10.1.State the postulates of quantum mechanics introduced
throughout the chapter in your own words.
10.2 Wavefunctions
10.2.What are four requirements for any acceptable wave-
function?
10.3.State whether the following functions are acceptable
wavefunctions over the range given. If they are not, explain
why not.
(a)F(x) x^2 1, 0 x 10
(b)F(x) x1, x
(c)f(x) tan(x),
x
(d)ex
2
, x
(e)ex
2
, x
(f)F(x) sin 4x,
x
(g)xy^2 , x 0
(h)The function that looks like this:
(i)The function that looks like this:
10.3 Observables and Operators
10.4.What are the operations in the following expressions?
(a) 2 3
(b) 4 5
(c)ln x^2
(d)sin (3x3)
(e)eE/kT
(f) (^) ddx (^) 4 x^3 7 x (^7) x (^)
10.5.Evaluate the operations in parts a, b, and f in the pre-
vious problem.
f(x)
x
f(x)
x
10.6.The following operators and functions are defined:
Aˆ (^) x ( ) ˆBsin ( ) Cˆ (^) (^1 ) ˆD 10 ( )
p 4 x^3 2 x^2 q0.5 r 45 xy^2 s 23
x
Evaluate: (a)Aˆp(b)Cˆq(c)Bˆs(d)ˆDq(e)Aˆ(Cˆr) (f)Aˆ(ˆDq)
10.7.Multiple operators can act on a function. If Pˆx acts on
the coordinate xto yield x, Pˆy acts on the coordinate yto
yield y, and Pˆz acts on the coordinate zto yield z, evalu-
ate the following expressions written in terms of 3-D Cartesian
coordinates:
(a)Pˆx(4, 5, 6) (b)PˆyPˆz(0, 4, 1)
(c)PˆxˆPx(5, 0, 0) (d)PˆyPˆx( , /2, 0)
(e)Does PˆxPˆyequal PˆyPˆxfor any set of coordinates? Why or
why not?
10.8.Indicate which of the following expressions yield eigen-
value equations, and indicate the eigenvalue.
(a) (^) dd (^) xsin
2 x (b) (^) ddx
2
2 sin
2
x
(c)i (^) (^) xsin
2 x (d)i (^) x eimx, where mis a constant
(e) (^) x (ex^2 ) (f) 2 m
2
(^) ddx
2
2 0.5sin
2
3
x
10.9.Why is multiplying a function by a constant considered
an eigenvalue equation?
10.10.Relating to the question above, some texts consider
multiplying a function by zero to be an eigenvalue equation.
Why might this be considered a problematic definition?
10.11.Using the original definition of the momentum oper-
ator and the classical form of kinetic energy, derive the one-
dimensional kinetic energy operator
ˆK
2 m
2
d
d
x
2
(^2)
10.12.Under what conditions would the operator described
as multiplication by i(the square root of 1) be considered a
Hermitian operator?
10.13.A particle on a ring has a wavefunction
(^) ^12 eim
where equals 0 to 2 and mis a constant. Evaluate the an-
gular momentum pof the particle if
pˆi
How does the angular momentum depend on the constant m?
Exercises for Chapter 10 311
EXERCISES FOR CHAPTER 10