , ) ( , )i x t H x tt

0( , )x t to be specified: initial conditions

Assume Ĥ is time-independent

As in chapter 2 we consider separation of variables

Try ( , ) ( ) ( )x t x t

,x t ˆ( ) ( ) ( )x i t H xt

ˆ ( )

/ ( )( )

H xi t E

t x

: both are constant

ˆ ( ) ( )n n nH x E x

( )ni E tt

( )

niE t

t e

Special solutions: Stationary states

0( )

( , ) ( )niE t t

nx t x e

These states are called stationary states because



2

( , ) ( )* ( )n niE t iE t

n nx t x e x e

2

( )n x independent of t

Also 0

ˆt t

A A and even probability to find eigenvalue a are independent of time:

ˆ ( ) ( )a aAX x aX x

( ) ( )* ( ) niE t

a a nC t X x x dx e

2

( ) * *(0) (0) (0)a a a a a a aP t c c c c c P independent of time!!

But : stationary states are very special

The general solution is linear combination:

0( )

( , ) ( )niE t t

n n

n

x t c x e

0( )

( )niE t t

n n n

n

i c x E et

0( )

ˆ( ( ))niE t t

n n

n

c H x e

0( )

ˆ ( )niE t t

n n

n

H c x e

(reminder Ĥ independent of t , no electromagnetic fields!)

0( ) ( )n nn

t t c x

specify 0( )t t

0*( ) ( , )n nc x x t t dx

( , )x t is determined for all time, very simple!!

Probability to find nE ?

0( )

( ) *( ) * ( )miE t t

n n m m

m

c t x c x e

0( )niE t t

nc e

2 22 (0)n n nc c c



independent of time E is conserved = n nn

P E

Other properties, eg. ˆ ˆ*( )* ( )x t x t dx

oscillation in time

Further remarks:

All depends on initial wave function, which is arbitrary in principle [compare to classical

mechanics specify (0)X , (0)P for all particles ( )X t , ( )P t follows from equation of motion]

Stationary States: ( 0) ( )nt x energy eigenstates

S.E. Predicts: excited states live forever

Q.M: does not predict thermal equilibrium over time

need to combine quantum mechanics with

a) Statistical mechanics

b) Electromagnetic field (even in a vacuum)

Excited states decay: From quantum electrodynamics (Dirac 1927, Feynman, Schrodinger,

Tomonaga 1949…)

Time-dependence of expectation values

ˆ( ) *( , ) ( , )A t x t A x t dx

* ( , )ˆ ˆ( ) ( , ) *( ) ( ) *( )

d d A d x tA t A x t dx x x x A

dt dt t dt

ˆ ( )i Ht

ˆ ( )

iH

t

ˆ* ( ) *i Ht

* ˆ *i H

t

Assume: 0A

t

Â : Hermitian

ˆ ˆ( , ) * *( , )d

A x t A dx x t A dxdt t t

ˆ ˆˆ ˆ( , ) * *( ) ( )i i

x t A H dx x A H x dx



ˆ ˆˆ ˆ( , ) ( , ) * ( , )* ( , )i ix t AH x t dx x t AH x t dx

ˆˆ*( ) ( , ) *( ) ( , )i i

x HA x t dx x AH x t dx

*( )( ) ( , )i

x AH HA x t dx

Or

ˆ ˆ[ ]t

d Ai AH

dt

Difficult Step: General produce of 2 Hermitian operators

* ˆ ˆ ( )f ABg x dx

*

ˆˆ ( )Bg x Af dx Reverse Operators!

*

ˆˆ( )g x BAf dx

Examples: ,P

i P Ht

2 2

2, ( )

2

di V x

x m dx

P V

i it x

P V

t x

Same as from Classical Mechanics

2

2

v Vma m F

xdt

dP V

dt x

Also

2

2, ( )2

x di x V x

t m dx

2

m x

i

im x



i

Pm

Or

1x

Pt m

Again: Same as in Classical Mechanics

Ehrenfest theorem:

Expectation values in Quantum Mechanics obey relations from classical mechanics

chapter 4 postulates of quantum...

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