Problem Set #1 Answers
Note: Relevant Chapters in Mortimer are Chapters 14 and 15.
For the following three problems, please circle the correct answer.
1. The Compton Effect was first observed by studying the scattering of X-rays by a graphite target. Although the incident X-rays were monochromatic, the scattered X-rays contained an additional component of longer wavelength.
2. If the frequency of incident light is above the threshold frequency, then as the intensity of light increases, the kinetic energy of ejected electrons remains constant and the number of electrons increases.
3. If the frequency of incident light is above the threshold frequency, then as the frequency of light increases, the kinetic energy of ejected electrons increases and the number of electrons remains constant.
4. What is the effect of incident light striking a metal surface if the frequency of the light is below the threshold frequency for this metal?
No effect. Each individual photon lacks enough energy to remove an electron from the metal.
5. Define in a maximum of 20 words for each
(a) Correspondence Principle: Quantum mechanics approaches classical mechanics in the limit of large quantum numbers.
(b) Compton Effect: The elastic scattering of light by electrons. In the collision of an electron with a photon, the electron gains energy, while the photon loses energy and emerges with a lower frequency. Furthermore, the angles of scattering must be explained using the law of conservation of momentum, with the momentum of the photon given by .
(c) Photoelectric Effect: Involves shining light (of intensity I and frequency ) on a metal surface to remove electrons with kinetic energy (Ek) and flux (F, number of electrons emitted). The Ek increases linearly with , while F is unaffected by . Also F increases linearly with I, while Ek is unaffected by I.
6. The work function for metallic cesium is 3.43 ´ 10-19 J. Calculate the kinetic energy and the speed of the electrons ejected by light of 300 nm wavelength. [Hint: Planck's constant, h,is 6.626 ´ 10-34 J s; the speed of light in a vacuum, c, is 3.00 ´ 108 m/s; the mass of an electron, me, is 9.109 ´ 10-31 kg]
Note: W = the work function of the metal
7. The peak of the Sun's emission occurs at about 480 nm. Estimate the temperature on its surface. Hint: The Wien displacement law reduces to , where the constant
Bohr Atom: Spectral Lines
8. Calculate the wavelength of light emitted when an electron falls from the n = 100 to the n = 99 orbit of the hydrogen atom. [Hint: Rydberg's constant, R, is 1.097 ´ 107 m-1]
9. Electrons are accelerated by a 1000 V potential drop. Calculate the de Broglie wavelength. Also calculate the wavelength of the X-rays that would be produced when these electrons strike a solid. [Hint: The electron charge is 1.602 ´ 10-19 C; the mass of an electron, me, is 9.109 ´ 10-31 kg; Planck's constant, h,is 6.626 ´ 10-34 J s.]
Well-Behaved Wave Functions
10. List the characteristics of a well-behaved wave function.
11. Which of the following functions are well-behaved? For those functions that are not well-behaved, explain why not.
12. A particle can move only along the x-axis and has a wave function . Give the expression for
For any normalized , what is the probability that the particle is between and ? [Hint: this question has a numerical answer.]
For any normalized , the probability that the particle is between and is .
Harmonic Oscillators: Hermite Polynomials
13. Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33 ´ 10-26 kg and force constant 155 N/m. [Hint: Planck's constant, h,is 6.626 ´ 10-34 J s.]
At the zero-point energy, v = 0. Therefore, we obtain the following:
14. For the harmonic oscillator, substitute the wave function for the ground state into the Schrodinger equation and derive the expression for the ground state energy (also known as the zero-point energy). Hint: The wave function for the ground state is where . The expression for the ground state energy is where is the frequency.
This can be rewritten in the more familiar form or .
15. In the vibrational motion of HI, the iodine atom essentially remains stationary because of its large mass. Assuming that the hydrogen atom undergoes harmonic motion and that the force constant k is 317 N/m, what is the fundamental vibration frequency . [Hint: The mass of a proton is 1.67 ´ 10-27 kg.]