1-12.A Problems10
Chapter 1 Classical Waves and the Time-Independent Schr ¨odinger Wave Equation
4. For ψ to be acceptable, it must be single-valued, continuous, nowhere infinite, with a piecewise continuous first derivative. For most situations, we also require ψ to be square-integrable.
5. The wavefunction for a particle in a varying potential oscillates most rapidly where V is low, giving a high T in this region. The low V plus high T equals E. In another region, where V is high, the wavefunction oscillates more slowly, giving a low T , which, with the high V , equals the same E as in the first region.
1-12.A Problems10 1-1. Express A cos(kx) + B sin(kx) + C exp(ikx) + D exp(−ikx) purely in terms of cos(kx) and sin(kx).
1-2. Repeat the standing-wave-in-a-string problem worked out in Section 1-4, but clamp the string at x = +L/2 and −L/2 instead of at 0 and L.
1-3. Find the condition that must be satisfied by α and β in order that ψ (x) = A sin(αx) + B cos(βx) satisfy Eq. (1-20).
1-4. The apparatus sketched in Fig. 1-8 is used with a dish plated with zinc and also with a dish plated with cesium. The wavelengths of the incident light and the corresponding retarding potentials needed to just prevent the photoelectrons from reaching the collecting wire are given in Table P1-4. Plot incident light frequency versus retarding potential for these two metals. Evaluate their work functions (in eV) and the proportionality constant h (in eV s).
TABLE P1-4 Retarding potential (V) λ(Å)
Cs
Zn
6000
0.167
—
3000
2.235
0.435
2000
4.302
2.502
1500
6.369
1.567
1200
8.436
6.636
1-5. Calculate the de Broglie wavelength in nanometers for each of the following: a) An electron that has been accelerated from rest through a potential change of 500 V.
b) A bullet weighing 5 gm and traveling at 400 m s−1.
1-6. Arguing from Eq. (1-7), what is the time needed for a standing wave to go through one complete cycle?
10Hints for a few problems may be found in Appendix 12 and answers for almost all of them appear in Appendix 13.
Section 1-12 Summary1-7. The equation for a standing wave in a string has the form (x, t) = ψ(x) cos(ωt) a) Calculate the time-averaged potential energy (PE) for this motion. [Hint: Use
PE = − F d; F = ma; a = ∂2/∂t2.]
b) Calculate the time-averaged kinetic energy (KE) for this motion. [Hint: Use KE = 1/2mv2 and v = ∂/∂t.]
c) Show that this harmonically vibrating string stores its energy on the average half as kinetic and half as potential energy, and that E(x)avαψ2(x).
1-8. Indicate which of the following functions are “acceptable.” If one is not, give a reason.
a) ψ = x b) ψ = x2 c) ψ = sin x d) ψ = exp(−x) e) ψ = exp(−x2) 1-9. An acceptable function is never infinite. Does this mean that an acceptable function must be square integrable? If you think these are not the same, try to find an example of a function (other than zero) that is never infinite but is not square integrable.
1-10. Explain why the fact that sin(x) = −sin(−x) means that we can restrict Eq. (1-32) to nonnegative n without loss of physical content.
1-11. Which of the following are eigenfunctions for d/dx?
a) x2 b) exp(−3.4x2) c) 37 d) exp(x) e) sin(ax) f) cos(4x) + i sin(4x) 1-12. Calculate the minimum de Broglie wavelength for a photoelectron that is pro duced when light of wavelength 140.0 nm strikes zinc metal. (Workfunction of zinc = 3.63 eV.)
Multiple Choice Questions (Intended to be answered without use of pencil and paper.)
1. A particle satisfying the time-independent Schr¨odinger equation must have a) an eigenfunction that is normalized.
b) a potential energy that is independent of location.
c) a de Broglie wavelength that is independent of location.
Chapter 1 Classical Waves and the Time-Independent Schr ¨odinger Wave Equation d) a total energy that is independent of location.
e) None of the above is a true statement.
2. When one operates with d2/dx2 on the function 6 sin(4x), one finds that a) the function is an eigenfunction with eigenvalue −96.
b) the function is an eigenfunction with eigenvalue 16.
c) the function is an eigenfunction with eigenvalue −16.
d) the function is not an eigenfunction.
e) None of the above is a true statement.
3. Which one of the following concepts did Einstein propose in order to explain the photoelectric effect?
a) A particle of rest mass m and velocity v has an associated wavelength λ given by λ = h/mv.
b) Doubling the intensity of light doubles the energy of each photon.
c) Increasing the wavelength of light increases the energy of each photon.
d) The photoelectron is a particle.
e) None of the above is a concept proposed by Einstein to explain the photoelectric effect.
4. Light of frequency ν strikes a metal and causes photoelectrons to be emitted having maximum kinetic energy of 0.90 hν. From this we can say that a) light of frequency ν/2 will not produce any photoelectrons.
b) light of frequency 2ν will produce photoelectrons having maximum kinetic energy of 1.80 hν.
c) doubling the intensity of light of frequency ν will produce photoelectrons having maximum kinetic energy of 1.80 hν.
d) the work function of the metal is 0.90 hν.
e) None of the above statements is correct.
5. The reason for normalizing a wavefunction ψ is a) to guarantee that ψ is square-integrable.
b) to make ψ*ψ equal to the probability distribution function for the particle.
c) to make ψ an eigenfunction for the Hamiltonian operator.
d) to make ψ satisfy the boundary conditions for the problem.
e) to make ψ display the proper symmetry characteristics.
Reference [1] F. Flam, Making Waves with Interfering Atoms. Physics Today, 921–922 (1991).