School of Physics and Astronomy
PHS2061 Quantum Mechanics - Assignment 2
Question 1
A particle with energy E is incident on a step potential as shown in Fig. 1. The wavefunction in the two
regions is given by:
Calculate the reflection coefficient R. A numerical value is required for R.
Figure 1: A particle of energy E > 0 incident on a step potential.
Question 2
A particle of mass m and energy is incident on a P?schl-Teller potential (see Fig. 2) defined by:
where is a real constant and the hyperbolic secant function is
Figure 2: A particle of energy E > 0 is incident on a P?schl-Teller potential from the far left
The classical expression for the total energy of the particle is:
(a) Using the procedure of canonical quantisation, obtain the time-dependent Schr?dinger equation
that describes the corresponding one-dimensional quantum mechanical system. Take care to show
your intermediate steps.
(b) The solution to the time-independent Schr?dinger equation for the P?schl-Teller potential is:
where C is a complex constant and the hyperbolic tangent function is:
In the regions the solutions can be written as:
Obtain expressions for the amplitudes A and B.
(c) Write an expression for the transmission coefficient T in terms of the amplitudes A and B (and any
other relevant constants).
(d) Show that the transmission coefficient T is equal to unity for the P?schl-Teller potential. Why is
this potential called reflectionless?
Question 3
Consider a hydrogen-like atom in two dimensions (2D) in which an electron with charge !e and mass m
is bound to a positive charge +e. The interaction between the electron and the positive charge is given by
the Coulomb potential:
where r is the separation between the two charges and is the vacuum permittivity constant. Such a 2D
quantum system could be realised by an electron in a thin semiconducting layer interacting with an ionised
impurity atom, such as phosphorous.
The time-independent Schr?dinger equation for the electron is written in polar coordinates as:
where denotes the two-dimensional eigenfunction and E is the energy eigenvalue.
(a) Use the technique of separation of variables to show that the time-independent Schr?dinger equation
can be written as two independent ordinary differential equations:
(1)
and
(2)
where l denotes the separation constant. Take care to show all intermediate steps.
Hint: Write the eigenfunction in the separable form
(b) Show that the following eigenfunction is a solution to the angular part of the time-independent
Schr?dinger equation (2):
(c) The eigenfunction must be single-valued, which requires Use this condition
to determine the allowed values of l.
(d) The ground state radial eigenfunction has the form , where A is a real normalisation
constant and is the Bohr radius of the two-dimensional hydrogen atom. Using this
eigenfunction, obtain an expression for the ground state energy of the two-dimensional hydrogen
atom.
Hint: Substitute R(r) into the radial time-independent Schr?dinger equation (1) and set l = 0.
(e) The energy of an electron in a two-dimensional (2D) hydrogen atom is given by:
Calculate the ionisation energy of a 2D hydrogen atom. Compare this with the ionisation energy
of the three-dimensional (3D) hydrogen atom. Provide a brief qualitative explanation for any
differences.
Question 4
A diatomic molecule can be modelled by a Morse potential defined by:
where and are real constants, and r is the displacement of the diatomic molecule from its equilibrium
bond length.
(a) Show that for small oscillations, about the equilibrium position r = 0, the time-independent
Schr?dinger equation has the following form:
where M is the reduced mass of the diatomic molecule.
Hint: Consider an expansion of the exponential function, i.e.,
(b) Write expressions for the raising operator and lowering operator which allows the
eigenvalue problem to be solved using the technique of operator factorisation.
(c) Show that the Hamiltonian operator can be written in terms of the raising and lowering operators
as:
(d) When the lowering operator acts on the ground state eigenfunction we obtain:
Use this result to determine the ground state energy of the diatomic molecule.
(e) Use the properties of the raising/lowering operators to determine the explicit form of the ground
state eigenfunction of the diatomic molecule. Take care to show all steps leading to your
expression for