Ph2b Quiz - 2. Instructions

Transcription

1 Ph2b Quiz - 2 Instructions 1. Your solutions are due by Monday, February 26th, 2018 at 4pm in the quiz box outside 201 E. Bridge. 2. Late quizzes will not be accepted, except in very special circumstances. 3. The quiz is open textbook (Griffiths), open lecture notes, open section notes, open homework sets and open homework solutions. Please check the collaboration policy on the course website phys002 for more details. 4. Calculators may be used. Computers and smart devices, other than your personal brain are not allowed. 5. There are a total of 4 questions and you have 2 hours (in a single sitting) to work on the quiz. Not all questions are of equal length, so please go through them all before you start and allocate your time appropriately. 6. Please justify your answers and show all work. Time Limit: 2 hours Please write your name, section number, and your TA s name on the front sheet of your submission. Good Luck! 1

2 1 Mix-and-Match - I (5 points) Match the low energy wavefunctions given to the right with the potentials numbered 1 to 5 shown on the left in Figure 1, by writing the appropriate letter for each case. There is a one-to-one correspondence, no wavefunction works twice. Figure 1: Potentials and Wavefunctions for Problem 1. 2

3 2 Mix-and-Match - II (4 points) Given the following potential in Figure 2, match the wavefunctions with their correct energy. Note that not all are valid wavefunctions: there are 6 wavefunctions given and only 4 of them can be matched up (in a one-to-one correspondence) with the energies E 1, E 2, E 3, E 4. Do not worry about normalization of any of the wavefunctions. Figure 2: The Potential and Wavefunctions for Problem 2. 3

4 3 The Life of a Quantum Harmonic Oscillator (10 points) Let us study how a wavefunction evolves in time in the quantum harmonic oscillator. (a.) [1 point] Write down the most general solution Ψ(x, t) to the time-dependent Schödinger equation in terms of the of the ψ n (x) solutions (n = 0, 1, 2,...) of the time-independent Schödinger equation for the harmonic oscillator potential, ( Hψ n (x) = ω n + 1 ) ψ n (x). (1) 2 (b.) [4 points] Compute x for the state Ψ(x, t). You should get a bunch of terms that oscillate in time; at what frequency are these oscillations? What is the average over one time period? Hint: Write x in terms of the raising and lowering operators a ± and then compute x. (c.) [3 points] We can imagine initializing the wavefunction at t = 0 in various ways. For each way listed below (for some complex numbers c n s), state whether x will oscillate or not: 1. Ψ(x, 0) = n even c nψ n (x) 2. Ψ(x, 0) = n odd c nψ n (x) 3. Ψ(x, 0) = n prime c nψ n (x) (d.) [2 points] Now say we measured the energy of the oscillator at some time t = t 1 > 0 and got a measurement value of ( m + 1 2) ω for a fixed positive integer m. What is the state of the oscillator for t > t 1 i.e. what is Ψ(x, t > t 1 )? Does x oscillate in time when the state is Ψ(x, t > t 1 )? 4

5 4 Fun with Commuting Observables! (10 points) In this question, feel free to use results from earlier parts (of this same question) even if you are unable to prove/verify them. Let  and ˆB be a pair of commuting observables, [Â, ˆB] = 0. (2) Prove the following assertions: (a.) [1 point] If Ψ is an eigenvector of Â, then ˆB Ψ is also an eigenvector of  with the same eigenvalue. (b.) [1 point] If ψ 1 and ψ 2 are two eigenvectors of  with different eigenvalues, then ψ 1 ˆB ψ 2 = 0. (c.) [1 point] If  has non-degenerate eigenvalues, then its eigenvectors are also eigenvectors of ˆB. If two operators commute, then their corresponding observables are compatible, that is, both observables can be known to arbitrary precision. Let s consider this for the parity operator P, P ψ(x) = ψ( x), (3) which flips the wavefunction in position. (d.) [1 point] What is P 2 ψ(x)? (e.) [2 points] What are the possible eigenvalues of P? (f.) [2 points] If H = p2 2m + V (x) where V (x) = V ( x), then what is [H, P ]? (g.) [2 points] Show that if ψ(x) is a stationary state of the Hamiltonian H = p2 2m + V (x) where V (x) = V ( x), then so is ψ( x). If the energy of ψ(x) is E, then what is the energy of ψ( x)? 5