Mastering Quantum Mechanics: Challenge Exam Guide
School
ICS CANADA**We aren't endorsed by this school
Course
FINANCE CORPORATE
Subject
Chemistry
Date
Dec 10, 2024
Pages
4
Uploaded by ProfBoarPerson1248
Name: Quantum Mechanics Challenge ExamDate: December 5, 2022Time: 2 hoursTotal Score: 100Instructions:1. Answer all questions in the space provided or on a separate sheet of paper if needed.2. Show all your work for calculation problems for full credit.3. Circle your answers for multiple-choice questions.4. You may use a calculator for any calculations.5. The exam is closed-book, closed-notes, and no electronic devices are allowed.6. Please write your name and student ID number on your exam paper.1. Which of the following statements is correct about the de Broglie wavelength of an electron?A. It is inversely proportional to its momentum.B. It is proportional to its mass.C. It is proportional to its energy.D. It is independent of its velocity.2. What is the Schrödinger equation for a particle in an infinite square well?A. -ħ^2/2m(d^2/dx^2) + V(x)= EψψψB. -ħ^2/2m(d^2/dx^2) = EψψC. ħ^2/2m(d^2/dx^2) = EψψD. -ħ^2/2m(d^2/dx^2) + V(x)= 0ψψ3. Consider a one-dimensional harmonic oscillator with mass m and spring constant k. Whatis the energy of the nth excited state?A. (n + 1/2)ħωB. nħωC. (n - 1/2)ħωD. n(1/2)ħω4. The wave function of a particle in a box is (x) = A sin(nx/L). What is the probability ψπdensity at the boundaries of the box?A. ZeroB. MaximumC. Half of the maximumD. Quarter of the maximum
5. What is the uncertainty principle for position and momentum expressed in terms of their standard deviations ()?σA. xp ≥ ħ/4σ σπB. xp ≥ ħ/2σ σπC. xp ≥ ħ/σ σπD. xp ≥ ħσ σ6. A hydrogen atom is in the 3p state. What is the degeneracy of this state?A. 3B. 6C. 9D. 127. The ground state energy of a hydrogen atom is -13.6 eV. What is the energy of the first excited state?A. -3.4 eVB. -1.51 eVC. -3.4 eV/4D. -1.51 eV/48. A photon of wavelength is incident on a metal surface. The emitted photoelectron has λkinetic energy K. What is the threshold frequency for this metal?A. K/ħB. K/ħλC. ħ/KλD. ħK/λ9. A particle is in a superposition of two energy eigenstates with energy eigenvalues E1 and E2. What is the average energy of the particle in this superposition?A. (E1 + E2)/2B. √(E1E2)C. (E1 - E2)/2D. √((E1 - E2)/2)10. A quantum system is described by the wave function (x) = e^(-|x|/a). What is the ψprobability current density at x = a?A. 1/(2aħ)B. 1/aħC. a/(2ħ)D. a/ħ11. The energy spectrum of a quantum system is given by E_n = n^2ħ. Calculate the ωaverage energy of the system in the state with quantum number n = 2.
Answer: (5/3)ħω12. A particle of mass m is in a one-dimensional box of length L. The ground state energy is E_1. Calculate the energy of the second excited state.Answer: E_2 = (9/4)E_113. A quantum system is in a superposition of two states with probabilities p and 1-p. Calculate the expectation value of an operator O for this system.Answer: O = pO_1 + (1-p)O_2⟨ ⟩14. A quantum system undergoes a transition from an initial state with energy E_i to a final state with energy E_f, emitting a photon of frequency . What is the energy of the emitted νphoton?Answer: h= |E_i - E_f|ν15. A hydrogen atom is in the 2s state. Calculate the expectation value of the angular momentum squared (L^2) for this state.Answer: L^2 = 0⟨⟩16. A particle is described by the wave function (x) = Ce^(-|x|/a). Calculate the probability ψcurrent density J(x) for this wave function.Answer: J(x) = ħ/(2m)(1/a^2 - 2x/a^3)e^(-|x|/a)17. A quantum system is in a superposition of two states with energies E_1 and E_2. The system interacts with a time-dependent potential V(t). Calculate the time-evolution of the system.Answer: (t) = e^(-iEt/ħ)(0)ψψ18. A quantum system is described by the Hamiltonian H = p^2/2m + V(x). Calculate the energy levels of the system in the presence of a harmonic potential V(x) = (1/2)kx^2.Answer: E_n = (n + 1/2)ħ, where = √(k/m)ωω19. A quantum system undergoes a transition from an initial state with energy E_i to a final state with energy E_f, emitting a photon of wavelength . Calculate the frequency of the λemitted photon.Answer: = |E_i - E_f|/hν20. A particle is described by the wave function (x) = Ae^(-x^2/2a^2). Calculate the ψexpectation value of the position x for this wave function.⟨ ⟩Answer: x = 0⟨ ⟩21. A quantum system is in a superposition of two states with probabilities p and 1-p. Calculate the variance of the energy for this system.
Answer: _E^2 = (1-p)p(E_2 - E_1)^2σ22. A particle of mass m is in a one-dimensional box of length L. The ground state energy is E_1. Calculate the energy of the third excited state.Answer: E_3 = (25/16)E_123. A quantum system is described by the wave function (x) = Ae^(-|x|/a). Calculate the ψprobability current density J(x) for this wave function.Answer: J(x) = ħ/(2m)(1/a^2 - 2x/a^3)e^(-|x|/a)24. A quantum system undergoes a transition from an initial state with energy E_i to a final state with energy E_f, emitting a photon of wavelength . Calculate the frequency of the λemitted photon.Answer: = |E_i - E_f|/hν25. A hydrogen atom is in the 2p state. Calculate the expectation value of the angular momentum squared (L^2) for this state.Answer: L^2 = ħ^2(l(l + 1)), where l = 1⟨⟩26. A quantum system is in a superposition of two states with energies E_1 and E_2. The system interacts with a time-dependent potential V(t). Calculate the time-evolution of the system.Answer: (t) = e^(-iEt/ħ)(0)ψψ27. A quantum system is described by the Hamiltonian H = p^2/2m + V(x). Calculate the energy levels of the system in the presence of a harmonic potential V(x) = (1/2)kx^2.Answer: E_n = (n + 1/2)ħ, where = √(k/m)ωω28. A particle is described by the wave function (x) = Ce^(-|x|/a). Calculate the probability ψcurrent density J(x) for this wave function.Answer: J(x) = ħ/(2m)(1/a^2 - 2x/a^3)e^(-|x|/a)29. A hydrogen atom is in the 2s state. Calculate the expectation value of the angular momentum squared (L^2) for this state.Answer: L^2 = 0⟨⟩30. A particle is described by the wave function (x) = Ae^(-x^2/2a^2). Calculate the ψexpectation value of the position x for this wave function.⟨ ⟩Answer: x = 0⟨ ⟩