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3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
COMMUTATION RELATIONS AND THEIR CONSEQUENCES
5.1 Commutators and Compatibility

The commutator of two operators Q and R is defined as
   RQ
 .
 Q , R  QR
To evaluate the commutator we operate on some arbitrary wave function with it.

Clearly if the commutator is zero, the order in which the operators are written is
unimportant and they commute. For example,
 P , P   0
x
y
and the operators for two different components of linear momentum commute.

In contrast position and momentum operators do not necessarily commute. For
two different components
Px , y  0 ,


whereas for the same components
 P , x  i .
x
Thus the x-component of position and the x-component of momentum do not
commute.

The significance of the commutation relations is that commuting operators
represent compatible dynamical variables that can be simultaneously measured
without uncertainty. This is because commuting variables share the same
eigenfunctions.

For example the wave function
u(x , y , z )  exp(ik . r )  expi[k 1 x  k 2 y  k 3 z ]
is simultaneously an eigenfunction of Px , Py and Pz . There are well defined
components of momentum in all three direction ( k 1 etc.) but the position in
space is infinitely uncertain ( x   ).
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3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
5.2 The Generalised Uncertainty Principle

If  q and  r are, as usual, the root-mean-square uncertainties in the
measured values of the quantities represented by the operators Q and R , then
the Generalised Uncertainty Principle states that
 q.  r 
1
2
 Q , R 
.
In this inequality, the expectation value is evaluated as usual:
 Q , R 

   RQ
    . d ,
  *  QR
all space
where the wave function  is the same for the whole set of measurements used
to obtain the uncertainties.

Thus if Q and R commute, this product of uncertainties can be zero. This will
happen if the wave function of the system is an eigenfunction of the operators.
Of course if the initial wave function is not one the eigenfunctions, the product
of uncertainties will be greater than zero.

In contrast, if Q and R do not commute then the product of uncertainties will
be larger than some minimum value. For example, we have seen that
 P , x  i
x
so that
1
 px .  x   .
2
This relation is true for all possible wave functions.
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3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
5.3 Commutation Relations for the Angular Momentum Operators

It is most convenient to work in Cartesian co-ordinates to derive the
commutation relations between the various angular momentum operators.
Classically,
Lr  P
so by the Correspondence Principle (Postulate P3) we know that
L  r  P  L x i  L y j  L z k
where
 
 
L x  yPz  zPy  i y
z
 , etc.
 y
 z

Then these operators for the components of the angular momentum satisfy the
following cyclic commutation relations:
 L , L   iL
 L , L   iL
 L , L   iL
x
y
z
y
z
x
z
x
y

Because these commutation relations are all nonzero they represent incompatible
observables. This means that two different components of angular momentum
cannot be measured simultaneously without uncertainties, since they do not have
the same eigenfunctions.

However consider the operator for the square of the total angular momentum.
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Classically L  L . L , so that the operator
L2  L2x  L2y  L2z .
It can be shown that
 L , L   0
2
z
and this is left for a workshop exercise.

This means that the z-component of angular momentum is compatible with the
total square angular momentum, and they have the same eigenfunctions. We
have met these eigenfunctions before, they are the Spherical Harmonics
Ylm ( ,  ) .
In fact any one, but only one, component of angular momentum can be
simultaneously defined with the total square because there is no correct z-axis in
the system until the symmetry is broken (e.g. by placing a magnetic field across
an atom).
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3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
WORKSHOP QUESTIONS
Hand your solutions to the following questions to Dr. Mulheran at the end of the first
workshop in week 6. Some of your solutions will be marked as part of the continuous
assessment of this course which contributes 20% of the overall module grade. Your
solutions must be well presented; untidy work will be penalised.
5.1
(a) Show that the commutation relation between the x-component of
momentum and the Hamiltonian operator obeys
ˆ   i  V  .
Pˆx , H
x 


[3 marks]
(b) Under which circumstances may the x-component of momentum be known
simultaneously with a particle’s total energy without uncertainty?
[2 marks]

5.2
(a)

Derive the Energy-Time Uncertainty Principle by using the total energy
operator

.
E tot  i
t
(b)
5.3
(The operator for time is just time t itself when the wave function is
written as a function of time).
[3 marks]
Excited atomic energy levels decay spontaneously due to the quantum
fluctuations of the electromagnetic field, and have typical half-lives of
about 10-8 s. What is the uncertainty in the value of these energy levels
and what is the typical natural line width you expect to observe in
spectroscopic measurements?
[2 marks]


(a) Using Lˆx , Lˆy  iLˆz , etc, show that
2
2
Lˆ y , Lˆ x   Lˆ z , Lˆ x  i Lˆ y Lˆ z  Lˆ z Lˆ y .

(b) Hence prove that
 

Lˆ , Lˆ  0


[3 marks]
x
2
.
What is the consequence of this result?
[2 marks]
5.4
A particle has the wave function
u( x, y, z )  Ax exp   x 2  y 2  z 2 
where  is a real number and the normalisation constant A does not have to be
evaluated. Using Cartesian co-ordinates, show that this wave function is an
2
eigenfunction of both L and L̂ x , and find the corresponding eigenvalues.
[5 marks]

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
3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
WORKSHOP SOLUTIONS
5.1
The kinetic energy operator and that for the x-component of momentum
commute:

i 3   2
 


Px , T  
   2
   0.

2   x
x 

However Px does not necessarily commute with the potential energy term in
the Hamiltonian:
 P , H    P ,V (x, y, z )  i x (V )  V . x  

x

x
V 
  i 

  x
This is only zero if
V
 0,
x
i.e. if the potential energy is independent of x.
5.2
(a)

 E , t   i t (t . )  t .  t   i ,
i.e.  E , t   i .

tot
tot
Thus by the Generalised Uncertainty Principle
E . t 
(b)
t  108 s , E 
1
.
2

 10 26 J .
8
210
.
s
Uncertainty in the frequency of the emitted light when an electron moves
8 1
from an excited state to the ground state is   10 s . This is in
15 1
comparison to the frequency of the light   10 s , so that the
uncertainty in the light’s wavelength is
 
5.3
c

2
.   10 14 m.
Start from the commutator between L̂ x and L̂ y operators:
Lˆ , Lˆ  iLˆ iLˆ  Lˆ Lˆ  Lˆ Lˆ .

x
y

z
z
x
y
Operate on this with L̂ y ,
iLˆ y Lˆ z  Lˆ y Lˆ x Lˆ y  Lˆ y Lˆ y Lˆ x
and then operate on L̂ y :
iLˆ z Lˆ y  Lˆ x Lˆ y Lˆ y  Lˆ y Lˆ x Lˆ y .
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y
x
3/PH/SB Quantum Theory - Week 5 - Dr. P.A. Mulheran
Adding these equations we find:
i Lˆ y Lˆ z  Lˆ z Lˆ y  Lˆ x Lˆ y Lˆ y  Lˆ y Lˆ y Lˆ x   Lˆ2y , Lˆ x .




Similarly for the other relation. Thus
2
2
2
Lˆ2 , Lˆ x  Lˆ y , Lˆ x  Lˆ z , Lˆ x  Lˆ x , Lˆ z  0 .

5.4
u 1

   2x u ,
x x

 
 
 

u
u
 2  y u ,
 2zu .
z
y
 u
u 
  i 2yz  2yz .u  0.u
Lˆ x u  i y
z
 y 
 z
Thus u is an eigenfunction of L̂ x with an eigenvalue of zero.
 iz
Lˆ y u 
u , the wave function is not an eigenfunction of L̂ y .
x
 
   iz 
2

Lˆ y u  i z
x
u
 z  x 
 x
   2 z 2
 
2
 x

 iz 2
 
u  i 2  i u 

 x
 
  2u
2
2
Similarly Lˆ Z u   2 u , Lˆ x u  0 , so that
L2 u  2 2 u ,
2
2
and the wave function is an eigenfunction of L with eigenvalue 2 (the
quantum numbers l  1 and m  0 ).
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