Activity: Representations of the Infinite Square Well

Quantum Fundamentals 2022 (3 years)
• group Small Group Activity schedule 120 min. build Tabletop Whiteboard with markers, Computers with Maple, Voltmeter, Coordinate Axes, A handout for each student description Student handout (PDF)

Representations of the Infinite Square Well

Consider three particles of mass $m$ which are each in an infinite square well potential at $0<x<L$.

The energy eigenstates of the infinite square well are:

$E_n(x) = \sqrt{\frac{2}{L}}\sin{\left(\frac{n \pi x}{L}\right)}$

with energies $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$

The particles are initially in the states, respectively: \begin{eqnarray*} |\psi_a(0)\rangle &=& A \Big[ 2i \left|{E_4}\right\rangle - 3\left|{E_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[ i \sqrt{\frac{8}{L}}\sin{\left(\frac{4\pi x}{L}\right)} - \sqrt{\frac{18}{L}}\sin{\left(\frac{10\pi x}{L}\right)} \right]\\[6pt] \psi_c(x,0) &=& C x(x-L) \end{eqnarray*}

For each particle:

1. Determine the normalization constant.
2. At $t=0$ what is the probability of measuring the energy of the particle to be $\frac{8\pi^2\hbar^2}{mL^2}$?
3. Find state of the particle at a later time $t$.
4. What is the probability of measuring the energy of the particle to be the same value $\frac{8\pi^2\hbar^2}{mL^2}$ at a later time $t$?
5. What is the probability of finding the particle to be in the first half of the well?

• assignment Wavefunctions

assignment Homework

Wavefunctions
Quantum Fundamentals 2022 (2 years)

Consider the following wave functions (over all space - not the infinite square well!):

$\psi_a(x) = A e^{-x^2/3}$

$\psi_b(x) = B \frac{1}{x^2+2}$

$\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)$ (“sech” is the hyperbolic secant function.)

In each case:

1. normalize the wave function,
2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
3. find the probability that the particle is measured to be in the range $0<x<1$.

• assignment Quantum concentration

assignment Homework

Quantum concentration
bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side $L$; the concentration in effect is $n=L^{-3}$. Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature $kT$. (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration $n_0$ thus defined is equal to the quantum concentration $n_Q$ defined by (63): $$n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}$$ within a factor of the order of unity.
• assignment One-dimensional gas

assignment Homework

One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of $N$ particles, each of mass $M$, confined to a one-dimensional line of length $L$. The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature $T$. You may assume that the temperature is high enough that $k_B T$ is much greater than the ground state energy of one particle.
• assignment Spin Fermi Estimate

assignment Homework

Spin Fermi Estimate
Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
• group Time Evolution of a Spin-1/2 System

group Small Group Activity

30 min.

Time Evolution of a Spin-1/2 System
Quantum Fundamentals 2022 (3 years)

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
• assignment Ideal gas in two dimensions

assignment Homework

Ideal gas in two dimensions
Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
1. Find the chemical potential of an ideal monatomic gas in two dimensions, with $N$ atoms confined to a square of area $A=L^2$. The spin is zero.

2. Find an expression for the energy $U$ of the gas.

3. Find an expression for the entropy $\sigma$. The temperature is $kT$.

• face Fermi and Bose gases

face Lecture

120 min.

Fermi and Bose gases
Thermal and Statistical Physics 2020

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
• group Quantum Expectation Values

group Small Group Activity

30 min.

Quantum Expectation Values
Quantum Fundamentals 2022 (3 years)
• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

Sinusoidal basis set
Computational Physics Lab II 2022

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• assignment Unknowns Spin-1/2 Brief

assignment Homework

Unknowns Spin-1/2 Brief
Quantum Fundamentals 2022 (2 years) With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\left|{\psi_3}\right\rangle$ and $\left|{\psi_4}\right\rangle$.
1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?

Learning Outcomes