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MENG 213 2021 Short-answer questions (minimal to no calculations; 4 questions total)

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MENG 213 2021

1. Short-answer questions (minimal to no calculations; 4 questions total). Each question in this section is worth 4 points.
   1. Wavefunction basics. Suppose we have two different wavefunctions ψ₁(x) and ψ₂(x) that differ only by a complex phase: ψ₂(x)= e^iθ(x)ψ₁(x) . Here, θ(x) is some real-valued function of x. What could we measure (in general) to try and determine whether we are in state 1 or 2? When is it impossible to distinguish these two states? You don’t need to do any detailed calculations here, just fully explain your reasoning.
   2. Free particle basics. In class, we discussed the evolution of the probability density P(x,t) of a Gaussian wavepacket (Gaussian probability blob) for a free particle (no forces). Suppose this wavepacket starts at t = 0 with an average position x₀ and an average momentum p₀. In a total of five sentences or less, explain
      1. What aspects of this evolution matched the expectations of classical physics for how a free particle would move?
      2. What aspects of this evolution did not match our classical expectations?
   3. Double slit. In our discussion of the quantum double slit experiment, we saw that quantum particles produced a pattern very reminiscent of classical wave interference. It is tempting to conclude that we should thus treat particles like electrons exactly like we treat classical waves. Explain in three sentences or less why we can’t do this.
   4. Infinite square well. In lecture, we discussed the energy eigenstates of a particle confined to an infinite square well potential in one dimension. If we treated this system classically (i.e. at time t put a classical particle with some energy E in the box), and then measured x or p at a random time, we could also describe the system using probabilities. Let’s call these probabilities P_cl(x) and P_cl(p) . In five sentences of less, discuss:
      1. One feature of the quantum solutions that match the probabilities we’d get classically.
      2. One feature of these quantum solution that is completely different from the classical probabilities.
2. Short Calculation Questions (5 questions total)
   1. Basic manipulations of a wavefunction (8 points). Consider an electron in one dimension which is described at t = 0 by a wavefunction

       if −L/2 ≤ x < L/2

ψ

0            otherwise

where ψ₀ and L are constants. Do not worry about the fact that this function is discontinuous at x = L/2 and x =−L/2.

1. 1. 1. Find the constant ψ₀ by insisting the wavefunction is normalized
      2. What is the probability that if you try to measure the particle between x = 0 and x = L/3, you do not find it?
      3. Calculate the average value of x in this state ψ(x) as well as the position uncertainty ?x.
      4. Use the uncertainty principle, derive a minimum possible value on the momentum uncertainty ?p in the state ψ(x).
   2. Tunneling basics (6 points) .
      1. Suppose we are given a potential energy V(x) and told that our particle (mass m) has an energy E. According to classical physics, there are forbidden regions where the particle is not allowed to be. Where are these regions? Explain your reasoning in 2-3 sentences at most.
      2. Consider now a constant potential energy V(x)= V₀ where V₀ > 0. Suppose we want to find a state ψ(x) for our particle that has a definite energy E that is smaller than V₀. Write the general form of the solution to the time-independent Schrodinger equation for this problem.
      3. Explain why even though we can mathematically solve the equation, we can never get a physically reasonable solution, and hence there are no energy eigenstates with E <V₀ for this system.
   3. Momentum and energy operators (6 points) .
      1. A particle in one dimension is described (at t = 0) by a wavefunction ψ(x) that is purely real and that is properly normalized. Show that the average momentum in this state must be zero. (Hint: use integration by parts).
      2. A particle in one dimension is described by a wavefunction ψ₂(x). One finds that in this state, the particle has a definite, completely certain value of p² (momentum squared), but an uncertain value of momentum p. Write the most general form for the state ψ₂(x) (without worrying about normalization).
   4. Intereference (6 points) . A free particle (no potential energy!) moving in one dimension is described by a wavefunction ψ(x,t) that has the form

ψ(x,t)=ψ₁(x,t)+ψ₂(x,t)

where ψ₁(x,t) describes a particle with a definite momentum p₁, and ψ₂(x,t) describes a particle with a definite momentum p₂.

1. 1. At t = 0 , one finds that the probability density P(x,0) associated with ψ(x,0) oscillates as a function of x. One finds that P(x,0) is periodic in x with a period d, i.e. P(x+d,0)= P(x,0) . Use this to find p₁− p₂. Explain your reasoning.
   2. One also finds the the probability density at x = 0, P(0,t) oscillates as a funciton of time. Find the period of these oscillations in terms of d.

5. Time-indep Schrodinger equation (6 points). A particle in 1D having a mass m is described by a wavefunction:

ψ(x,t = 0)= Ae−x2/L2

This wavefunction corresponds to a definite energy E = 0. Using this information, what is the potential energy V(x)? Express your answer using some or all of the constants m,A,L.