Exercises

19.3 EXERCISES
that would involve a number of non-trivial integrals if tackled using explicit
wavefunctions.
Given that the first-order change in the ground-state energy of a quantum system when
it is perturbed by a small additional term H in the Hamiltonian is 0|H |0, find the firstorder change in the energy of a simple harmonic oscillator in the presence of an additional
potential V (x) = λx3 + µx4 .
From the definitions of A and A† , equation (19.39), we can write
1
(A + A† )
x= √
2mω
⇒
H =
λ
µ
(A + A† )3 +
(A + A† )4 .
(2mω)3/2
(2mω)2
We now compute successive values of (A + A† )n |0 for n = 1, 2, 3, 4, remembering that
√
and
A† |n = (n + 1) |n + 1 :
A |n = n |n − 1
(A + A† ) |0 = 0 + 1/2 |1,
√
(A + A† )2 |0 = |0 + 2 |2,
√
(A + A† )3 |0 = 0 + 3/2 |1 + 23/2 |1 + 6 3/2 |3
√
= 33/2 |1 + 6 3/2 |3,
√
√
√
† 4
(A + A ) |0 = 32 |0 + 18 2 |2 + 18 2 |2 + 24 2 |4.
To find the energy shift we need to form the inner product of each of these state vectors
with |0. But |0 is orthogonal to all |n if n = 0. Consequently, the term 0| (A + A† )3 |0
in the expectation value is zero, and in the expression for 0| (A + A† )4 |0 only the first
term is non-zero; its value is 32 . The perturbation energy is thus given by
0 | H | 0 =
3µ2
.
(2mω)2
It could have been anticipated on symmetry grounds that the expectation of λx3 , an
odd function of x, would be zero, but the calculation gives this result automatically. The
contribution of the quadratic term in the perturbation would have been much harder to
anticipate! 19.3 Exercises
19.1
19.2
19.3
Show that the commutator of two operators that correspond to two physical
observables cannot itself correspond to another physical observable.
By expressing the operator Lz , corresponding to the z-component of angular
momentum, in spherical polar coordinates (r, θ, φ), show that the angular momentum of a particle about the polar axis cannot be known at the same time as
its azimuthal position around that axis.
In quantum mechanics, the time dependence of the state function |ψ of a system
is given, as a further postulate, by the equation
∂
|ψ = H|ψ,
∂t
where H is the Hamiltonian of the system. Use this to find the time dependence
of the expectation value A of an operator A that itself has no explicit time
dependence. Hence show that operators that commute with the Hamiltonian
correspond to the classical ‘constants of the motion’.
i
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QUANTUM OPERATORS
For a particle of mass m moving in a one-dimensional potential V (x), prove
Ehrenfest’s theorem:
7
6
dx
dpx px dV
and
=−
=
.
dt
dx
dt
m
19.4
19.5
Show that the Pauli matrices
0 1
0
, Sy = 12 Sx = 12 1 0
i
−i
0
,
Sz = 12 1
0
0
−1
,
which are used as the operators corresponding to intrinsic spin of 12 in nonrelativistic quantum mechanics, satisfy S2x = S2y = S2z = 14 2 I, and have the
same commutation properties as the components of orbital angular momentum.
Deduce that any state |ψ represented by the column vector (a, b)T is an eigenstate
of S2 with eigenvalue 32 /4.
Find closed-form expressions for cos C and sin C, where C is the matrix
1
1
C=
.
1 −1
Demonstrate that the ‘expected’ relationships
cos2 C + sin2 C = I
19.6
and
sin 2C = 2 sin C cos C
are valid.
Operators A and B anticommute. Evaluate (A + B)2n for a few values of n and
hence propose an expression for cnr in the expansion
(A + B)2n =
n
cnr A2n−2r B 2r .
r=0
Prove your proposed formula for general values of n, using the method of
induction.
Show that
n
∞ dnr A2n−2r B 2r ,
cos(A + B) =
n=0 r=0
19.7
19.8
where the dnr are constants whose values
determine.
you should
0 1
By taking as A the matrix A =
, confirm that your answer is
1 0
consistent with that obtained in exercise 19.5.
Expressed in terms of the annihilation and creation operators A and A† discussed
in the text, a system has an unperturbed Hamiltonian H0 = ωA† A. The system
is disturbed by the addition of a perturbing Hamiltonian H1 = gω(A + A† ),
where g is real. Show that the effect of the perturbation is to move the whole
energy spectrum of the system down by g 2 ω.
For a system of N electrons in their ground state |0, the Hamiltonian is
H=
N
N
p2xn + p2yn + p2zn V (xn , yn , zn ).
+
2m
n=1
n=1
Show that p2xn , xn = −2ipxn , and hence that the expectation value of the
double commutator [ [ x, H ] , x ], where x = N
n=1 xn , is given by
0 | [ [ x, H ] , x ] | 0 =
672
N2
.
m
19.3 EXERCISES
Now evaluate the expectation value using the eigenvalue properties of H, namely
H|r = Er |r, and deduce the sum rule for oscillation strengths,
∞
(Er − E0 )| r | x | 0 |2 =
r=0
19.9
N2
.
2m
By considering the function
F(λ) = exp(λA)B exp(−λA),
where A and B are linear operators and λ is a parameter, and finding its
derivatives with respect to λ, prove that
eA Be−A = B + [ A, B ] +
Use this result to express
exp
19.10
1
1
[ A, [ A, B ] ] +
[ A, [ A, [ A, B ] ] ] + · · · .
2!
3!
iLx θ
Ly exp
−iLx θ
as a linear combination of the angular momentum operators Lx , Ly and Lz .
For a system containing more than one particle, the total angular momentum J
and its components are represented by operators that have completely analogous
commutation relations to those for the operators for a single particle, i.e. J 2 has
eigenvalue j(j + 1)2 and Jz has eigenvalue mj for the state |j, mj . The usual
orthonormality relationship j , mj | j, mj = δj j δmj mj is also valid.
A system consists of two (distinguishable) particles A and B. Particle A is in
an = 3 state and can have state functions of the form |A, 3, mA , whilst B is
in an = 2 state with possible state functions |B, 2, mB . The range of possible
values for j is |3 − 2| ≤ j ≤ |3 + 2|, i.e. 1 ≤ j ≤ 5, and the overall state function
can be written as
jm
|j, mj =
CmA jmB | A, 3, mA | B, 2, mB .
mA +mB =mj
jm
The numerical coefficients CmA jmB are known as Clebsch–Gordon coefficients.
Assume (as can be shown) that the ladder operators U(AB) and D(AB) for
the system can be written as U(A) + U(B) and D(A) + D(B), respectively, and
that they lead to relationships equivalent to (19.34) and (19.35) with replaced
by j and m by mj .
(a) Apply the operators to the (obvious) relationship
|AB, 5, 5 = |A, 3, 3 |B, 2, 2
to show that
|AB, 5, 4 =
6
10
|A, 3, 2 |B, 2, 2 +
4
10
|A, 3, 3 |B, 2, 1.
(b) Find, to within an overall sign, the real coefficients c and d in the expansion
|AB, 4, 4 = c|A, 3, 2 |B, 2, 2 + d|A, 3, 3 |B, 2, 1
by requiring it to be orthogonal to |AB, 5, 4. Check your answer by considering
U(AB)|AB, 4, 4.
(c) Find, to within an overall sign, and as efficiently as possible, an expression
for |AB, 4, −3 as a sum of products of the form |A, 3, mA |B, 2, mB .
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