Winter 2013
Chem 350: Introductory Quantum Mechanics
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
 x
 
u  xi  yj  zk   y 
z
 
Or
e1 , e2 , e3
Or
 ux 
 
u   uy 
u 
 z
 vx 
 
v   v y   vx i  v y j  v z k
v 
 z
Vector has both direction and a length
Vector Addiction u  v
 u x  vx 


u  v   u y  vy 
u v 
z 
 z
Multiplication by a scalar
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Chem 350: Introductory Quantum Mechanics
Winter 2013
  ux 
 u    u y  Multiply length by 
u 
 z
Inner Product
u  v  u x vx  u y v y  u z vz
ei  i , e2  j , e3  k
ei  e j   ij
( ui ei   v j e j )   ui v j  ei e j   ui v j  ij   ui vi
i
j
i, j
i, j
Length
u  u v
Angle
u  v  u v  cos
2
 
u   1
3
 
i
 1
 
v  1 
 2 
 
u  22  12  32  14
v  12  12  22  6
(u  v)  2  1  (1)  1  3  (2)  5
5
cos  
  ....
14  6
Examples of use of inner product in physics
Work  F  l  F l cos 
F l cos  mgh
h  l cos
Or: Dipole moment
u   qi ri
Charges qi at position r
i
r
2
 r 

 2 
 x  q   q  
qr points in the positive direction from – to +
If we apply an electric field E
Chapter 4 – Postulates of Quantum Mechanics
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Chem 350: Introductory Quantum Mechanics
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: u v sin  : sign from right hand rule
Mathematical formula
e1
ux
vx
e2
uy
vy
e3
uz
vz
(determinant)
u  v  e1 (u y vz  u z v y )  e2 (u x vz  u z vx )  e3 (u x v y  u y vx )
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
Also:
Change direction of velocity, not speed
Angular Momentum
L  r  p  pr
( yPx  xPy )e1  ( zPx  xPz )ez  ( xPy  yPx )e3
Lz  xPy  yPx
Ly  zPx  xPz
Remember
Cyclic Permutation
Lx  yPz  zPy
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 ai of the operator  can be obtained
Operators in quantum mechanics are obtained by replacing
Px  i



, Py  i
, Pz  i
x
z
y
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Chem 350: Introductory Quantum Mechanics
A function of position, eg. V ( x ) is interpreted as multiplication by V ( x)  Vˆ ( x)
 To know possible outcome from experiment:
ˆ ( x )  a  ( x )
Solve A
a
a
Eigenvalue
With operator  we often need to impose boundary conditions
Observable
Operator
Position
x, y , z
Momentum
Px , Py , Pz
Px 2
2m
P2
T
2m
Tx 
Kinetic energy
xˆ , yˆ , zˆ
multiply



i
, i
, i
x
z
y
2
2m x 2
2
 2
2
2 
T 




2m  x 2 y 2 z 2 
2


2
2m
ˆ
V ( x)
V ( x)
Potential Energy
2
: multiply by Vˆ ( r )
Hˆ  Tˆ  Vˆ
Total Energy
E
Lˆx
lx  yPz  zPy
Angular momentum
l y  zPx  xPz
lz  xPy  yPx
Lˆ y
2
 2  Vˆ (r )
2m
 
 
 i  y  z 
y 
 z
 
 
 i  z  x 
z 
 x

 
 
Lˆz  i  x  y 
x 
 y
Sˆ , Sˆ , Sˆ
x
Spin
y
z
Sˆ  Sˆx  Sˆ y  Sˆz 2
2
2
2
We have discussed linear operators and eigenvalue equations. What does Hermitian mean?
An operator  is Hermitian if for any pair of functions f ( x ) and g ( x ) in the domain of the
operator (ie. Satisfying the boundary conditions)
Chapter 4 – Postulates of Quantum Mechanics
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Chem 350: Introductory Quantum Mechanics
Winter 2013
it is true that

b
a




*
ˆ ( x) dx  b g ( x) Af
ˆ ( x) dx
f * ( x) Ag

a
In higher dimensions

domain


ˆ ( ) d 
f * ( ) Ag

domain


*
ˆ ( ) d
g ( ) Af
The domain of integration is part of the definition of Hermitian operator (Boundary conditions)
In words: act with  on g and integrate with f * should be equal to act with  of f , take
complex conjugate, and integrate against g ( ) .
We can write the latter expression also as

 
 
ˆ * d   g ( ) * Af
ˆ d  *
g ( ) Af

 
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!)
Px  i




:
x
P
g 
f *

f * ( x)  i
g ( x)dx (partial integration)
dx   i f * ( x) g ( x)   P   i
x 
x

f *
 i
g ( x )dx
x
f 

  g ( x)  i
* dx
x 

  g ( x)( Pˆx f ( x)) * dx
Likewise 
2
and other listed operators are Hermitian
2m x 2
2
Operators in Quantum Mechanics are required to be Hermitian because these operators have very nice
mathematical properties:
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Chem 350: Introductory Quantum Mechanics
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  ( x) 
 C  ( x)
n n
n

with Cn 
 * ( x) ( x)dx
domain n
Let us denote eigenvalues of  to be an with eigenfunctinos n ( x)
Then: Eigenvalues are real

n
* ( Aˆn )dx   n ( Aˆn ) * dx
an  n *( x)n ( x)dx  an *  n *( x)n ( x)dx
an  an * , an is real
2) Eigenfunctions corresponding to different eigenvalues are orthonormal

k
* ( x) Aˆl ( x)dx   l ( Aˆk ( x)) * dx
(al  ak *) k * ( x)l ( x)dx  0
ak *  ak

al  ak
 * ( x)l ( x)dx  0
domain k
k , l are orthogonal if ak  al
3) If k ( x) and l ( x) have the same eigenvalue (degenerate) then any linear combination
is also eigenfunction with the same eigenvalue
Aˆ ( x)  Aˆ (Ck k ( x)  Cll ( x))
 Ck Aˆk ( x)  Cl Aˆl ( x) [Linear]
 Ck ak ( x)  Cl al ( x) [Same Eigenvalue]
 a(Ck k ( x)  Cl l ( 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
* ( x)l ( x)dx  c
l new  l ( x)  ck ( x)
k normalized
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Chem 350: Introductory Quantum Mechanics

*
k
( x)(l ( x)  ck ( x))dx
 normalize l new
 c  c 1  0
This is true in general.
With any Hermitian operator we can associate a set of orthonormal functions k ( x)
Aˆk ( x)  ak k ( x)
Earlier

 * ( x)l ( x)dx   kl
domain k
k l
k l
 1
0
5) Any (suitable) wave function can expressed as a linear combination of the k ( x)
 ( x )   C k k ( x )
k
Completeness Property
This statement is hard to prove and we assume it to be true
 Further consequences
5a) Normalization
  * ( x) ( x)dx


    cl * l * ( x)   ck k ( x) dx
k
 l

  cl *  ck  l * ( x)k ( x)dx
l
k
  cl *  ck  lk
l
k
  cl * (0  0  ...cl  0  ...)
l
  cl * cl   cl
l
2
l
Normalized Wavefunction:
c
l
2
1
l
5b) Expectation Values
  * ( x) Aˆ ( x)dx


    ck * k * ( x)Aˆ  cl l ( x) dx
l
 k

  ck *  cl al  k * ( x)l ( x)dx
k
l
Chapter 4 – Postulates of Quantum Mechanics
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Chem 350: Introductory Quantum Mechanics
Winter 2013
  ck *  cl al  kl
k
l
 0  0  ....ck ak  ...
  (ck * ck )ak  A   Pk ak
k
k
5c)
A2  ....   cl al 2  A2   Pk al 2
2
l
l
From this we deduce that Pk the probability to find ak is given by ck * ck if
 ( x)   ck k ( x)
k
This assumes we know the coefficient. We can calculate it!
ck   k * ( x) ( x)dx   k * ( x) cl l ( x)dx
l
  cl  k * ( x)l ( x)dx   cl  kl
l
l
 0  0  ...  ck  0  0....
 ck
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
Aˆk ( x)  ak k ( x)
 Type of Measurement  Â , { ak k ( x) }
If we measure A for individual quantum system, we get always an eigenvalue (rolling the dice)
Possible Outcomes a1 , a2 , a3 .....
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Chem 350: Introductory Quantum Mechanics
Probability:
N1
N2
 P1 (a1 ) ,
 P2 (a2 )
Ntot
Ntot
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 ak with probability Pk
Pk  ck * ck  ck
2
ck   k * ( x) ( x)dx
How do we prepare an ensemble described by wavefunction  ( x ) ?
Could be the ground state of a molecule (low T )
 ( x)   0
Hˆ  0 ( x)  E0 0 ( x)
This is like ‘measuring’ the energy first, then measure Â
We could alternatively measure B̂ and collect all Microsystems with eigenvalue bk
Take those systems that yield eigenvalue bk . They are described by  bk ( x)
Bˆ bk ( x)  bk bk ( x)
Why? If I measure B again I would measure bk with certainty (if it does not change over time)
Cbk    bk * ( x) ( x)dx  1
  ( x)   bk ( x)  (ei )
 A measurement is the standard procedure to prepare an ensemble of microscopic systems in a welldefine wavefunction (takes some effort!)
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Discussion of Uncertainty Relations
If I measure  for an ensemble, can I get a sharp value for  ?
 Yes: use  ( x)  a ( x)
 Â  a ,
Â2  a 2 ,  A  0
Next question, if I would measure  and B̂ for an ensemble, can I create a sharp value for both  and
B̂ ?
If  ( x ) is eigenstate of  and eigenstate of B̂
Aˆ ( x)  a ( x)
Bˆ ( x)  b ( x)
 ( x )  a , b ( x )
When can I create a sharp value for measuring  and B̂ , for each compatible value of a and b ?
If  and B̂ have a complete set of common eigenstates
Aˆa ,b ( x)  aa ,b ( x)
Bˆa ,b ( x)  ba ,b ( x)
Completeness Property:
Any  ( x ) :
 ( x)   cabab ( x)
( ab )
ˆ ˆ ( x)  Aˆ c b ( x)  c ab ( x)
AB
 ab ab
 ab ab
( ab )
( ab )
ˆ ˆ ( x)  Bˆ  c a ( x)   c ba ( x)
BA
ab
ab
ab
ab
( ab )
( ab )
ˆ ˆ  BA
ˆ ˆ ) ( x)  0
 ( AB

ˆ ˆ  BA
ˆ ˆ  0 then
Hence, If and only if : [ Aˆ , Bˆ ]  AB
 and B̂ have complete set of common eigenfunctions
And: We can create samples to measure sharp values of both  and B̂
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If  and B̂ commute measuring B̂ does not destroy the value measured for  (compare to my
example of measuring lunchboxes for schoolkids)
After sample preparation: we can obtain completely sharp values  A   B  0
What if  and B̂ do not commute?

Mathematical uncertainty Principle
AB 

1
  Aˆ , Bˆ  
2
Depends on the state ( ) in general
If  and B̂ have no common eigenstate then:
One cannot generate an ensemble in which one obtains sharp values for both  and B̂
Most famous example: Measure position and momentum
 




[ xˆ, Pˆx ]  x  i
x ( x) 
 ( x)   i
x 
x



 
 
 i x
  i  ( x)  i x

x 
x 
 i  ( x )
[ xˆ, Pˆx ]  i
xp 
1
1
 i  
2
2
Chapter 4 – Postulates of Quantum Mechanics
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Chem 350: Introductory Quantum Mechanics
Heisenberg Uncertainty Relation
Measure x and p for ensemble
x p 
1
2
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

Pˆx  i
x
Consider
Pˆx eikx  keikx k can have any value
Eigenfunctions e  ikx
but  ( x)  eikx cannot be normalized :


(1d)



(eikx ) * eikx dx


eikx eikx dx   1dx   !

And orthonormality is ill-defined



e
 ik1x ik2 x
e

dx   e
 i ( k1  k2 ) x

1
e  i ( k1  k2 ) x
dx 
i (k1  k2 )


Function keeps oscillating, does not vanish at infinity…
Continuous ‘spectra’ require special care
ck 
1

2


eikx ( x)dx
Converges if  ( x ) is normalized
Other example position x̂
xˆa ( x)  xa ( x)  aa ( x)
( x  a)a ( x)  0
 a ( x)  0 if x  a
Funny function!!
a ( x)  ? if
xa
a ( x  a)   ( x  a)
  ( x  a)  1
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 ( x  a)
Dirac delta function
a.k.a. normalization



 ( x  a) ( x)dx   (a)
P ( a ) : the probability to find particle at a
P(a)   ( x) dx
2
P( x)dx   ( x) dx
2
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 ) :
i

( x, t )  Hˆ  ( x, t )
t
 ( x, t0 )  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 )

 ( x)i
  (t ) Hˆ  ( x)
t
i

i

Hˆ  ( x)
/  (t ) 
 E : both are constant
t
 ( x)
Hˆ n ( x)  Enn ( x)

 En  (t ) 
t
 (t )  e
 iEnt
Special solutions: Stationary states
 ( x, t )  n ( x)e
 iEn ( t t0 )
These states are called stationary states because
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 ( x, t )  n ( x) * e
2
 n ( x )
Also A
t
 Aˆ
t0
 iEnt
2
n ( x)e
iEnt
independent of t
and even probability to find eigenvalue a are independent of time:
ˆ ( x)  aX ( x)
AX
a
a
Ca (t )   X a ( x) *n ( x)dx  eiEnt
Pa (t )  ca
2
 ca * ca  ca * (0)ca (0)  Pa (0) independent of time!!
But : stationary states are very special
The general solution is linear combination:
 ( x, t )   cnn ( x)e
 iEn ( t t0 )
n
i

  cnn ( x)En e
t
n
 iEn ( t t0 )
  cn ( Hˆ n ( x))e
 iEn ( t t0 )
n
 Hˆ  cnn ( x)e
 iEn ( t t0 )
n
(reminder Ĥ independent of t , no electromagnetic fields!)
 (t  t0 )   cnn ( x)
n
 specify  (t  t0 )
 cn   n * ( x) ( x, t  t0 )dx
  ( x, t ) is determined for all time, very simple!!
Probability to find En ?
cn (t )   n * ( x) cm * m ( x)e
 iEm ( t t0 )
m
 cn e
 iEn ( t t0 )
cn 2  cn  cn (0)
2
2
Chapter 4 – Postulates of Quantum Mechanics
61
Winter 2013
Chem 350: Introductory Quantum Mechanics
 independent of time E is conserved =
P E
n
n
n
Other properties, eg. xˆ 


 *(t ) * xˆ (t )dx  oscillation in time

Further remarks:
All depends on initial wave function, which is arbitrary in principle [compare to classical
mechanics  specify X  (0) , P (0) for all particles  X (t ) , P (t ) follows from equation of motion]
Stationary States:  (t  0)  n ( 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

A(t )    *( x, t )Aˆ ( x, t )dx

 d *


d
A
d ( x, t )
A(t )  
Aˆ ( x, t )dx    * ( x)  ( x)    * ( x)Aˆ
 dt


dt
t
dt

 i ˆ
i
 Hˆ ( )
 H ( )
t
t
  
ˆ
i
 *  ( H ) *
 t 
i
Assume:
A
0
t
 * ˆ
 H *
t
 : Hermitian

d
 
A    ( x, t )  Aˆ

dt
 t



ˆ  dx
 * dx   * ( x, t ) A
t

i
 i

 ( x, t )  Aˆ Hˆ   * dx    * ( x) Aˆ Hˆ ( x)dx


Chapter 4 – Postulates of Quantum Mechanics
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Winter 2013
Chem 350: Introductory Quantum Mechanics

i
ˆ ˆ  ( x, t )  *dx 
ˆ ˆ  ( x, t ) dx
  ( x, t )  AH
  ( x, t ) * AH
i

i
ˆ ˆ ( x, t )dx 
  * ( x)HA
  * ( x) AH ( x, t )dx
i

i
  * ( x)( AH  HA) ( x, t )dx
Or
i
Difficult Step:
d A
ˆ ˆ]
 [ AH
dt
t
General produce of 2 Hermitian operators

ˆ ˆ ( x)dx
f * ABg
  dx
ˆ ˆ  dx
  g ( x)  BAf
ˆ
ˆ ( x) Af
  Bg
*
Reverse Operators!
*
Examples:
i
P

t
 P, H 


  2 d2
  i
,
 V ( x)  
2
x  2m dx


i
P
 i
t
V
x
P
V

t
x
Same as from Classical Mechanics
2v
V
ma  m 2  F  
x
dt
dP
V


dt
x
Also
i
  d2

x
  x,
 V ( x) 
2
t
 2m dx

 


m x
2
i

i
m
x
Chapter 4 – Postulates of Quantum Mechanics
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Winter 2013
Chem 350: Introductory Quantum Mechanics

i
P
m
Or
x
1

P
t
m
Again: Same as in Classical Mechanics
Ehrenfest theorem:
Expectation values in Quantum Mechanics obey relations from classical mechanics
Chapter 4 – Postulates of Quantum Mechanics
64