## Activity: Hydrogen Probabilities in Matrix Notation

Central Forces 2023 (2 years)

For the state $\left|{\Psi}\right\rangle = \sqrt{\frac{7}{10}} |2, 1, 0\rangle + \sqrt{\frac{1}{10}} |3, 2, 1\rangle + i\sqrt{\frac{2}{10}} |3, 1, 1\rangle$

Convert the state to matrix notation. Then, staying in matrix notation, calculate

1. $\mathcal{P}_{L_z=\hbar}$;
2. $\langle E\rangle$

• assignment Quantum Particle in a 2-D Box

assignment Homework

##### Quantum Particle in a 2-D Box
Central Forces 2023 (3 years) You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length $L$ are $\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}$. If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)
1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length $L_x$ in the $x$-direction and length $L_y$ in the $y$-direction.
2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. $$\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)$$ Using your expressions from part (a) above, write out all the terms in this sum out to $n=3$, $m=3$. Arrange the terms, conventionally, in terms of increasing energy.

You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

4. Find a formula for the $c_{nm}$s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.
• assignment Bottle in a Bottle 2

assignment Homework

##### Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy 2021 (2 years)

Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was $p=106$ Pa and its temperature $T=300$ K. Approximate the helium gas as an ideal gas of equations of state $pV=Nk_BT$ and $U=\frac32 Nk_BT$.

1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

2. Compute the integral $\int \frac{{\mathit{\unicode{273}}} Q}{T}$ and the change of entropy $\Delta S$ between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

• keyboard Position operator

keyboard Computational Activity

120 min.

##### Position operator
Computational Physics Lab II 2022

Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

assignment Homework

Energy and Entropy 2021 (2 years)

The isothermal compressibility is defined as $$K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ $K_T$ is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as $$K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}$$ and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that $$\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}$$ Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• assignment Frequency

assignment Homework

##### Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2022 (2 years) Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ where $c$ is real and positive. Let the initial state of the system be $\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle$, where $\left|{m_1}\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? This frequency is the Bohr frequency.
• group Quantum Measurement Play

group Small Group Activity

30 min.

##### Quantum Measurement Play
Quantum Fundamentals 2022 (2 years)

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
• group Spherical Harmonics

group Small Group Activity

5 min.

##### Spherical Harmonics
Central Forces 2023 (3 years)
• assignment Visualization of Wave Functions on a Ring

assignment Homework

##### Visualization of Wave Functions on a Ring
Central Forces 2023 (3 years) Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

Learning Outcomes