10.4 Uncertainty Principle
10.14.Calculate the uncertainty in position, x, of a baseball
having mass 250 g going at 1602 km/hr. Calculate the un-
certainty in position for an electron going at the same speed.
10.15.For an atom of mercury, an electron in the 1sshell has
a velocity of about 58% (0.58) of the speed of light. At such
speeds, relativistic corrections to the behavior of the electron
are necessary. If the mass of the electron at such speeds is
1.23 me(where meis the rest mass of the electron) and the
uncertainty in velocity is 10,000 m/s, what is the uncertainty
in position of this electron?
10.16.How is the Bohr theory of the hydrogen atom incon-
sistent with the uncertainty principle? (In fact, it was this in-
consistency, along with the theory’s limited application to
non-hydrogen-like systems, that limited Bohr’s theory.)
10.17.Though not strictly equivalent, there is a similar un-
certainty relationship between the observables time and energy:
E
t 2 In emission spectroscopy, the width of lines (which gives a
measure of E) in a spectrum can be related to the lifetime
(that is, t) of the excited state. If the width of a spectral line
of a certain electronic transition is 1.00 cm^1 , what is the min-
imum uncertainty in the lifetime of the transition? Watch your
units.
10.5 Probabilities
10.18.For a particle in a state having the normalized wave-
function
(^2) a sin (^) a x
in the range x0 to a, what is the probability that the par-
ticle exists in the following intervals?
(a)x0 to 0.02 a (b)x0.24ato 0.26a
(c)x0.49ato 0.51a (d)x0.74ato 0.76a
(e)x0.98ato 1.00a?
Plot the probabilities versus x. What does your plot illustrate
about the probability?
10.19.A particle on a ring has a wavefunction eim,
where 0 to 2 and mis a constant.
(a)Normalize the wavefunction, where dis d. How does
the normalization constant depend on the constant m?
(b)What is the probability that the particle is in the ring in-
dicated by the angular range 0 to 2 /3? Does this answer
make sense? How does the probability depend on the con-
stant m?
10.20.A particle having mass mis described as having the
(unnormalized) wavefunction k, where kis some con-
stant, when confined to an interval in one dimension, that in-
terval having length a(that is, the interval of interest is x 0
to a). What is the probability that the particle will exist in the
first third of the interval, that is, from x0 to (1/3)a? What
is the probability that the particle will be in the third third of
the box, that is, from x(2/3)ato a?
10.21.Consider the same particle in the same box as in the
previous problem, but the (unnormalized) wavefunction is dif-
ferent. Now, assume kx, where the value of the wave-
function is directly proportional to the distance across the box.
Evaluate the same two probabilities, and comment on the dif-
ferences between the probabilities in this case and the previ-
ous one.
10.6 Normalization
10.22.What are the complex conjugates of the following
wavefunctions? (a) 4 x^3 (b)() ei
(c) 4 3 i
(d)isin^32 x(e)eiEt/
10.23.Normalize the following wavefunctions over the range
indicated. You may have to use the integral table in Appendix 1.
(a)x^2 , x0 to 1
(b)1/x, x5 to 6
(c)cos x, x
/2 to /2
(d)er/a, r0 to , ais a constant, d 4
r^2 dr
(e)er
(^2) /a
, rto , ais a constant. Use dfrom
part d.
10.24.For an unbound (or “free”) particle having mass min
the complete absence of any potential energy (that is, V0),
the acceptable one-dimensional wavefunctions are
Aei(2mE)1/2x/Bei(2mE)1/2x/, where Aand Bare constants and
Eis the energy of the particle. Is this wavefunction normaliz-
able over the interval x? Explain the significance
of your answer.
10.7 The Schrödinger Equation
10.25.Why does the Schrödinger equation have a specific
operator for kinetic energy and only a general expression, V,
for the potential energy?
10.26.Explain the reason that the kinetic energy operator
part of the Schrödinger equation is a derivative whereas the
potential energy operator part of the Schrödinger equation is
simply “multiplication times a function V.”
10.27.Use the Schrödinger equation to evaluate the total en-
ergy of a particle having mass mwhose motion is described
by the constant wavefunction k. Assume V0. Justify
your answer.
10.28.Evaluate the expression for the total energies for a par-
ticle having mass mand a wavefunction 2 sin x, if the
potential energy Vis 0 and if the potential energy Vis 0.5 (as-
sume arbitrary units). What is the difference between the two
eigenvalues for the energy, and does this difference make sense?
10.29.Explain how the Hamiltonian operator is Hermitian.
(See section 10.3 for the limitations of Hermitian operators.)
10.30.Verify that the following wavefunctions are indeed
eigenfunctions of the Schrödinger equation, and determine
their energy eigenvalues.
(a)eiKxwhere V0 and Kis a constant
(b)eiKxwhere Vk, kis some constant potential energy,
and Kis a constant
(c) (^2) a sin (^) a x, where V 0
312 Exercises for Chapter 10