Formal quantum mechanics
The formal basis of quantum mechanics
• Overview of the postulates of quantum mechanics
• Linear Hermitian Operators
eigenvalues and eigenvectors
orthonormality and completeness
• Predicting results of measurements
expectation values
collapse of the wavefunction
• Commutation relations
compatible observables
uncertainty principle
• Wavepackets
1
Formal basis of quantum
mechanics
This section puts quantum mechanics onto a more formal
mathematical footing by specifying those postulates of the theory
which cannot be derived from classical physics.
Main ingredients:
1. The wave function (to represent the state of the system)
2. Hermitian operators and eigenvalues (to represent observables)
3. A recipe for finding the operator associated with an observable
4. A description of the measurement process, and for predicting the
distribution of possible outcomes
5. The time-dependent Schrödinger equation for evolving the
wavefunction in time
2
The wave function
Postulate 1: For every dynamical system, there exists a wavefunction Ψ that
is a continuous, square-integrable, single-valued function of the coordinates of
all the particles and of time, and from which all possible predictions about the
physical properties of the system can be obtained.
Examples of the meaning of “The coordinates of all the particles”
For a single particle moving
in one dimension:
  x, t 
For a single particle moving
in three dimensions:
  r,t 
For two particles moving in three
dimensions:
  r1 , r2 ,t 
Square-integrable means that the normalization integral is finite
If we know the wavefunction we know everything it is possible to know.
3
Observables and operators
Postulate 2a: Every observable is represented by
a Linear Hermitian Operator (LHO).
An operator L is linear
if and only if
^
^
^
L[c1 f1  c2 f 2 ]  c1 L[ f1 ]  c2 L[ f 2 ]
(for arbitrary functions f1 and f 2 and
constants c1 and c2 )
Examples: which of the following operators are linear?
Lˆ1[ f ]  f  2
Lˆ [ f ]  xf
2
Lˆ3 [ f ]  x
df
Lˆ4 [ f ] 
dx
Note: the operators involved may or
may not be differential operators
(i.e. they may or may not involve
4
differentiating the wavefunction).
Hermitian operators
An operator O is
Hermitian if and only if





*
fi (O f j )dx    f j (O fi )dx 
 


*
*



*
f j (O fi * )dx

for all functions fi fj which vanish at infinity

Special case.
If operator O is real, this is


fi * (O f j )dx 

Compare the definition of a
Hermitian matrix M


f j (O f i* )dx
M ij   M ji 
*

Analogous if we identify a
matrix element with an integral:
M ij


f i* (O f j )dx
5
Hermitian operators:
examples



*


*
fi (O f j )dx    f j (O f )dx 
 


*
*


 *

The operator x is Hermitian   f j ( xfi )dx    fi * ( x* f j )dx   fi* ( xf j )dx

 
 
d
The operator
is not Hermitian
dx
*
*

  
  * 
* 
*
f j fi  fi 
f j dx 
  f j ( fi )dx    
x
 x  
 
   x

but -i



d
is Hermitian
dx
d2
The operator 2 is Hermitian
dx
6
Eigenvectors and eigenfunctions
Postulate 2b: the eigenvalues of the linear Hermitian operator give the possible
results that can be obtained when the corresponding physical quantity is measured.
Definition of an eigenvalue for a general linear operator
aˆn  nn
operator
eigenvalue
eigenfunction
Compare definition of an eigenvalue of a matrix
Example: the time-independent
Schrödinger equation:
Mx   x
2
2


d
ˆ
H ( x)   
 V ( x)  ( x)  E ( x)
2
 2m dx

Important fact: The eigenvalues of a Hermitian operator are real
(like the eigenvalues of a Hermitian matrix). Proof later.
7
Identifying the operators
Postulate 3: the operators representing the position and momentum of a particle are
x̂  x

pˆ x  i
x
rˆ  r
(one dimension)
 


pˆ  i i  j  k   i 
y
z 
 x
(three dimensions)
Other operators may be obtained from the corresponding classical quantities by
making these replacements everywhere.
Examples:
Kinetic energy
2
2
p 2x
1




ˆ =
Kx =
 K
x
 i
 
2m
2m 
x 
2m x 2
Hamiltonian (Energy)
2
2
2
p2

H
 V  x   Hˆ  
V  x
2
2m
2m x
Angular momentum (see Section 5)
ˆ  i r  
L  rp  L
8
Example: Momentum eigenfunctions
Momentum
eigenfunction
Eigenfunction equation
pˆ x p  x   p p  x 
Momentum
operator
pˆ x  i

x
Eigenvalue
= the momentum

 i
 p  x   p p  x 
x
Eigenfunctions are plane waves
 p  x   eikx  eipx /
p = ħk from the de Broglie relation
 ikx
ikx
ikx
i
e

hk
e

p
e






x
9
Important properties of
Linear Hermitian Operators
In the eigenvalue equation
Qˆn  qnn
qn  q*n
(i)
The eigenvalues are real
(ii)
Different eigenfunctions are orthogonal

*

 m  x n  x  dx  0, (m  n)

(iii)
The eigenfunctions form a complete set
  x, t    an (t )n ( x)
n
10
Important properties of
Linear Hermitian Operators (2)
Qˆn  qnn
Qˆ  q 
Proof of (i) and (ii)
m
Reminder: Hermitian property
(1)

(2)
m m


ˆ dx  q   dx


(1)


Q
n
n

 n
*
*
*
m
m
m
*

^


*
*
fi (Q f j )dx    f j (Q fi )dx 
 

^
*
*
   (2)     Qˆm dx   q*  *n dx
m 
m


 n

*
n
*

q
*

q
n
m
Use the Hermitian property to show

Case 1: n = m
   dx  0  q
*

n
n
n

Case 2: n ≠ m and
qn  qm 
 qn
*
*

 n dx  0

m
    dx  0
*

m
n

Can choose normalized
eigenfunctions
*

 n dx  1

Case 3: n ≠ m but
n
11
qn  qm next!
Important properties of
Linear Hermitian Operators (3)
Case 3: n ≠ m but qn  qm (degenerate eigenvalues)
Any linear combination of degenerate
eigenfunctions is also an eigenfunction
with the same eigenvalue:
^
^
^
Q  c11  c22   c1 Q[1 ]  c2 Q[2 ]
 c1q1  c2 q2
 q  c11  c22 
So we are free to choose two linear combinations that are orthogonal, e.g.
a  1
b  c11  c22
Two coefficients and two constraints:
normalization and orthogonality
If the eigenfunctions are all orthogonal and
normalized, they are said to be orthonormal.



m*n dx   mn  1, if m  n
 0, if m12 n
Orthonormality example: Infinite well
Consider the two lowest energy eigenfunctions of the time-independent
Schrödinger equation for an infinite square well
Normalized eigenstates are
1
 
cos 
x
a
 2a 
1
 2 
 2  x 
sin 
x
a
 2a 
1  x 
 2  x
1  x
x = -a
x=a
1 a
  x   2 x 
cos 
 sin 
 dx  0


a
a
 2a   2a 
We have the integral of an odd function
over an even region, which is zero.
The eigenstates are orthogonal because
their positive and negative regions give
13
cancelling contributions to the integral.
Orthonormality example: Infinite well (2)
General case
1
 n 
cos 
x  , n  1,3,5 
a
 2a 
1
 n 
 n  x 
sin 
x  , n  2, 4, 6 
a
 2a 
 n  x 
Can easily prove orthonormality using trigonometry formulas
1 a
 n x   m x 
sin

 sin 
 dx   mn


a
a
 2a   2a 
1 a
 n x 
 m x 
cos 
 cos 
 dx   mn


a
a
 2a 
 2a 
1 a
 n x   m x 
cos

 sin 
 dx  0


a
a
 2a   2a 
These results are already
14
familiar from Fourier series
Complete sets of functions
The eigenfunctions φn of a Hermitian operator
form a complete set. This means that any
other function satisfying the same
boundary conditions can be expanded as:
 ( x)   ann ( x)
n
This expansion is a generalization of the Fourier series.
This sum of different eigenstates is called a superposition.
If the eigenfunctions are orthonormal, the
coefficients an can be found as follows (in 1D)

an   n*  x   x  dx

Proof



  x   x  dx   n*  x   amm  x  dx
*
 n

 a
m
Orthonormality
m


  x m  x  dx  an
*
m  n



n*m dx   mn
15
These expansions are very important in describing the measurement process.
Completeness for a continuum
Particles can have a discrete set of eigenvalues (like the harmonic oscillator or
infinite potential well) or they can have a continuum of energies (e.g. a free particle).
For a continuum, use an integral instead of a sum in the wavefunction expansion
 ( x)   ann ( x)    x  
n
ak  

 a  k   k , x  dk


*

  k , x   x  dx

E.g. Free particles: Use momentum eigenstates

eikx
  x   a k 
dk ,
2

a k  



e  ikx
  x  dx
2
16
This is just a Fourier decomposition
Expansion in complete sets: examples
A particle is in an infinite well from –a to a. For the wavefunctions given, find the
coefficients an in an expansion using the Hamiltonian eigenstates (the wavefunctions
are zero outside the well of course).
1)
2)
  x 
1  x 
 2 x 
 5 x  
cos

3
sin

5
cos





 
3 a   2a 
 2a 
 2a  
1
  x 
2a
Hamiltonian eigenstates
1
 n 
cos 
x  , n  1,3,5 
a
 2a 
1
 n 
 n  x 
sin 
x  , n  2, 4, 6 
2
a
a


 n  x 
 ( x)   ann ( x)
n
an   n*  x  17x  dx
a
a
Expansion in complete sets: examples
Plot of partial expansions of   x  
1
2a
First term
First 5 non-zero terms
First 15 non-zero terms
18
Eigenfunctions and measurement
Postulate 4a: When a measurement of the observable Q is made on a
normalized wavefunction ψ, the probability of obtaining the eigenvalue qn
is given by the modulus squared of the overlap integral

Pr  qn   an , an   n*  x   x  dx
2

This corresponds to expanding the wavefunction in
the complete set of eigenstates of the operator for the
physical quantity we are measuring and interpreting
the modulus squared of the expansion coefficients as
the probability of getting a particular result. This is the
general form of the Born interpretation
 ( x)   ann ( x).
n
Corollary: if a system is definitely in the eigenstate φn, the result of
measuring Q is definitely the corresponding eigenvalue qn.
The meaning of these “probabilities” for a single system is still a matter for debate.
The usual interpretation is that the probability of a particular result determines the
19
frequency of that result in measurements on an ensemble of similar systems.
Expectation values
The expectation value is the average (mean) value of many measurements.
It is the sum of all the possible results times the corresponding probabilities:
Q   Pr  qn  qn   an qn
2
n
We can also
write this as:
Expand Ψ in eigenstates of Q
n
Qˆn  qnn
 dx  
*
n m
  mn

Q    *  x  Qˆ  x  dx
Proof
  x    ann  x 
n



* * ˆ 
Q     amm  Q   ann  dx
m
  n



* * 
    amm    an qnn  dx
m
 n

  am* an qn  m* n dx   an qn
2
m
n
n
20
Wavefunction Normalization

The normalization of the wavefunction is
We can also write this in terms of
the expansion coefficients
N    *  x   x  dx  1

 an  1
2
for a normalized
wavefunction
n
This is consistent with the probability
interpretation for expansion coefficients
Pr  qn   an   an  1
2
2
n
Can prove this using the expectation value of the operator Q = 1!
The eigenvalues of Q = 1 are qn = 1 so we have
Q 

  x  Qˆ  x  dx,
*


1    *  x 1  x  dx  1,

Q   an q n
2
n
1   an  1
2
n
21
Expectation Values:
examples
1
 n 
cos 
x  , n  1,3,5 
2
a
a


1
 n 
 n  x 
sin 
x  , n  2, 4, 6 
a
 2a 
 n  x 
1) A particle is in the ground state of an infinite well from –a to a.
What is the expectation value of the position and the momentum?
n2 2 2
En 
8ma 2
2) For the same infinite well, a particle has wavefunction
  x 
1  x 
 2 x 
 5 x  
cos

3
sin

5
cos





 
3 a   2a 
 2a 
 2a  
Check that this is correctly normalized.
What is the expectation value of the energy?
22
Expectation Values: examples
3) A particle is in the ground state of a harmonic oscillator potential
of frequency ω:
1/ 4
 m 

 
 0  x  
exp  m x 2 / 2

Calculate the average value of its kinetic energy. You may use:



exp   x / a dx  a 
2
2
a3 
 x exp   x / a dx  2

2
2
2
23
Collapse of the wavefunction
Postulate 4b: Immediately after a measurement, the wavefunction is an
eigenfunction of the operator corresponding to the eigenvalue just obtained
as the measurement result.
 ( x)   ann ( x)   ( x)  n ( x)
n
Pr  an
2
This is the famous collapse of the wavefunction and is
an idea mainly due to John von Neumann in 1932.
This ensures that we are guaranteed to get the same result
if we immediately re-measure the same quantity.
 ( x)  n ( x)  Pr(qn )  an  1
2
Problem: This is a different time-evolution from the Schrödinger equation.
How do we know when to use the Schrödinger equation and when to use
collapse, i.e. what constitutes a measurement?
24
Evolution of the system
Postulate 5: Between measurements (i.e. when it is not disturbed by
external influences) the wavefunction evolves with time according to the
time-dependent Schrödinger equation.

i
 Hˆ 
t
Hamiltonian operator.
This is a linear, homogeneous differential equation, so the linear
combination of any two solutions is also a solution.
This is the superposition principle.
1
 Hˆ 1
t
 2
i
 Hˆ  2
t
i
  1   2  ˆ
i
 H  1   2 
t
25
Time dependent expansions
We can expand the full time-dependent wavefunction
using time-dependent expansion coefficients.
  x, t    an (t )n  x 
n
We can work out how these evolve using the TDSE
for Ψ(x,t) and the overlap integral.

an (t )   n*  x    x, t  dx

Simple special case:
Suppose the Hamiltonian is time-independent.
We know that separated solutions of the TDSE
exist in the form:
( x, t )  exp(iEnt / ) n ( x)
Hˆ n  x   En n  x 
The eigenfunctions of the TISE form a complete
set, so we can expand the initial wavefunction as
 ( x, 0)   an (0) n ( x)
n
Hence we can find the complete time
dependence from the superposition principle
( x, t )   an (0) exp(iEnt / ) n ( x)
n
an (t )
26
Commutators
In general operators do not commute:
the order in which the operators act on
functions matters.
µµy ¹ RQ
µµy (in general)
QR
Example, position and momentum operators:
xx
p x  i

x
 


x p x  x  i



i
x


x
x


 


p x x   i
x



i


i
x




x
x


We define the commutator as the
difference between the two orderings:
Q, R   QR  RQ


Two operators commute only if their
commutator is zero.
For position and momentum:
 x, p x   i 


 x, p x   i


27
Compatible operators
Two observables are compatible if their operators share the
same eigenfunctions (but not necessarily the same eigenvalues).
Consequence: two compatible observables can have
precisely-defined values simultaneously.
Start with general wavefunction
Qˆn  qnn
Rˆ  r 
n
n n
 ( x)   ann ( x).
n
Measure observable
Q. Get result qm with
(probability |am|2)
Wavefunction collapses
to corresponding
eigenfunction φm
Measure observable R.
Definitely get rm
(eigenvalue of R for φm)
 ( x)   ann ( x)  m ( x)
n
Wavefunction
is still φm
Re-measure Q.
Definitely get qm again
 ( x)  m ( x)
28
For simplicity we only consider the non-degenerate case here.
Compatible operators (2)
Compatible operators commute
Proof
Expand ψ in the set of simultaneous eigenfunctions
QR  RQ    a QR  RQ 
  a  Qr   Rq  
n
n
n
n
n n
n n
  x    ann  x 
n
Qˆn  qnn
Rˆn  rnn
n
  an  rn qnn  qn rnn   0
n


 QR  RQ   0
 Q, R   0


Can also prove the converse (see Rae Chapter 4) :
if two operators commute then they are compatible.
29
Example: position and momentum
x and px do not commute.
There are no functions which are simultaneous
eigenfunctions of the position and momentum operators
This is directly related to the uncertainty principle.
If we measure x we lose information about px and vice versa
But now consider
 x, p x   i


xpx 
2
 x, p y     x  i     i   x  


y  
y




 i x
i x
0
y
y
 x, p y   0


So x and py commute. The x position and y momentum are compatible.
We can know x and py at the same time with arbitrary accuracy.
30
Commutation relations and
the Uncertainty Principle
How does  x, p x   i


relate to the Uncertainty Principle?
xpx 
2
Outline derivation of the UP (see Rae §4.5)
Use Schwarz’s Inequality to obtain
Define rms deviations
 x 
2
 px 

2

x  x
1 
1

xpx 
x, p x  i


2
2
2
 px  px 
2
In general we get an uncertainty relation
for any two incompatible observables,
i.e. whose corresponding operators do
not commute
 xpx 
2
For general non-commuting operators Qˆ , Rˆ
qr 
1
2
Qˆ , Rˆ 


31
Wavepackets and the
Uncertainty Principle
Wavepackets are the best way of describing a quantum system
with both particle-like and wave-like characteristics.
We cannot have absolute certainty of both position and momentum.
But we can construct a wavepacket which is localized in both
position and momentum
 ( x)
E.g. real space probability density
2
Width  
  x   eik x exp   x 2 / 4 2 
0
  x   exp   x 2 / 2 2 
2
x
Write this as a Fourier transform
(expansion in momentum eigenstates)
1
  x 
2

 dk A  k  e
A(k )
A k   e
2
ikx
2 2  k  k0 
2
Width  1/ 
2

 A k   e
  2  k  k0 
2
k0
32
k
Wavepackets and the
Uncertainty Principle (2)
Rough uncertainty in postion given
from the point where the Gaussian
falls to 1/e of its peak value
Similarly, rough uncertainty
in momentum:
k 
x  2
1
2
2
  x  e
2
A k   e
2
 x 2 / 2 2
2 2  k  k0 
2
 p  k 
1
2 2
Hence the product of uncertainties is a constant, independent of σ
px 
1
2 2
2 2 
NB: The Uncertainty relation is usually evaluated using rms
widths rather than our 1/e estimate. In that case we get
So the Gaussian is actually a minimum uncertainty wavepacket
px 
2
33
2
Summary of the
Uncertainty Principle
xpx 
We have now seen three ways of thinking about the Uncertainty principle:
(1) As the necessary disturbance of the system due to measurements
(e.g. the Heisenberg microscope)
(2) Arising from the properties of Fourier transforms (narrow spatial
wavepackets need a wide range of wavevectors in their Fourier
transforms and vice versa)
(3) As a fundamental consequence of the fact that x and p are not
compatible quantities so their corresponding Hermitian operators do
not commute. They do not share any eigenvectors and therefore
cannot have precisely defined values simultaneously.
For general non-commuting operators Qˆ , Rˆ
qr 
1
2
Qˆ , Rˆ 


34
2
Evolution of expectation values
Consider the rate of change of the
expectation value of an observable Q
for a time-dependent wavefunction
d Q
i
dt

Q(t )   *  x, t  Qˆ   x, t  dx


d
* ˆ
i

(Q )dx

dt 


i





Q

(Q )   * (i
 )   * (Qi
)dx
t
t
t
( Hˆ  * )(Q )   * (QH  )dx  i




 ˆ
 H
t
*
i
 Hˆ  *
t
*
 * H (Q )   * (QH )dx  i

  H , Q  i


Q
t
Q
t
Q
t
i




fi ( H f j )dx 
*


f j ( H f i* )dx
d Q
i  ˆ ˆ
Qˆ
 H , Q 
dt
t
35
Ehrenfest’s theorem
Example: conservation of energy
Consider the rate of change of the mean energy

d E

   * H dx
dt
t - 
d Q
i  ˆ ˆ
Qˆ
 H , Q 
dt
t
i 
H


H, H 


t
The Hamiltonian is independent of time
Everything commutes with itself!
Hˆ
0
t
H , H   0


d E
dt
0
Although the energy of a system may be uncertain (in the sense that
measurements made on many copies of the system may give different
results) the average energy is always conserved with time.
36
Example: position and momentum
Consider the rate of change of the mean position

d x

i
x
  * xdx   Hˆ , x  
dt
t -
t
d x

i  pˆ 2


V
(
x
),
x
 2m

dt


2
i 
d2 

  2m dx 2 , x 


pˆ
i d

 x
m dx
m
Can also show similarly that
x
0
t
 A  B, C    A, C    B, C 
V ( x), x  0
 d2 
d
,
x

2
 dx 2 
dx


d px
dV ( x)

dt
dx
These look very like the usual classical expressions relating position and
velocity and Newton’s second law. So we recover classical mechanics-like
37
expressions for the evolution of expectation values.
Summary (1)
There is a wavefunction

Linear Hermitian Operators represent observables
Eigenvalues give possible measurement results
Qˆn  qnn
Orthonormality of eigenfunctions





fi (O f j )dx 
*


*
f j (O f i * )dx

m*n dx   mn
Completeness and the overlap integral
  x, t    an (t )n ( x)
n

an (t )   n*  x    x, t  dx

Position and momentum operators
Other operators: use these in the classical expression
Collapse of the wavefunction
at a measurement
xˆ  x
p x  i
 ( x)   ann ( x)   ( x)  n ( x)
n

x
Pr  38
an
2
Summary (2)
Expectation values
and Ehrenfest’s theorem
Q 


2
 Qˆ dx   an qn
*
n
d Q
i  ˆ ˆ
Qˆ
 H , Q 
dt
t
 an  1
2
Normalization
n
Time-dependent Schrödinger equation
Commutation relations
and the Uncertainty principle

i
 Hˆ 
t
Q, R   QR  RQ


qr 
Compatible observables:
 x, p x   i


1
2
Qˆ , Rˆ 


Commute
Have simultaneous eigenfunctions
Can be uniquely determined simultaneously
39