Solution to Participation Questions

1. Position Measurements
   Q: A measurement of the position of a particle is made, and it is found to 
      be at point P. Where was the position of the particle just before the 
      measurement was made? 
   A: We will discuss this more later in the course, but for now I will
      just say the "orthodox" answer that the particle did not have a
      well defined position prior to the measurement is correct. 
      (see pages 3-5)

2. Class Pictures
   Q: In the Resources section I have posted a photo I took of the class 
      with each student numbered. I would like to use this to learn the 
      names of the students. Please indicate your number here: ____. 
      If you are not in the picture just enter zero.

3. Commutator
   Q: In class we showed that the commutator of x and p was [x,p] = i hbar . 
      What is the commutator [x^2,p]?
   A: The momentum operator is (-i hbar d/dx).  Consider an arbitrary f(x):

      [x^2,p]f = x^2 p f - p x^2 f
               = - i hbar ( x^2 f' - 2x f - x^2 f') 
               = 2i hbar f,

      where f' is df/dx.  Since this is true for any f, [x^2,p] = 2i hbar f.
   Q: I am interested in attending a workshop/presentation on using Matlab 
      outside of class?
   A: Everyone who responded received credit for this question.  Most people
      were interested in a session outside of class.

4. Harmonic oscillator
   Q: What is the expectation value of p^2 for the first excited state of 
      the harmonic oscillator, psi _1?
   A: For the harmonic oscillator, p = i sqrt(hbar m omega/2) (a+ - a-).
      This means that p^2 is (hbar m omega/2)(a+a- + a-a+ - a+^2 - a-^2).
      Because of the orthogonality of the eigenstates, the a+^2 and a-^2
      terms do not contribute to the expectation value of p^2.  
      a+a- psi_1 = psi_1 and a-a+ psi_1 = 2 psi_1 so the expectation value
      of p^2 is (3/2) hbar m omega .

5. Dimensionalysis
   Q: If the potential energy of as oscillator was (1/2) K x^4 instead of 
      (1/2) k x^2, what would the characteristic length scale of the 
      oscillator be in terms of K, hbar, and m?
   A: The kinetic energy term of the Schrodinger equation has units:
     
      [hbar ^2/m] L^-2 [psi],
 
      where [ ] means "units of".  The potential energy term has units:

      [K] L^4 [psi] .

      Setting these two equal to each other we obtain,

      L^6 = hbar^2/(m K) or L = (hbar ^2/(m K))^1/6 .

6. Probability current
   Q: What is the probability current for the wave function B exp(- rho x)?
   A: The probabilty current is zero because of the wave function is real
      and psi = psi^*.  (The coefficients B and B* can be pulled out in front.)
      On the Sept. 23 lecture we showed the B exp(rho x) + B' exp(-rho x)
      can carry probability current, but only if both B and B' are non-zero.

7. Gaming
   Q: Recently it was announced that people playing a computer game called 
      FoldIt solved the structure of a retrovirus enzyme that had stumped 
      researchers for a decade.  There have also been various attempts to 
      "gamify" education and teaching. 
         How - if at all - do you think computer games could be used in 
      teaching advanced physics such as this quantum mechanics class?
   A: The concensus seems to be that while gaming will not replace
      traditional teaching, there might be a place for games to help in
      visualization and developing intuition.  As such, gaming might be
      placed in a larger set of tools that includes simulations and 
      visualization software.  Several students also mentioned the role of 
      competition in gaming, and the ability to accumulate rewards over time.

8. Delta function potential
   Q: Near x = 0, a solution to the time independent Schrodinger equation has 
      the form:
         psi(x < 0) = A + Bx
         psi(x > 0) = A + Cx .
      What is the prefactor of the delta function potential at the origin?
   A: hbar^2 (C-B)/(2mA)