Midterm practice problems

[This is a "live post", meaning i will continue to edit both this top part and add or modify problems.]

The other thing I would mention, is that there are a lot of posts on this blog and it might be valuable for you to take the time to review them all and try to determine: which ones you understand, which ones are most important, and which ones are likely to be relevant for the midterm. Feel free to post questions, a summary, or related stuff here.

Also, on your "card" for the midterm, in addition to the Schroedinger wave eq., p as a derivative, a and a+ in terms as x and p, x in terms of a and a+, and p in terms of a and a+ (and as a derivative wi respt to x (ok, i said that twice), eig states: for inf sq well, 4 eig states for 1D HO, and the ground state for delta function, any thing else?, (and the boundary condition for the delta function) etc., please also include hbar in eV-seconds just in case there is a problem with actual numbers to work out.
I probably left some things out. This list is not comprehensive --just what popped into my head when I was wanting to mention that about hbar. Oh yeah. Length scales for each potential. (or inverse length scales).

Make sure you read all the problems and do as many as you can. You can ask questions.

Practice problems:
1. A particle of mass m (pomm) is in the ground state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? As much detail as you can... what depends on what?...
Perhaps start with a grand statement regarding: what is the central issue of this problem, then get into details and how things depend on other things and so on.

2. A particle of mass m (pomm) is in the 1st-excited state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? same comments as 1.

3. Consider a particle of mass m (pomm) in a 1D harmonic oscillator potential. Suppose that the particle is in a superposition state (mixed state) that is an equal combination of the mth and nth (energy) eigenstates. Calculate the expectation values of: x, x^2, p, p^2, V, T... (anything else?).

4. (More specific). For a pomm in a 1D HO, suppose state is an equal mix of |0> and |1> .
(all states are normalized). a) Calculate the expectation values of x and x^2 and thereby obtain delta x. (i hope this isn't too difficult.) b) Graph delta x (as a function of what?). Discuss, if it is interesting.

5. Suppose a pomm is initially (for t less than 0) in the ground state of a harmonic oscillator potential v=(1/2) k x^2 , and that at t=0 the potential suddenly morphs into a 2-delta function potential with two attractive delta functions of identical strength, one at x = +d/2, the other at x = -d/2 . Discuss what happens, and especially describe what you would find regarding the nature of that state function in the general vicinity of the origin (from like -3d to +3d or so) a long time later!
You may include a discussion of dependencies on strength, separation,...

b) DO the same thing for the case where one delta function is at x=0 and one is at x=+d. Does that make a difference. What might you find to be dramatically different, if anything?

Saturday afternoon additions:

6) Suppose that for t less than zero, an electron is in the ground state of an attractive delta function potential (adp) located at x=0, and that at t=0 two more adp's appear in addition to the first one), one at x= 2/k, the other at x=-2/k.
a) (quick calculation of expectation value at t=0+) Sketch the state function and V(x) at t=0+, and calculate the expectation value of the potential energy of the electron (at t=0+). How does it compare with the value of the expectation value of the potential energy of the electron before t=0?
b) (though question and essay) Describe what happens as a function of time after that. (You may focus on describing the time dependence of the "probability density".)

7. Consider a potential V(x) consisting of 2 identical attractive delta functions. When they are far apart there are two bound states, right? Show that the 2nd bound state ceases to exisit at a specific value of the ratio of the length scales k^-1 and d, where d is the separtation between the two delta functions and k^-1 is the length scale associated with the g.s. of a single delta function potential. (Is this true?) What is that ratio??
b) Describe in words several ways of looking at what happens as you approach this point from the part of parameter space in which there is an odd bound state. (to E, to k ...) (what is happening to the state???

8. (is phase important?)
Suppose you had one ensemble of sq well systems with a pomm in the state (psi_0+psi_1 )/sqrt(2), and another in the state (psi_0 + i psi_1 )/sqrt(2) . Could you distinguish these two cases via energy measurement? explain. how could you distinguish?

9, (phase) For an electron in a infinite square well, discuss the similarities and differences between the state: sqrt(2/L) Sin(2 pi x/L), (the 1st excited state), and the state which is the absolute value of that. Could you distinguish these by energy measurement (of separate ensembles of each)? Discuss that? Are the energy expectation values the same or different? Discuss other similarities or differences and end with a cogent paragraph, imagining that you ar the teacher of 139a, that explains the meaning and significance of phase in QM.
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10. In 1D QM, which is easier: to be given the potential and asked to find the ground state, or to be given the ground state and asked to find the potential?

oh, i just remembered two more problems (closely related to each other)

11. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite harmonic oscillator (1D).

12. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite square well (1D).
(Is it clear what you are expected to do? Can anyone suggest a better way to phrase these questions?)

Do we need questions in other categories for balance? What are we leaving out?