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School of Physics and Astronomy
PHS2061 Quantum Mechanics - Assignment 1
Question 1
An electron in the ground state of a hydrogen atom is described by the radial coordinate r. The ground state
eigenfunction is given by:
where is a real constant (the Bohr radius) and A is a real normalisation constant. In the ground state
the eigenfunction has spherical symmetry, i.e., it does not depend on the angular coordinates or
(a) Write an expression for the wavefunction describing the stationary ground state. Take
care to define any symbols you introduce.
(b) The radial probability density of the electron in the ground state of a hydrogen atom is denoted by
. This is defined so that is the probability of finding the electron with radial
coordinate between r and r + dr, i.e.,
Determine the radial position at which the electron is most likely to be found.
(c) By imposing the normalisation condition on the eigenfunction determine the constant A.
Hint: The following integral will be required:
Question 2
In 2013 Eibenberger and his colleagues at the University of Vienna carried out a double slit experiment
with very large molecules. In one such experiment they used the fluorous porphyrin molecule L12
, which consists of 810 atoms. Figure 1 shows a simplified schematic of the
experimental set-up. A coherent beam of L12 molecules was collimated and sent through a double slit. The
kinetic energy of the L12 molecules was 0.38 eV. After passing through the double slit the L12 molecules
were recorded by a detector D, located at a distance from the double slit.
Figure 1: Double slit experiment using fluorous porphyrin molecules L12. The point P is equi-distant
from Slits 1 and 2.
(a) Figure 2 shows the count rate, as a function of distance z, for the detection of L12 molecules in the
double slit experiment.
What aspect of Figure 2 cannot be explained by classical physics? How does quantum mechanics
provide an explanation of the observed count rate?
Figure 2: Count rate of L12 molecules in the double slit experiment.
(b) The wavefunctions describing the two possible paths, corresponding to an L12 molecule passing
through Slit 1 or Slit 2 and being detected at D, are given by:
where A is a complex constant and are real phase angles.
Write an expression for the probability density recorded at the detector D in this double slit
experiment.
(c) Show that the count rate R is given by:
where
Hints: You may assume an ideal detector for which the count rate is equal to the probability
density.
The following identities may be required:
(d) Consider the points P and Q in Figure 2. If the point P is the same distance from both slits (see
Figure 1), how much further from Slit 1 than from Slit 2 is the point Q? Express your answer
symbolically in terms of the de Broglie wavelength. A numerical answer is not required.
Question 3
A particle of mass m is confined to a one-dimensional (1D) infinite potential well (i.e., a 1D box) of size
L. The energy eigenvalues and normalised eigenfunctions are given by:
At t = 0 the particle is described by the wavefunction:
where is a complex coefficient.
(a) Is an eigenfunction of the linear momentum operator You must justify your answer.
(b) Is an eigenfunction of the Hamiltonian operator for the particle in the 1D box? You must
justify your answer.
(c) Calculate the numerical value of
(d) What is the probability that a measurement (at t = 0) will find the particle in the state
(e) What is the probability that a measurement (at t = 0) will find the particle in the state
(f) What is the probability that a measurement (at t = 0) will find the particle in an eigenstate with
energy
(g) Calculate the expectation value of a set of measurements of the energy in the state Express
your answer in terms of and L.
(h) The position of the particle in the 1D box is measured at t = 0. What is the probability that the
particle is found in the region You must justify your answer.
Question 4
(a) Consider two operators and associated with the observables A and B. Write the mathematical
expression corresponding to the situation in which the two operators do not commute.
(b) What is the physical significance of two operators not commuting? Your answer must comment on
the link between the commutation relation and the uncertainty principle.
(c) The translation operator is defined by the following action on an eigenfunction:
where a is a displacement in the x-direction. It can be shown that:
where is the linear momentum operator. Evaluate the commutator
Hints: The following operator expansion may be useful:
where is the identity operator and you may assume