HW 5. Due Monday May 6. With solutions.

HW5 soln link
https://drive.google.com/file/d/1asYW5eZ73eUyvEtGpD9KUsHxAWnfuaWc/view?usp=sharing

I would recommend starting on this soon and working on it some each day so you have time to ruminate and ask questions as needed. I think that is what a lot of people are doing and having really good success with that approach. I don't believe that you can do this all at one time and really learn a lot. Just my 2 cents. 

Additionally, I think it will really help your success if you ask questions when you are stuck. You can ask questions here or by email. Actually, sometimes posting your question here and asking me by email is good. Engagement and discussion are encouraged for this assignment and all assignments.

1.  a) Show that \(\psi_k = e^{ikx}\) is an energy eigenstate of the Schrodinger equation (TISE) for a free electron: \( - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = E \psi(x) \) What is its energy eigenvalue, \(E_k\)?
 b) Also, show that \(\psi_k = e^{ikx} e^{-iE_kt/\hbar}\) is solves the Schrodinger wave equation for a free electron: \( - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = i\hbar \frac {\partial \psi(x)}{\partial t} \)

2. Consider a state of the form,
 $$\Psi(x,t) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\phi(k) \, e^{ikx} \,e^{-iE_kt/\hbar} dk$$
This is similar to non-stationary states we have looked at before, e.g., in the previous homework, however, instead of involving just two energy eigenstates, it involves an infinite number of energy eigenstates.  Explain the nature and meaning of each of the 3 terms in this integrand, and why the integral is over k.

3. Let's consider the specific case of: \(\phi(k)= A_a e^{-(k-k_o)^2a^2}\).
a) Determine \(A_a\) using a normalization constraint* on \(\phi(k)\) and show that, at t=0, integrating the equation above with this specific \(\phi(k)\) leads to:
\(\Psi(x,0) =\frac{1}{(a\sqrt{2 \pi})^{1/2}} e^{-x^2/(4a^2)} e^{ik_o x}\).
b) Describe this wave-function. Check its normalization.
c) Sketch a graph of the real part of this wave-function for a=1 nm and \(k_o = 6.28 \: nm^{-1}\).
d) Sketch a graph of the imaginary part of this wave-function for a=1 nm and \(k_o = 6.28 \: nm^{-1}\).
e) Sketch a graph of the probability density.
f) I would suggest a domain of about -4 to 4 nm for your plots.

4. Kinetic energy. This problem is triple credit because it is very important.
a) Calculate the expectation value of the kinetic energy at t=0 for an electron in this state \(\Psi(x,0) =\frac{1}{(a\sqrt{2 \pi})^{1/2}} e^{-x^2/(4a^2)} e^{ik_o x}\).
b) Discuss the character and nature of the different parts of the kinetic energy using terms like velocity, momentum, confinement and .... (You could back to this question after doing the next problem.)

5.  Time dependence. To understand the nature of the kinetic energy, it is helpful to examine time dependence of an electron in this state. Perhaps we can see how fast the electron is moving. Using the same integral form of \(\Psi(x,t)\) and the same \(\phi (k)\). 
a) Show that integrating over k leads to:
 \(\Psi(x,t) =\frac{1}{(a\sqrt{2 \pi} (1+it/t_c))^{1/2}} e^{-(x-2 k_o a^2 t/t_c)^2/(4a^2(1+t^2/t_c^2)}e^{ik_o x/(1+t^2/t_c^2)} e^{ix^2t/t_c/(4a^2(1+t^2/t_c^2))} e^{-i(k_o a)^2t/t_c/(1+t^2/t_c^2)}\)
where  \(t_c=(\frac {\hbar} {2ma^2})^{-1}\) is a characteristic time scale for this wave-packet state.
b) Describe this wave-function or, if you prefer, you can describe \(|{\Psi (x,t)}|^2\).
For for a=1 nm and \(k_o = 6.28 \: nm^{-1}\):
c) Plot the probability density for this electron at t=0, at \(t=t_c\)  and at \(t= 2 t_c\).  (3 different graphs).  What is \(t_c\) in femto-seconds?
d) With what speed is the electron wave-packet is traveling?
extra credit: e) Guess and then calculate the value of the expectation value of x as a function of time. What is the value of the expectation value of x at t= 100 femto-seconds?
f) How does the "size" of the electron seem to be evolving with time?

6. Consider an electron, one electron, in the wave packet state:  \(\Psi(x,t) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k)\, e^{ikx} \,e^{-iE_kt/\hbar} dk\),
where \(\phi(k)= \frac{\sqrt{a}}{(\pi/2)^{1/4}}  e^{-k^2a^2}\).
a) Calculate \(\Psi(x,0)\). Is it normalized correctly?
b) Graph  \(\Psi(x,0)\). What is its value at x=0?
c) Calculate the expectation value of the kinetic energy for this electron at t=0.
d) How does this kinetic energy EVal compare with that of the electron from problems 4-7?
e) extra credit: how does the size of this electron evolve as a function of time?

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Some terminology for scattering:  This is an informal, non-technical version of some terminology.
free state, free electron state, which is usually short for free-electron energy-eigenstate:
This refers to an energy eigenstate of the potential V(x) =0 which has the form \(e^{ikx}\)
unbound state is kind of like a free state, except that V(x) is not exactly zero everywhere.
bound state: A state the diminishes exponentially at large x and is normalizable, like the energy-eigenstates of the HO or finite square well.
Any energy eigenstate should be classifiable as either bound or unbound. Free state lies within the unbound category.
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7. Scattering. Consider an electron and an attractive delta-function potential, \(V(x)=-\alpha \delta (x)\). One can use boundary conditions and the TISE to construct incoming and outgoing waves in the region \(x<0\), and an outgoing "transmitted" wave in the region \(x>0\). (These names only really make sense in the context of wave-packet constructions, however, it is a useful mathematical exercise to construct an unbound state with these 3 parts. It may help contextualize this formalism when we learn to understand localization and movement of a free electron.) This single-electron psuedo-state is an energy eigenstate, but is not normalizable so the coefficients in front of each of the three terms can't be called normalization parameters like we did for bound states. Let's call them pre-factors and set the prefactor of the incoming wave equal to 1. (This is an arbitrary convention, but I imagine that it will be useful.) In this nomenclature, the state of a single electron which extends through all space can be written as:\( e^{ikx} + B e^{-ikx} \:\: x<0\)
\( C e^{ikx} \:\: x>0\)
a) Find B and C as a function of the strength of the delta-function and k.
b) Calculate the reflection and transmission coefficients, R and T.
c) Graph R & T as a function of k. Over what range of k is T less than 0.5? Over what range of k is R greater than 0.5?
d) Over what range of k is T greater than 0.5? Over what range of k is R less than 0.5?
e) At what value of k is R=T=0.5?
f) For \(\alpha= 76\: meV*nm\), what value of k in \(nm^{-1}\) leads to T=0.5? What is 1/k? How does this particular value of 1/k compare with the length scale associated with the bound state of this attractive delta-function potential?
f) T approaches one at very high energy. Show that T can be viewed as crossing over from a low energy region in which T is less than 1/2, to a high energy region where T is greater than 1/2 . What would you pick as the crossover energy? How does that compare with the energy of the ground state of this potential?
g) Now let's consider a positive delta function. Do all the same things and see if they are different or the same. This might not take much extra work.
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8. optional. If you are really into scattering, it might be interesting to do the same things as problem 10 for a positive or negative positive square well.I think there may be "resonances" where T=1 at particular k values. You'll have to define a notation with more terms...