Problem Set 1

Problem Set 1#

Problem 1 (20 points equally distributed among parts a through d)#

Recall the energy eigenfunctions for the particle-in-a-box from class have the form

\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi}{L}x\right). \]

  • (a) Are the energy eigenfunctions given by \(\psi_n(x)\) also eigenfunctions of the momentum operator, \(\hat{p} = -i\hbar \frac{d}{dx}\)? It will probably be helpful to justify your answer mathematically and with a sentence or two of explanation.
    Answer:

  • (b) Assuming the length of the box is \(L = 10 \times 10^{-10} \: \text{m}\), what is the momentum expectation value of the ground-state of the particle in a box? You can use the formula below to compute this expectation value, and you may use Wolfram Alpha or other tools to evaluate the final integrals.

\[ \langle p \rangle = \langle \psi_1(x)| \hat{p} | \psi_1(x) \rangle = -i \hbar \int_0^L \psi_1^*(x) \, \frac{d}{dx} \, \psi_1(x) \, dx \]

  • (c) Assuming the length of the box is \(L = 10 \times 10^{-10} \: \text{m}\) and the mass of the electron is \(9.109 \times 10^{-31} \: \text{kg}\), what is the energy expectation value of the ground-state of the particle in a box? You can use the formula below to compute this expectation value. You should not need to use Wolfram Alpha to compute the integrals in this case.

\[ E_1 = \frac{1^2 \pi^2 \hbar^2}{2mL^2} \]

  • (d) For the ground state wavefunction of a particle in a box, the momentum expectation value is zero. But what about the expectation value of the square of the momentum, i.e. \(\langle p^2 \rangle\)? Is it zero or nonzero? Use this expression to explain why or why not:

\[ \langle p^2 \rangle = \langle \psi_1(x)| \hat{p}^2 | \psi_1(x) \rangle = -\hbar^2 \int_0^L \psi_1^*(x) \, \frac{d^2}{dx^2} \, \psi_1(x) \, dx \]

Problem 2 (20 points equally distributed among parts a through d)#

Let us again consider a quantum particle in a box, with box length \(L = 10 \times 10^{-10}\) m.

  • (a) What is the wavelength of the particle in the ground state?

  • (b) What is the wavelength of the particle in the second excited state?

  • (c) What is the momentum of the particle in the second excited state? You can use the relationship \(p = \frac{h}{\lambda}\), where \(h\) is Planck’s constant.

  • (d) What is the kinetic energy of the particle in the second excited state? You may use either the quantum mechanical formula for \(E_n\) or the classical mechanics formula \(E = \frac{p^2}{2m}\).

Problem 3 (20 points equally distributed among parts a through d)#

Let’s examine the time evolution of a quantum particle in a box. Suppose the initial state of a particle is the following linear combination of two eigenstates:

\[ \Psi(x,0) = \frac{1}{\sqrt{2}} \left[ \psi_1(x) + \psi_2(x) \right] \]

  • (a) What is the wavefunction at a later time \(t\)?

  • (b) What is the probability density \(|\Psi(x,t)|^2\)?

  • (c) Plot \(|\Psi(x,t)|^2\) at \(t = 0\) and at some later time \(t = t_1\) such that the wavefunction has noticeably changed. You can use Python or any other tool. Label your axes.

  • (d) Is the probability density \(|\Psi(x,t)|^2\) time-independent or time-dependent? Explain why this happens in terms of the energy eigenfunctions and time evolution.

Problem 4 (20 points equally distributed among parts a through d)#

This problem explores how quantum mechanics explains the color of light emitted by hydrogen atoms. Use the Bohr model of the hydrogen atom and the following energy level equation:

\[ E_n = -\frac{13.6 \, \text{eV}}{n^2}, \quad n = 1, 2, 3, \dots \]

  • (a) What is the energy (in eV) of the photon emitted when an electron transitions from \(n=3\) to \(n=2\)?

  • (b) What is the wavelength (in nm) of this photon? Use \(E = \frac{hc}{\lambda}\).

  • (c) What part of the electromagnetic spectrum does this correspond to (e.g., infrared, visible, ultraviolet)?

  • (d) What color does this correspond to in visible light, and how does that explain what we see when looking at hydrogen’s emission spectrum?

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