HW 5, due Friday 3PM. We need more peer-to peer discussion. Please join the discussion (e.g., regarding problem 5).

This assignment looks difficult to me. I would suggest starting very soon. Also, I encourage you to use comments to clarify what is asked in these problems. Especially for something like problem 5, this is a good forum to discuss what one should actually calculate and develop a consensus on that. For problem 5 b) I would recommend factoring out an overall \(e^{-i E_1 t/\hbar}\) factor. That may make it a little easier to see what that graph looks like.


Problem order was modified slightly on Feb 11. (Problem 4)
Commentary (on 5) added Feb 12.

1. If we go back to separation of variables, then we see that states have a time dependence that multiplies their spatial wave-function. So far we have been concentrating on just the spatial wave function. Going forward, however, we will begin to include time dependence where appropriate. Figuring when and how time dependence effects the wave-function, the "probability density" and expectation values is an important part of this endeavor.
a) For an electron in the ground state of a harmonic oscillator, what is the full time-dependent wave-function?
b) Plot the wave-function as a function of x: at t=0, \(t=\pi \hbar/(4E_1)\), and at times when the time dependent part of the wave-function is -i and -1 respectively (4 plots).  What are those times?

2. Consider an electron in a 1D harmonic oscillator potential. Suppose the electron is in a state that is an equal mix of the ground state and 1st excited state. At t=0:
\( \psi  =  \frac {1} {\sqrt {2}} \psi_{1} +  \frac {1} {\sqrt {2}} \psi_{2}  \).
a) Calculate \(\langle x \rangle\).
b) Is this state time dependent in a way that might effect \(\langle x \rangle\)? Is \(\langle x \rangle\) time dependent?
c) Is  \(\langle x \rangle\) real, imaginary or both in this case?
d) Calculate and plot \(\langle x\rangle\) as a function of ___. (Fill in the missing word.)
e) (extra credit) Calculate  \(\langle p\rangle\) and graph it.

3. Consider an electron in a hydrogen atom. The electron is in a state that is an equal mix of the ground state and a 1st excited state \(\psi_{21x}\).  That is, at t=0, \( \psi  =  \frac {1} {\sqrt {2}} \psi_{100} +  \frac {1} {\sqrt {2}} \psi_{21x}  \).
a) Calculate \( \langle x \rangle\). Are there any important time dependences that go with this state. Does it effect your result for the expectation value?
b) Earlier in the quarter, we combined states to make hybrids (such as \(\frac {-1} {\sqrt {3}} \psi_{200} +  \frac {\sqrt{2}} {\sqrt {3}} \psi_{21x} \)), however, we never worried about time dependence of  any expectation values. Were we ignoring time dependence that was really there?  Is this combination state different in some fundamental way from the combination states, such as sp^2 states, that we examined earlier in the quarter?
 
4.  I moved this to lower down so people don't get bogged down in it.

5. [For this problem, part d is the key part. a, b and c lead you to d).] Consider an electron in a 1D harmonic oscillator potential. Suppose the electron is in a state that is an equal mix of the ground state and 2nd excited state. That is, at t=0, \( \psi  =  \frac {1} {\sqrt {2}} \psi_{1} +  \frac {1} {\sqrt {2}} \psi_{3}  \).
a) Calculate \(\langle x \rangle\).
b) Graph \(|\Psi (x,t)|^2\) as a function of x for t=0 and \(t = \hbar \pi /(E_3 - E_1) \). [ I would recommend factoring out an overall \(e^{-i E_1 t/\hbar}\) factor. That may make it a little easier to see what that graph looks like.]
c) Look at those graphs. How are they different?  What expectation value do you imagine might be time dependent?
d) Calculate that.

6. [For this problem also, part d) is the most important part. c) is the concept, d) is the implementation.] Consider a single electron in a double square well potential. (Aside: the double dot qubit, which is a "popular" approach to creating quantum computers of the future, is essentially a double square well based system.) Suppose both square wells are 16 eV deep and have the same width L. Let s (for separation) be the distance between these two identical wells (near edge to near edge).
a) Sketch the square well potential.
b) Sketch the ground state and 1st excited state for a single electron in this system.
Suppose that after hours of calculation you know everything about the one-electron states of this 2 well system.*
c) How would you model an electron starting out at t=0 mostly on one side of the well?  How would you calculate how long it takes an electron to mostly get to the other side
d) How long does it take for the electron to get from one side to the other?
e) Is this tunneling? What is tunneling?

* Everything includes the energies of the states and the characteristics of the spatial wave functions. Like in problem 5 from HW 2, except for the double well where the wave-function in the spatial region between the two wells is of the form:
\(cosh(x/a_n)\) for symmetric states,
\(sinh(x/a_n)\) for anti-symmetric states.
For the ground state and the 1st excited state, the form of the wave-function in the right-hand well, I think could be written as:
\(\psi_n (x) = B_n cos(k_n (x-(L+s)/2)\)
and outside the well I think we could write:
\(\psi_n (x) = C_n e^{-x/a_n}\)
[this detail may be distracting. What do you really need to know about the states?]

4. Consider an electron in a hydrogen atom. The electron is in a state that is an equal mix of the ground state and a 1st excited state \(\psi_{211}\).  That is, at t=0, \( \psi  =  \frac {1} {\sqrt {2}} \psi_{100} +  \frac {1} {\sqrt {2}} \psi_{211}  \).
a) What is  \( \psi_{211}  \)?
b) Calculate \( \langle x \rangle\).
c) Calculate \( \langle y \rangle\).
d)  Draw the vector \( \langle \vec{r}  \rangle = (\langle x  \rangle, \langle y \rangle, \langle z \rangle ) \).  What plane is it in? Describe its time dependence.

7. (extra credit) For an electron in the same state as in problem 2:
a) calculate the expectation value of the momentum as a function of time.
b) Plot \(\langle x \rangle\) and \(\langle p \rangle\) vs time. What is their relationship?
(Should this be extra credit? How difficult are those derivatives? How many terms do you get?)

8. extra work/extra credit: For an electron in the state given in problem 5:
a) graph and discuss the relationship between K.E. and P.E. (as a function of time). Do whatever calculations are interesting or necessary to aid and inform this discussion.