HW 4. with solutions added at end

This assignment includes problems illustrating the nature of tunneling and exploring why the wave-function squared is known as a "probability density". 
1. Consider a potential V(x) that is infinite for x less than 0 nm and \(V(x) = kx^2/2\) for x greater than zero with k= 1 \(eV/nm^2\).  (Sketch V(x)
a) Find the normalized ground state wave function. (If it satisfies the TISE and BCs and has no nodes it is the GS.) What is its value (include units) at x = 1nm?
b) Calculate the size of an electron in the GS of this potential. (Note: this includes a calculation of x and a calculation of x^2, right?) Size is \(\Delta x\).
c) Calculate the EVals of the KE and PE respectively. What is the ratio of PE to KE?

2. Consider a potential V(x) that is infinite for x less than 0 nm, -0.3 eV for x between 0 and 1 nm, and 0 eV for x greater that 1 nm. (sketch V(x))
a) Find the normalized ground state wave function. (If it satisfies the TISE and BCs and has no nodes it is the GS.) What is its value (include units) at x = 1nm?
b) Calculate the size of an electron in the GS of this potential.
c) Calculate the EVals of the KE and PE respectively. What is the ratio of PE to KE? (a negative number in this case.)

3. Consider delta function potential (attractive), that is, \(V(x) = -\alpha \delta(x)\) with \(\alpha= 8 \: eV*nm\). Explain the units of \(\alpha\).
a) Find the normalized ground state wave function. (If it satisfies the TISE and BCs and has no nodes it is the GS.) What is its value (include units) at x = 1nm?
b) Calculate the size of an electron in the GS of this potential. (Note: this includes a calculation of x and a calculation of x^2, right?) Size is \(\Delta x\).
c) Calculate the EVals of the KE and PE respectively. What is the ratio of PE to KE? (a negative number in this case.)

4. Evaluate the "uncertainty related" product \(\Delta x \Delta p\) for each of the systems above (3 separate calculations). You can use the KE multiplied by 2m to get the EVal of p^2, so you don't have to do another integral for that. You can also show that the EVal of p is zero, so \(\Delta p\) is the square root of the EVal of KE multiplied by 2m. Does that make sense? (multiply by 2m before you take the square root, right?).

5. Suppose an electron is in the ground state of a FSW that is 0.3 eV deep and 2nm wide. Suppose a 2nd well suddenly forms 1 nm away. How long does it take the electron to tunnel to the 2nd well?
(This is what I would call a "high-concept" problem. An electron is in one well at t=0. Can it get to a well nearby by traveling through a "forbidden region"? If so, how long does it take? In a ideal world it might be nice to just think about problem 5 for a week or so, but instead I am going to outline a method for you here. The method outlined here in problem 6 uses energy eigenstates, and their time dependence. The tunneling takes place via an interference-related phenomenology. There is another method involving matrices and "hopping" that looks a lot more like tunneling which we could try to cover as well at some point.)

6. Specifically, let's say that for t<0, V(x) = -0.3 eV between x= 0.5 nm and x= 2.5 nm and zero everywhere else (sketch it).
Then for  t>0,  V(x) changes and becomes: V(x) = -0.3 eV between x=0.5 nm and 2.5 nm, and also between x= -2.5 nm and -0.5 nm and zero everywhere else (sketch this too).) Let's use \(\psi_1=C cosh(bx)\) between the wells (why?) and \(\psi_1 = B e^{-b(x-2.5 nm)}\) outside the well to the right (x > 2.5 nm), along with an appropriately centered and specified cosinusoidal function in the well with a coefficient of A.
a) What are the "appropriately centered" sinusoidal function we should use inside each well for the ground state?
b) Why is b the same parameter in the center as it is outside the wells? Why do we use cosh for the GS? Sketch a graph of your best guess as to the nature of the ground state. Sketch a graph of your best guess as to the nature of the 1st excited state. How are these wave functions relevant to fining a tunneling time?
c) Determine the k, b and the energy E for the ground state and first excited state.** What did you use instead of cosh for the 1st excited state? Why? Does your 1st excited state have exactly one node?
d) What are the normalization parameters for the ground state and 1st excited state?**
e) Do the same thing for the 2nd and 3rd excited states. Do these states have exactly 2 and 3 nodes, respectively? Why do these states tend to come in pairs?
f) You can use these states to calculate how long it takes for the electron to tunnel from one well to the other well.

7. Suppose that an electron is in the ground state of a harmonic oscillator potential with k=.076 eV/nm^2 which suddenly collapses at t=0 leaving only \(V(x) = -\alpha \delta(x)\) with alpha = 8 eV as the potential function for t > 0.
a) What is the probability that the electron remains bound (trapped) in the ground state of the delta function potential?
(See the hint below for how to approach this problem using an expansion involving energy eigenstates. Feel free to approximate any difficult integral you encounter. You may be able to do that if the size of the delta function bound state is much much smaller than the size of the intitial wavefunction. I would suggest making this as simple as possible to capture the essence of what this problem can teach us. )

8. Suppose that an electron is in the ground state of a harmonic oscillator potential with k=.076 eV/nm^2 which suddenly collapses at t=0 leaving only \(V(x) = -\alpha \delta(x-1 nm)\) with alpha = 8 eV as the potential function for t > 0.
a) What is the probability that the electron remains bound (trapped) in the bound state of the delta function potential?
b) How is this probability different from that from problem 7? Would it be accurate to say that the probability of an electron being captured in the delta function ground state is proportional to the value the probability density (at the location of the delta function) at t=0?

9. Consider a finite square well 0.3 eV deep and 2nm wide. Suppose an electron is in the mixed state given by:
\(\Psi(x,t) = \frac{1}{\sqrt{2}}[\Psi_1(x,t) -\Psi_2(x,t)] \)
a) At what time will the EVal of x have a maximum value? (just one time is fine)
b) At that time, what fraction of the PD integral lies outside the well to the right? At that time, what fraction of the PD integral lies outside the well to the left?

10. Same as problem 9 (same questions and well), but suppose instead that the state is:
\(\Psi(x,t) = \frac{1}{\sqrt{2}}[\Psi_1(x,t) + i \Psi_2(x,t)] \)
a) same as 9
b) same
c) What makes the time different?

hint for 7 & 8. Imagine writing the state for t > 0 as a sum or superposition*:
\(\Sigma_{n=1}^{\infty} c_n \Psi_n (x,t)\).
This is like a fourier series and the coefficients are given by:
\(c_n=\int_{-\infty}^\infty \Psi(x,0) \psi_n(x)dx\)

So for the ground state of the delta function, which is its only bound (localized) state, the relevant coefficient is \(c_1\). You can calculate:
\(c_1=\int_{-\infty}^\infty \Psi(x,0) \psi_1(x)dx\)
to get results for problems 7 and 8.
(Perhaps it is more nuanced for the non-local states of this system, but I think this will work for \(c_1\). Feel free to comment here.)

** There is a separate post where we can work on and discuss this interesting and challenging problem.
Solutions.
part 1
https://drive.google.com/file/d/12kRFc4N4GZTi-uV0vq1h87rxIPEQeKts/view?usp=sharing

part 2