EE 4395 Special Topics Applied Quantum Mechanics for Electrical Engineers Homework Problems

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1 EE 4395 Special Topics Applied Quantum Mechanics for Electrical Engineers Homework Problems Part II: Electromagnetic Waves 2.1 Use the relativistic formulas for total energy (γmc 2 )and momentum (γmv) to prove the following relation: 2.2 (a) Use the relativistic formulas for total energy (γmc 2 )and momentum (γmv) to prove the following relation: (b) If the mass of a relativistic particle is equal to zero, what expression can be used to determine the energy of the particle? Use the relationship in part (a) to answer this question. (c) If the mass is equal to zero, what is the velocity of the particle? Use the relationships in problems 2.1 and 2.2 part (a) to answer this question. 2.3 Determine the energy of an electron and a photon if each has momentum equal to 1.07 x kgm/s. 2.4 A photon is emitted as a result of an electron transition from a higher energy level to a lower energy level. Indicate the type of photon that is emitted if its energy is equal to (a) 2.49 ev and (b) 1.38 ev. (Hint: Solve for the wavelength and use table 2.1 in the course notes to determine the type of photon). 2.5 Use the de Broglie relation λ=h/p to find the wavelength of electrons with kinetic energy 550 ev. 2.6 (a) Determine the velocity of an electron with wavelength equal to 2 x m. Explain how you determined whether to use classical vs relativistic momentum calculations. (b) What bias is required to accelerate an electron to this velocity? 2.7 Graph y = A sin(wt kx) for a range of x between 0 and 2 m in increments of 0.1 for the following conditions: (a) A = 1 m, f = 1 s -1, λ = 1 m, t = 0 (b) A = 1 m, f = 1 s -1, λ = 1 m, t = 0.5 (c) What is the value for τ for the waves in (a) and (b)? (d) What is the value for t/τ for the waves in (a) and (b)? (e) What happens if you add wave (a) to wave (b) and plot the resulting wave? Use Excel or matlab. [50 points] 2.8 The minimum frequency required to eject photoelectrons from metal X is 5 x s -1. (a) What is the work function of this metal in ev? (b) What type of radiation has this frequency? [20]

2 2.9 The maximum wavelength required to eject photoelectrons from copper is 264 nm. What is the wavelength of photons required to eject photoelectrons with a velocity of 2.5 x 10 7 m/ /s?[30] 2.10 Sodium has a work function of 2.3 ev. (a) What is the wavelength of photons with this energy? [10] (b) If photons with wavelength equal to 400 nm strike sodium, what is the kinetic energy (in ev) and the velocity (m/s) of the photoelectrons being ejected? [25] 2.11 Photoelectrons are ejected from platinum with a velocity of 5 x 10 5 m/s. (a) What is the stopping potential required to eliminate the photoelectric current? (b) If the work function of platinum is 6.4 ev, what is the photon energy required to eject photoelectrons at this velocity? What type of radiation has this energy? [40] 2.12 Determine the slope of the data corresponding to the kinetic energy of photoelectrons as a function of photon frequency as a result of a photoelectric experiment. Use 5 significant figures in your calculation. [20] Frequency,s -1 Kinetic Energy, J E E E E E E E E E E-19 (Source data: A photon with energy equal to 0.7 MeV collides with a free electron resulting in a scattered photon at an angle 100 with the incident photon and an electron moving at some velocity (recoil electron). (a) What is the energy of the scattered photon in ev? (b) What is the kinetic energy of the recoil electron? and (c) What is the velocity of the recoil electron? [30] 2.14 The following eigenfunction is a typical solution of the time-independent Schrödinger equation and satisfies boundary conditions for a particle in a confined space of a certain length. [30] (a) Plot the wave function as a function of x for L = 10 cm and n = 1, 2 and 3 (3 plots on the same graph). (b) On a separate graph, plot ψ 2 as a function of x using the conditions specified in part (a) (3 plots on the same graph). (c) Report your observationss for parts (a) and (b)

3 2.15 Classify the following statements as True or False. [24] In an electron diffraction experiment (like the 2 slit experiment) an electron will always fall in the same spot for the same set of conditions. The probability density is a function of the square of the probability amplitude (ψ). We square the probability amplitude (ψ) in order to get all positive numbers. Using the Schrödinger equation, I can determine the exact position of an electron. The wave function Ψ used in quantum mechanical calculations has no physical meaning and cannot be measured. The number of photons in an electromagnetic wave is proportional to the square of the wave amplitude used to model the wave. If we add up the wave functions in an electron diffraction experiment we can determine the resulting interference pattern by taking the absolute value of the squaring the sum of the wave functions. The lower the value of the ψ 2 for a given location, the higher the probability that a particle is located at that location. Each particle is described by its own wave function. Boundary conditions are required in order to determine the solution to Schrödinger s equation. Eigenvalues represent the quantized energy values based on solutions to the time-dependent Schrödinger equation. In the double slit experiment, if a small particle is replaced with a heavier particle (all other parameters held constant), the distance between the fringes would increase A 2-slit experiment involves electrons that are accelerated with energy equal to 1,000 ev, with the following conditions: Slit separation of 2.5 nm Slit-screen separation of 10 cm (a) Calculate the kinetic energy of the electrons in units of ev. (b) Calculate the velocity of the electrons in m/s. (c) Calculate the wavelength of the electrons in nm. (d) Calculate the distance between the fringes in cm. [40]

4 2.17 In a 2-slit experiment involving electrons, the slit separation is 2 nm and the slit-screen separation is 25 cm. (a) What does the velocity of the electrons need to be in order to produce a fringe separation of cm? (b) What is the acceleration bias required to achieve this electron velocity? [40] 2.18 (a) Calculate A n values for n = 1, 2, 3, 4, 5, 9, 10 if a = λ/2, where 2 sin (b) Calculate the k n values for the same n values if λ = 1 m, where (c) Calculate the corresponding λ values where 2 λ λ 2 Complete the following table in Excel and copy the table into a word file. (This is part of the solution to Workshop VIII). [30 Workshop pts] n A n k n λ n, m The uncertainty in position when measuring the position of a 65 gram object is estimated to be ± 1 µm. [45] (a) What is the uncertainty in momentum corresponding to these measurements? (b) What is the uncertainty in velocity corresponding to these measurements? (c) Determine the kinetic energy of the particle based on the uncertainty in velocity assuming the object is very close to stationary. (d) Determine the distance the object will move in 1 year based on the velocity calculated in part (c). (e) Is the uncertainty principle important in this case? Explain.

5 2.20 The velocity of a bullet is measured at 100 m/s, ±0.001 m/s. If the mass of the bullet is 20 g, (a) determine the uncertainty in the bullet s position. (b) Is the uncertainty in position significant in this case? [25] 2.21 The position of a proton is estimated with accuracy of ±6 fm which is the approximate radius of a large nucleus. (a) Determine the uncertainty in momentum, (b) the uncertainty in the velocity of the proton, (c) the ratio of Δv/c. (d) Is the uncertainty in velocity significant at this scale? The mass of a proton is equal to x kg. [40] References 1 D.A.B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge University Press, New York, A. Beiser, Concepts of Modern Physics, McGraw Hill, New York, F.W. Sears, Zemansky, Young, Addison Wesley Education Publishers, 1991.

Physics 111 Homework Solutions Week #9 - Friday

Physics 111 Homework Solutions Week #9 - Friday Tuesday, March 1, 2011 Chapter 24 Questions 246 The Compton shift in wavelength for the proton and the electron are given by Δλ p = h ( 1 cosφ) and Δλ e