HW 1a: quantum bound states.

(Quantum review homework)
*For problems 1.4 and 2.1, let's all use the harmonic oscillator wave-functions posted by Nicolas Blanc in the comments below.
For this homework assignment our main focus is on results and relationships. Integrals can be done via wolfram alpha (see video above); there is no point in doing them by hand or showing work. Please comment and ask questions below. (Extra credit for finding errors, ambiguities, etc..)

Section 1: quantum length scales
1.1 Consider an electron in a 1D quantum potential \(U(x) =\frac{1}{2} k x^2\) where k= 100 \(eV/nm^2\).  Suppose that the electron is in a state of the form \(\psi(x) = A e^{-x^2/(2a^2)}\).
a) Find A. Calculate the expectation value of \(x\). Calculate the expectation value of \(x^2\).
b) What is the expectation value of the U ?
c) Calculate the expectation value of the kinetic energy, T . What is the relationship between T and the expectation value of \(p^2\) ?
d) Graph the expectation values of T and U as a function of a. What is each one for a = 1 nm? What about for a = 2 nm?
e) What value of a gives you the lowest value for the total energy, T + U ? How does this value of a depend on each member of the "quantum triumvirate" m, k and \(\hbar\)?

1.2 For the results for problem 1.1, calculate the product of the expectation value of \(p^2\) times the expectation value of \(x^2\). On what does this depend? Discuss.

1.3 a) For an electron in the ground state of a hydrogen atom, express the quantum length scale known as the Bohr radius in terms of m, e and \(\hbar\).
b) For an electron in the ground state of a 1D harmonic oscillator, express the quantum length scale, "a", in terms of m, k and \(\hbar\).
c) discuss.
 1.3e (see below)

1.4  a) For the 3 lowest energy states of a 1D harmonic oscillator, write the normalized energy eigenstate wave-functions. (Please use the HO quantum length scale to simplify both the normalization factor and the exponential.)
[Please post your answer here. That way we can all converge on the same thing.]*
(see comments!)
b) Plot each of these wave functions as a function of x.
c) Using wolfram alpha to save time, calculate the expectation value of \(x^2\) for each of these states. I am thinking that your answer will be in the form of a number times \(a^2\).
d) Take the square root of that to get something with units of length. Plot that (3 points) as a function of the quantum number. What might this tell you about the nature of these states?

 Section 2: (time dependence)
2.1 Consider an electron in a harmonic oscillator potential (the same potential as in problem 1.1).  Suppose the electron is in a normalized state that is an equal mix of the ground state and 1st-excited state?
a) Write down a mathematical expression for that state including normalization and time dependence.
[Please post your answer here if you like. That may help us resolve any possible normalization or interpretation issues.]
b) For this state, calculate the expectation value of x.
c) Is the expectation value of x zero or non-zero. Does it depend on time or not?

Section 3: (square wells & multiple well systems) (This part is important for this class.)
3.1 Consider an infinite square well that is 0.1 nm wide.
a) What is the energy of the ground state?
b) What is the energy of the 1st excited state?
c) Is this kinetic or potential energy?
[For energy calculations like the ones in this problem I would suggest multiplying by \(c^2\) and using \(m_e c^2 = .51 \times 10^6 \: eV\) and \(\hbar c = 197 eV*nm\). (are those correct?)  Joules and kg don't work well in the quantum realm. eV and nm are much better.] (Also, I think, \(\hbar^2/m_e = 0.076 eV*nm^2\))

3.2  a) sketch the ground state for a finite square well that is 0.1 nm wide (and about 200 eV deep). How is it different from the infinite square well ground state (same width well)?
b) sketch the 1st excited state for a finite square well that is 0.1 nm wide and about 200 eV deep.
3.2 e (see below)

3.3 Consider a potential that involves two square wells each 0.1 nm wide and with a separation between them of about 0.05 nm. (Suppose the potential is zero between the wells and outside them, and that it is about -200 eV within each well.)
a) sketch the 4 lowest energy states for this two-well system.
b) Do the same thing for a system of 4 identical square wells, except in this case sketch the 8 lowest states.
c) think about and discuss the organizing principle for these states. Please feel free to post your thoughts on this below.
d) discuss the organization of the energies of these states. Might there be any clustering? How might that arrange itself? [Post here on clustering.]
3.3e (see below)

Section 4: Displaced functions. (There was an error in this problem. It was corrected on Friday.)
4.1 Suppose a= 0.5 nm.
a) Plot \(e^{-((x)^2/a^2}\)
b) Plot  \(e^{-((x-1 nm)^2/a^2}\) as a function of x.
c) Plot the function \(e^{-(x+1 nm)^2/a^2} + e^{-x^2/a^2} + e^{-(x-1 nm)^2/a^2}\) as a function of x.
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1.3e (hydrogen atom)  For a wave-function of the form \(\psi(r) = (\pi a^3)^{-1/2} e^{-r/a}\) calculate U and T (expectation values) and find the value of a that provides the lowest total energy (T + U).  You can post your results here if you like or send them to me by email.

3.2e (finite square well) Using some approximate numerical methods, or any approach you like, find the energies and wave-function of the bound states for a square well 0.1 nm wide and 200 eV deep. Assume the potential is centered at x=0 and that the ground state has the form \(A_1 cos (k_1 x) \) inside the well and \( B_1 e^{-(x-.05 nm)/a_1} \) outside the well on the right.  Note that from the Wave Equation (Schrodinger eqn) \(\hbar^2/2ma_1^2 = - E_1\) (which is positive), and \(\hbar^2k_1^2/2m = E_1-V_o\) where \(E_1\) is the ground state energy and \(V_o = - 200 eV\). (Please post results here. It may be useful for the class if someone does this problem because then we can use it as a model atom from which to construct a simple crystalline solid and it will provide us with a particular point of reference.)

3.3e (two square wells) (This is a difficult but not impossible problem that may be pretty interesting and relevant to crystal states.) Calculate the expectation values of the kinetic energy and potential energy for the ground state and first excited state of a two well potential. Call the center-to-center distance c and let's try c = 0.15 nm, L= 0.1 nm and let the well depth be -35 eV. [Suggestions: put the origin between the wells and then assume that in the space between the wells the wave function has the form \(C_1 cosh (x/a_1)\) for the ground state and  \(C_2 sinh (x/a_2)\) for the first excited state. That \(a_n\) parameter will have the same value as the one you use outside the wells. Why? Inside the wells, the form of the wave function is \(\pm A_n cos(k_n(x-c/2)\).