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104 General principles of quantum theory
Conversely, if B changes rapidly with time, then the period Ät for B to change
by one standard deviation will be short and the uncertainty in the energy of the
system will be large.
Problems
3.1 Which of the following operators are linear?
p
^
^
(a) (b) sin (c) xD x (d) D x x
3.2 Demonstrate the validity of the relationships (3.4a) and (3.4b).
3.3 Show that
^ ^ ^
^ ^
^
^
^ ^
[A,[B, C]] [B,[C, A]] [C,[A, B]] 0
^
^ ^
where A, B, and C are arbitrary linear operators.
^
^
^ 2
2
3.4 Show that (D x x)(D x ÿ x) D ÿ x ÿ 1.
x
2
^ 2
3.5 Show that xe ÿx 2 is an eigenfunction of the linear operator (D ÿ 4x ). What is
x
the eigenvalue?
^ 2
^ 2
3.6 Show that the operator D is hermitian. Is the operator iD hermitian?
x x
^
^
3.7 Show that if the linear operators A and B do not commute, the operators
^^
^ ^
^ ^
(AB BA) and i[A, B] are hermitian.
3.8 If the real normalized functions f (r) and g(r) are not orthogonal, show that their
sum f (r) g(r) and their difference f (r) ÿ g(r) are orthogonal.
2 ÿx=2
3.9 Consider the set of functions ø 1 e ÿx=2 , ø 2 xe ÿx=2 , ø 3 x e , ø 4
3 ÿx=2
x e , de®ned over the range 0 < x < 1. Use the Schmidt orthogonalization
procedure to construct from the set ø i an orthogonal set of functions with
w(x) 1.
3.10 Evaluate the following commutators:
^
^
2
(a) [x, ^ p x ] (b) [x, ^ p ] (c) [x, H] (d) [^ p x , H]
x
3
2
2
3.11 Evaluate [x, ^ p ] and [x , ^ p ] using equations (3.4).
x x
3.12 Using equation (3.4b), show by iteration that
n
[x , ^ p x ] i"nx nÿ1
where n is a positive integer greater than zero.
3.13 Show that
df (x)
[f (x), ^ p x ] i"
dx
2
2
3.14 Calculate the expectation values of x, x , ^ p, and ^ p for a particle in a one-
dimensional box in state ø n (see Section 2.5).
4
3.15 Calculate the expectation value of ^ p for a particle in a one-dimensional box in
state ø n .
^
3.16 A hermitian operator A has only three normalized eigenfunctions ø 1 , ø 2 , ø 3 ,
with corresponding eigenvalues a 1 1, a 2 2, a 3 3, respectively. For a
particular state ö of the system, there is a 50% chance that a measure of A
produces a 1 and equal chances for either a 2 or a 3.
