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Physics for Scientists and Engineers, 6e
Chapter 41 - Quantum Mechanics
Consider the wave function for the free particle,
Equation 41.3. At what value of x is the particle
most likely to be found at a given time?
1
1.
at x = 0
2.
at small nonzero values
of x
3.
at large values of x
4.
anywhere along the x
axis
2
3
4
5
25% 25% 25% 25%
1
2
3
4
The probability density for this wave function
is |ψ|2 = ψ*ψ = (Ae–ikx)(Aeikx) = A2, which is
independent of x. Consequently, the particle
is equally likely to be found at any value of x.
This is consistent with the uncertainty
principle—if the wavelength is known
precisely (based on a specific value of k in
Equation 41.3), we have no knowledge of
the position of the particle.
A particle is in a box of length L. Suddenly, the
length of the box is increased to 2L. What happens
to the energy levels shown in the figure?
1.
1
Nothing—they are
unaffected.
2.
They move farther apart.
3.
They move closer
together.
2
3
4
5
33%
1
33%
2
33%
3
According to Equation 41.12, if L is increased, all
quantized energies become smaller. Thus, the
energy levels move closer together. As L
becomes macroscopic, the energy levels are so
close together that we do not observe the
quantized behavior.
Which of the following will exhibit quantized energy
levels?
1
1.
an atom in a crystal
2.
an electron and a proton
in a hydrogen atom
3.
a proton in the nucleus of
a heavy atom
4.
all of the above
5.
none of the above
2
3
4
5
20% 20% 20% 20% 20%
1
2
3
4
5
The particles in all three parts 1, 2, and 3 are part of
a bound system.
Consider an electron, a proton, and an alpha particle
(a helium nucleus), each trapped separately in
identical infinite square wells. Which particle
corresponds to the highest zero-point energy?
1
1.
the electron
2.
the proton
3.
the alpha particle
4.
The zero-point energy is
the same in all three
cases.
2
3
4
5
25% 25% 25% 25%
1
2
3
4
In Equation 41.15, we set n = 1 for the zeropoint energy and see that the energy is
inversely proportional to the particle mass.
Consider the three particles in question 5 again.
Which particle has the longest wavelength when the
system is in the ground state?
1
1.
the electron
2.
the proton
3.
the alpha particle
4.
All three particles have
the same wavelength.
2
3
4
5
25% 25% 25% 25%
1
2
3
4
The wavelength is determined by the length L
of the well.
The wavelengths of the wave functions in Figure 41.8 are
longer than those in Figure 41.4 because the wave function
spreads out into the classically forbidden region. For an infinite
and a finite square well of the same length L, the quantized
energies of the particle in a finite well are
1
1.
the same as those for a
particle in an infinite well
2.
higher than those for a
particle in an infinite well
3.
lower than those for a
particle in an infinite well
4.
impossible to determine
2
3
4
5
25% 25% 25% 25%
1
2
3
4
The longer wavelength results in a smaller value
of the momentum from the de Broglie
relationship. If the momentum of the particle is
decreased, the energy also decreases.
For a particle undergoing simple harmonic motion in
the n = 0 state, the most probable value of x for the
particle according to quantum mechanics is
1
1.
x=0
2.
x = ±A
3.
All values of x are equally
likely.
2
3
4
5
33%
1
33%
2
33%
3
The ground state represents the largest
deviation from classical behavior. Classically,
the most probable values of x are x = ±A
because the particle is moving the slowest
near these points. As the top graph in Figure
41.15 shows, quantum mechanics predicts
that the maximum probability density in the
ground state is at x = 0