EP 307: Quantum Mechanics I
Tutorial Sheet 4
Problems in this tutorial sheet deal with the mathematical methods of quantum mechan-
ics as discussed in chapter 2 of the book by Cohen-Tannoudji et al. Read complement B
of chapter II to revise some important concepts related to linear operators, their functions,
and commutators.
1. |φn i are the eigenstates of a Hermitian operator H , which could be the Hamiltonian
of a physical system. Assume that the states |φn i form a discrete orthonormal basis.
The operator U (m, n) is de ned by
U (m, n) = |φm ihφn |.
(a) Calculate the Hermitian conjugate U † (m, n), of U (m, n)
(b) Calculate the commutator [H, U (m, n)]
(c) Prove that U (m, n)U † (p, q) = δnq U (m, p)
(d) Calculate Tr(U (m, n))
(e) Let A
Pbe an operator, with matrix elements Amn = hφm |A|φn i. Prove that
A= m,n Amn U (m, n)
(f) Show that Apq = Tr(AU † (p, q))
2. In a two-dimensional vector space, consider the operator whose matrix, in an orthonor-
mal basis {|1i, |2i}, is written as
0 −i
σy = .
i 0
(a) Is σy Hermitian? Calculate its eigenvalues and eigenvectors (giving their normal-
ized expansion in terms of the {|1i, |2i} basis.
(b) Calculate the matrices which represent projectors on to these eigenvectors. Then
verify that they satisfy the orthogonality and closure relations.
3. Let K be the operator de ned by K = |φihψ|, where |φi, |ψi ∈ E .
(a) Under what conditions is K Hermitian?
(b) Calculate K 2 . Under what conditions is K a projector?
(c) Show that K can always be written in the form K = λP1 P2 , where λ is a constant
to be calculated and P1 and P2 are projectors.
4. The σx matrix is de ned by
0 1
σx = .
1 0
Prove the relation
eiασx = I cos α + iσx sin α,
where I is the 2 × 2 identity matrix and α is a number.
1
5. Prove the following properties of commutators
(a) [A, B] = −[B, A]
(b) [A, B]† = [B † , A† ]
(c) [A, B + C] = [A, B] + [A, C]
(d) [A, BC] = [A, B]C + B[A, C]
(e) [AB, C] = A[B, C] + [A, C]B
(f) [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
6. Consider the Hamiltonian H of a particle in a one-dimensional problem de ned by
P2
H= + V (X),
2m
where X and P are the position and momentum operators which satisfy [X, P ] = i~.
Assume that H has a discrete spectrum described by the eigenvalue equation H|φn i =
En |φn i.
(a) by evaluating hφn |[X, H]|φn0 i, prove that
hφn |P |φn0 i = αhφn |X|φn0 i,
and determine the value of α
(b) From this, deduce, using the closure relation, the equation
X ~2
(En − En0 )2 |hφn |X|φn0 i|2 = hφn |P 2 |φn i
n0
m2
7. Let H be the Hamiltonian of a physical system with eigenspectrum de ned by H|φn i =
En |φn i.
(a) For an arbitrary operator A prove that hφn |[A, H]|φn i = 0.
(b) If the Hamiltonian in question represents a particle of mass m con ned along the
x direction with
2
P
H= + V (X),
2m
i. In terms of P , X , and V (X), compute the commutators: [H, P ], [H, X], and
[H, XP ].
ii. Show that hφn |P |φn i = 0.
iii. Establish a relation between K = hφn | 2m
P2
|φn i and hφn |X dX
dV
|φn i. How is K
related to hφn |V (X)|φn i, when V (X) = V0 X λ (λ = 2, 4, 6, . . . ; V0 > 0).
8. Using the relation hx|pi = (2π~)−1/2 eipx/~ , nd the expression hx|XP |ψi and hx|P X|ψi
in terms of ψ(x). Can these results be found directly by using the fact that in the |xi
representation, P acts like ~i dx
d
?
2
9. Consider a three-dimensional state space spanned by the orthonormal basis kets |u1 i, |u2 i,
and |u3 i. Consider two operators Lz and S de ned by
Lz |u1 i = |u1 i Lz |u2 i = 0 Lz |u3 i = −|u3 i
S|u1 i = |u3 i S|u2 i = |u2 i S|u3 i = |u1 i,
(a) Obtain the matrix representations of operators Lz , L2z , S , and S 2 with respect to
this basis. Are these operators observables?
(b) Give the form of the most general matrix which represents an operator which
commutes with Lz . Same question for L2z and S 2 .
(c) Do L2z and S form a C.S.C.O.? Give a basis of common eigenvectors.
