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D18b
Principles of Quantum Mechanics: Sheet 2
Michaelmas 2015
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
1. Starting from the appropriate equation of motion for either operators or states, verify
that
d
ih̄ hφ|A|ψi = hφ|[A, H]|ψi
dt
holds in both the Heisenberg picture and the Schrödinger picture.
2. (a) Find and solve the Heisenberg equations of motion for the position x̂(t) and
momentum p̂(t) of the harmonic oscillator. Write the solution in two forms, one
involving x̂(0) and p̂(0) and the other involving the corresponding annihilation
and creation operators a and a† (also defined at t = 0).
Show that for any complex number α there is a normalized state |ψα i such that
hψα |x̂(t)|ψα i = αe−iωt + α∗ eiωt ,
the general solution for the classical oscillator. [Use the results of example 6 on
sheet 1.]
(b) The Hamiltonian for a point particle of mass m and charge e in a constant electric
field E is
1 2
H(x̂, p̂) =
p̂ − e E· x̂ .
2m
Solve the Heisenberg equations of motion for x̂(t) and p̂(t), expressing your
answer in terms of x̂(0) and p̂(0). Verify that H(x̂(t), p̂(t)) = H(x̂(0), p̂(0)).
3. Define annihilation, creation operators ai , ai† for the three-dimensional harmonic oscillator with Hamiltonian
H = H1 + H2 + H3
where
Hi =
p̂2i
+ 1 mω 2 x̂2i
2m 2
(i = 1, 2, 3).
Use these to find the energy levels and the space of states on which H acts. Assuming
no internal structure, show that a complete set of commuting operators can be taken
to be {a†1 a1 , a†2 a2 , a†3 a3 }. Show that the degeneracy of the n-th excited state is 21 (n +
1)(n + 2).
4. Two quantum systems have spaces of states V and V ′ . Each space is two-dimensional
with an orthonormal basis for V given by {|1i, |2i} and for V ′ by {|1′ i, |2′ i}. An
operator A is defined on V by A|1i = |2i, A|2i = |1i and a similar operator A′ is
defined on V ′ by A′ |1′ i = |2′ i, A′ |2′ i = |1′ i. The identity operators on V and V ′ are
I and I ′ respectively.
(a) Consider the combined system whose space of states is represented by the tensor
product V ⊗ V ′ . With respect to a convenient basis, determine the matrix forms
of the operators
S = I ⊗ I ′ + λ A ⊗ A′ ,
T = A ⊗ I ′ + µ I ⊗ A′ ,
where λ, µ are real numbers. Find the eigenvalues of S.
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(b) Show that T 2 ∝ S if λ and µ are suitably related. Hence, or otherwise, find the
eigenvalues of T .
(c) Now suppose that V and V ′ are spaces of states for two indistinguishable fermions.
What are the implications for the eigenvalues of S?
5. (a) The trace of an operator A acting on a vector space V is defined as TrV (A) =
P
†
n hn|A|ni where {|ni} is any orthonormal basis for V . If a, a are annihilation,
creation operators and V is the usual space of states for an oscillator, show that
†
Tr(xa a ) = (1 − x)−1 for |x| < 1.
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
(b) Suppose V = V1 ⊗ V2 and A = A1 ⊗ A2 where A1 , A2 act on V1 , V2 respectively.
Show that TrV (A) = TrV1 (A1 )TrV2 (A2 ).
*(c) Let ai , ai† be a set of D annihilation, creation operators obeying [ai , aj† ] = δij ,
[ai , aj ] = 0, [a†i , aj† ] = 0. Show that
Tr(xN ) = (1 − x)−D
N =
where
D
X
ai† ai .
i=1
Find the degeneracy dn of the eigenvalue n of the operator N , by first explaining
why
X
Tr(xN ) =
d n xn .
n≥0
6. A particle of mass m is confined to a three-dimensional box with sides a < b < c (the
wavefunction for the particle vanishes on the boundary of the box). Show that, with
a suitable choice of coordinates, the allowed eigenfunctions and energy levels are
ψnpq (x, y, z) =
8 1
2
abc
sin
nπx
pπy
qπz
sin
sin
,
a
b
c
Enpq =
h̄2 π 2 n2 p2 q 2 + 2 + 2
2m a2
b
c
where n, p, q = 1, 2, . . . .
What are the two lowest energy levels, their degeneracies and associated wave functions, for two identical non-interacting spin-0 particles confined to the box? What
are the corresponding results for two non-interacting, identical, spin- 12 particles?
7. A harmonic oscillator of mass m, frequency ω and charge e is perturbed by a constant
electric field of strength E, resulting in a new term H ′ = −eE x̂ in the Hamiltonian.
Calculate the change in the energy levels to order E 2 and compare with the exact
result.
8. The quantum-mechanical observable Q has just three eigenstates, |1i, |2i, |3i which
are orthogonal and correspond to eigenvalues 2, 1, 1 respectively. The operator Q′ is
defined by Q′ = Q + ǫT, where


3 1 √0

6
hj|T |ii =  1 √2
 .
0
6 1
Deriving the necessary formulae from first principles, apply perturbation theory to
find two of the eigenvalues of Q′ correct to order ǫ and the third correct to order ǫ2 .
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Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
9. Two harmonic oscillators of frequency ω and mass m interact through a potential
mω 2 λx1 x2 where x1 and x2 are the displacements of the oscillators. Use perturbation
theory to calculate the ground state energy of the system correct to second order
in powers of λ by adding the interaction potential on to the Hamiltonian. Find
the energies of the first two excited states, working to lowest non-trivial order in
perturbation theory.
√
√
Verify your results by using the change of variables 2 q1 = x1 + x2 , 2 q2 = x1 − x2
to obtain an exact expression for the energy eigenvalues.
10. The first excited energy level of the hydrogen atom is four-fold degenerate (ignoring
electron spin). The four wave functions may be taken to be
ψ0 (x) =
1
(8πa3 )
1
2
r
1−
2a
e−r/2a ,
ψi (x) =
1
(32πa5 )
1
2
xi e−r/2a ,
a=
4πǫ0 h̄2
,
e2 m
where xi are the Cartesian components of the vector x. The atom is perturbed by
a weak electric field E parallel to the 3-axis. Show that the only non zero matrix
elements of the perturbation H ′ are hψ0 |H ′ |ψ3 i and hψ3 |H ′ |ψ0 i. Calculate the new
energy levels to first order in E.
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