U(x) Finite Well. E Re ψ(x) Classically forbidden

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1 Final Exam Physics 2130 Modern Physics Tuesday December 18, 2001 Point distribution: All questions are worth points 8 points. Answers should be bubbled onto the answer sheet. 1. At what common energy E do the wavelengths of electrons and photons dier by a factor of 2? (Hint: For example, let r = e =, solve for E in terms of r and then set r = 2.) a. 0.2 MeV. b. 0.6 MeV. c. 1.1 MeV. d. 4.2 MeV. 2. What is the dimension of a matter wavefunction (r;t) describing an electron in 3D? (L is length, V is velocity, M is mass.) (Hint: Probability is dimensionless.) a. dimensionless. b. L. c. L,3=2 d. V 2 L=M. 3. For a traveling wave on a string, let displacement be given by y(x; t) =A sin(kx,!t). What is the velocity of propagation of the wave? a.!=k in the +x direction. b.!=k in the,x direction. c. A! cos(kx,!t) in the +x direction. d. A! cos(kx,!t) in the,x direction. 4. In quantum mechanics, can a particle ever be exactly at rest if it is conned in a nite region? a. Yes, there is nothing to keep a particle from a good rest. b. No, velocity is not well dened in quantum mechanics. c. No, the probabilistic nature of the wavefunction guarantees that v>0. d. No, it would violate the Heisenberg Uncertainty Principle. 1

2 5. A string is clamped at both ends that are at x = 0 and x = 30 cm. It oscillates with an amplitude of 2 cm and a frequency of 40 Hz. What's the displacement y(x; t) in cm of its fundamental mode? a. 2 sin x cos 80t 30 b. 2 sin x cos 80t. 30 c. 2 sin( x 30, 80t)+sin( x t). d. 2 sin( x 20, 40t) + sin( x t) 6. What are the lowest three energies for an electron in a nite 1D box of width a =0:2 nm? a. 3eV,6eV,9 ev. b. 3eV,12 ev, 27 ev. c. 9.4 ev, 18.8 ev, 28.2 ev. d. 9.4 ev, 37.5 ev, 84.3 ev. 7. Which of the following functions is a solution to the dierential equation 00 (x)+k 2 (x) =0 where k and! are real numbers? a. e kx b. sin(k 2 x, wt) c. e ikx d. e,kx 8. Which of the plots in Figure 1 most closely resembles the probability density of the second excited state (n = 3) of a particle in a rigid box of width a? (a) (b) 0 a 0 a (c) (d) 0 a 0 a Figure 1: 2

3 9. Consider the vector model of angular momentum, ~L, for ` =2as shown in Figure 2. What is the minimum possible angle between L z and L? z θ L Figure 2: a b c. 30. d How many possible orientations of ~L can there be for ` =1? a. 1. b. 2. c. 3. d Consider the equation for the time-independent Schrodinger equation in 3D: 1 d sin d sin d! " # + `(` +1), m2 =0: d sin Which of the following is a solution to this equation if ` = m =0? a. sin. b. 2, 1. c. cos. d. any constant. 12. How many quantum states (n; `; m) are there with the same energy for n =1? (This is asking for the \degeneracy" of n = 1 quantum states.) a. 1. b. 3. c. 5. d. 7. 3

4 13. Generalizing, many quantum states (n; `; m) are there with the same energy for arbitrary n? a. n b. n(n, 1). c. 2n, 1. d. n Consider a nucleon that oscillates with simple harmonic motion with an amplitude of 4 fm and a peak kinetic energy of 20 MeV. About how many g 0 s of acceleration does this nucleon experience? (Recall that g = 9.8 m/s 2, is aunitof acceleration.) a g. b g. c g. d g. 15. For a particle subjected to a nite well, such as that shown in Figure 3, what condition is most responsible for determining the shape of the wavefunction for x>a. a. The wavefunction is continuous. b. The rst-derivative of the wavefunction is continuous. c. The wave function is bounded. d. The wavefunction is nowhere negative. U(x) Finite Well E Re ψ(x) Classically forbidden 0 a Figure 3: 4

5 The following gure (Fig. 4) is for #16 and #17. (a) P N (c) P N (b) P N (d) insufficient information Figure 4: 16. Which pair of functions (if any) in Figure 4 best represents the nuclear potentials experienced by the protons (P) and neutrons (N) in alightnucleus? 17. Which pair of functions (if any) in Figure 4 best represents the nuclear potentials experienced by the protons (P) and neutrons (N) in aheavy nucleus? 18. The exponential decay law for radioactive decay has the solution N(t) =N 0 exp(,rt), where N 0 is the original number of nuclei and r is the decay rate. Which of the following is also true (where t 1=2 is the half-life of the substance)? a. N = N 0 e t=t 1=2. b. N = N 0 e,t=t 1=2. c. N = N 0 2,rt. d. N = N 0 2,t=t 1=2. For questions #19 - #21 the following are the options. a. ssion. b.,. c.. d. fusion. 19. What kind of decay is 27Co! Ni + e, +? 20. What kind of decay is Pa! Ra + 4 2He + 1 1H? 21. What kind of reaction is 2 1H + 2 1H! 3 1H + 1 1H? 5

6 22. Which of the following will ssion when bombarded by a slow neutron? (1-238 U, U, Pu,4-228 Pa) a. 1, 2. b. 2, 3. c. 1, 2, 3. d. 2, Which of the following reactions are important in nuclear energy production in the Sun right now? a. 1, 4. b. 3. c. 1, 5, 6. d. 1, 2, H + 1 H! 2 H + e H + 2 H! 3 H + 1 H: 3 1 H + 12 C! 13 N + : 4 2 H + 3 H! 4 He + n: 5 2 H + 1 H! 3 He + : 6 3 He + 3 He! 4 He + 1 H + 1 H: 24. Which of the following are allowed nuclear decays? (Hint: Ignore the kinetic energy that might be released in the reaction.) a. 1. b. 2. c. 3. d p + 29Cu! 63 n Ni 2 n Zn! 2 1H Cu 3 n + 26Fe! Mn 4 n N! p C 6

7 25. What is the probability of the transmission of a particle of mass m across (from left to right) a potential barrier shown in the following gure? 2 K + h /2m U(x) K x 1 x Figure 5: This potential can be represented as follows 1, x x 1 2 if 0 x x 1 U(x) = K + h2 2m U(x) = 0 otherwise where K is the kinetic energy of the particle. a. e,x 1. b. e,azk,1=2 +b(zr) 1=2. c. 1/16. d. e,x2 1 =2h2. 7

(c) (d) insufficient information

(c) (d) insufficient information Final Exam Pysics 130 Monday December 16, 00 Point distribution: Te multiple coice questions (1-5) are wort 5 points eac and answers sould be bubbled onto te answer seet. Questions 6-8 are long-answer