Questions 10 and 11 (1st part)#

Problem 10 uses the following figure, which is not rendering for some people

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Problem 11 uses the following detailed problem description.

Modern displays often use quantum dot technology, where one (or more) electrons are confined to a region within a material. For an electron confined to a rectangular region in two dimensions, it is reasonable to approximate its motion perpendicular to the rectangle with harmonic confinement. The time-independent Schrödinger equation, eigenfunctions, and eigenvalues for a harmonically confined electron are, in atomic units:

\[ \left(-\tfrac{1}{2}\tfrac{d^2}{dz^2} + \tfrac{1}{2}\kappa z^2\right) \psi_k(z) = E_k \psi_k(z) \]

where the eigenenergies are

\[ E_k = \sqrt{\kappa}\left(k+\tfrac{1}{2}\right) \qquad \qquad k=0,1,2,\ldots \]

and the eigenfunctions are given in terms of the Hermite polynomials, \(H_k(z)\), as:

\[ \psi_k^{(\text{harm. osc.})}(z) =\frac{1}{2^k k!}\sqrt[4]{\frac{\kappa}{\pi}} e^{-\sqrt{\kappa}z^2/2}H_k\left(\sqrt[4]{\kappa}z\right) \]

The Hamiltonian for an electron confined to a rectangular region, in atomic units, is then:

\[ \hat{H} = -\frac{1}{2} \frac{d^2}{dx^2} -\frac{1}{2} \frac{d^2}{dy^2} -\frac{1}{2} \frac{d^2}{dz^2} + V_{a_x}(x) + V_{a_y}(y) + \frac{1}{2} \kappa z^2 \]

where

\[\begin{split} V_a(x) = \begin{cases} +\infty & x\leq 0\\ 0 & 0\lt x \lt a\\ +\infty & a \leq x \end{cases} \end{split}\]

11. ✍️🖩 What is the wavelength that corresponds to the lowest-energy excitation when \(a_x = 16\), \(a_y = 9\), and \(k_z = 4\)? Report your answer in nm. Hint: You may find it useful to recall that the speed of light is 137.036 in atomic units.

Questions 4 and 5 (3rd part)#

The figure didn’t render properly for everyone. Here is what you need.

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