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 and is defined as.
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,
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
,
whereas for the same components
.
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
is simultaneously an eigenfunction of , and . There are well defined components of momentum in all three direction ( etc.) but the position in space is infinitely uncertain ().
5.2 The Generalised Uncertainty Principle
/ If and are, as usual, the root-mean-square uncertainties in the measured values of the quantities represented by the operators and , then the Generalised Uncertainty Principle states that.
In this inequality, the expectation value is evaluated as usual:
,
where the wave function is the same for the whole set of measurements used to obtain the uncertainties.
/ Thus if and 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 and do not commute then the product of uncertainties will be larger than some minimum value. For example, we have seen that
so that
.
This relation is true for all possible wave functions.
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,so by the Correspondence Principle (Postulate P3) we know that
where
, etc.
/ Then these operators for the components of the angular momentum satisfy the following cyclic commutation relations:
/ 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. Classically , so that the operator
.
It can be shown that
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
.
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).
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.
[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
.
(The operator for time is just time t itself when the wave function is written as a function of time).
[3 marks]
(b) / 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]
5.3 / (a) Using , etc, show that
.
[3 marks]
(b) Hence prove that
.
What is the consequence of this result?
[2 marks]
5.4 / A particle has the wave function
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 eigenfunction of both and , and find the corresponding eigenvalues.
[5 marks]
Workshop Solutions
5.1 / The kinetic energy operator and that for the x-component of momentum commute:.
However does not necessarily commute with the potential energy term in the Hamiltonian:
This is only zero if
,
i.e. if the potential energy is independent of x.
5.2 / (a) / ,
i.e. .
Thus by the Generalised Uncertainty Principle
.
(b) / , .
Uncertainty in the frequency of the emitted light when an electron moves from an excited state to the ground state is . This is in comparison to the frequency of the light , so that the uncertainty in the light’s wavelength is
5.3 / Start from the commutator between and operators:
.
Operate on this with ,
and then operate on :
.
Adding these equations we find:
.
Similarly for the other relation. Thus
.
5.4 / , , .
Thus u is an eigenfunction of with an eigenvalue of zero.
, the wave function is not an eigenfunction of .
Similarly , , so that
,
and the wave function is an eigenfunction of with eigenvalue (the quantum numbers and ).
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