PH403: Quantum Mechanics I
Tutorial Sheet 7
Problems in this tutorial sheet deal with the quantum mechanics of a simple-harmonic
oscillator as discussed in chapter 5 of the book by Cohen-Tannoudji et al.
1. Obtain and integrate the equations of motion for the expectation values of the X and
P operators for a simple-harmonic oscillator of mass m and frequency ω .
2. Consider a harmonic oscillator of mass m and angular frequency ω . At time t = 0, the
state of this oscillator is given by
X
|ψ(0)i = cn |φn i,
n
where |φn i are stationary states with energies (n + 1/2)~ω .
(a) What is the probability P that a measurement of the oscillator's energy performed
at an arbitrary time t > 0, will yield a result greater than 2~ω ? When P = 0,
what are the nonzero coe cients cn ?
(b) From now on assume that only c0 and c1 are dierent from zero. Write the
normalization coe cient for |ψ(0)i and the mean value hHi of the energy in
terms of c0 and c1 . With the additional requirement hHi = ~ω , calculate |c0 |2 and
|c1 |2 .
(c) As the normalized state vector |ψ(0)i is de ned only to within a global phase
factor, we x this factor by choosing c0 to be real and positive. We set c1 = |c1 |eiθ .
We assume that hHi = ~ω and that
r
1 ~
hXi = .
2 mω
Calculate θ.
3. Two particles of the same mass m, with positions X1 and X2 and momenta P1 and P2 ,
are subject to the same potential
1
V (X) = mω 2 X 2 .
2
The two particles do not interact with each other.
(a) Show that the Hamiltonian H of the two-particle system can be written as
H = H1 + H2 ,
where H1 /H2 act on the state space of particle 1/2.
(b) Calculate the energies of the two-particle system, their degree of degeneracy, and
the corresponding wave functions.
1
(c) Do H , H1 , and H2 form a C.S.C.O? We denote by |Φn1,n2 i the eigenvectors
common to H1 and H2 . Write the orthonormalization and closure relation for the
states |Φn1,n2 i.
(d) Consider a system which, at t = 0, is in the state
1
|ψ(0)i = (|Φ0,0 i + |Φ1,0 i + |Φ0,1 i + |Φ1,1 i).
2
What results can be obtained, and with what probabilities, if at this time one
measures: (i) the total energy of the system, and (ii) energy of particle 1?
4. This is a continuation of the previous problem and, therefore, uses the same notations.
Assume that at t = 0, the system is in state |ψ(0)i given in the previous exercise. At
t = 0, one measures the total energy H and nds the result 2~ω .
(a) Calculate the mean values (expectation values) of the position, the momentum,
and the energy of particles 1 and 2 at an arbitrary t > 0.
(b) At t > 0, one measures the energy of particle 1. What results can be found,
and with what probabilities? Same question for a measurement of the position
of particle 1. Plot the curve for the corresponding probability density.
5. We know that the Hamiltonian of a simple-harmonic oscillator can be written as H =
~ω(a† a + 21 ), and its time evolution operator can be written as U (t, 0) = e−iHt .
(a) Consider the operators
ã(t) = U † (t, 0)aU (t, 0)
ㆠ(t) = U † (t, 0)a† U (t, 0)
By calculating their actions on the kets |φn i, nd the expressions for ã(t) and
ㆠ(t) in terms of a and a† .
(b) Calculate the operators X̃(t) and P̃ (t) de ned by
X̃(t) = U † (t, 0)XU (t, 0)
P̃ (t) = U † (t, 0)P U (t, 0).
How can the relations so obtained be interpreted?
(c) Show that U † 2ω π
, 0 |xi is an eigenvector of P and specify its eigenvalue. Simi-
larly, establish that U † 2ωπ
, 0 |pi is an eigenvector of X .
6. Consider the operator D(α) = eαa −α
† ∗a
(a) Prove that
2 /2 † ∗
D(α) = e−|α| eαa e−α a .
Hint: you should use Glauber's Formula
1
eA eB = eA+B e 2 [A,B] ,
which is valid when both the operators A and B commute with their commutator
[A, B].
2
(b) Prove that
D(α)|0i = |αi,
where |0i represents the harmonic oscillator eigenstate with n = 0, and |αi rep-
resents the coherent state.
(c) Prove that
0 2
|hα|α0 i|2 = e−|α−α | .
This shows that coherent states form a non-orthogonal set of vectors.
