## Activity: 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.

## Prerequisite Knowledge

• Spin 1/2 systems
• Familiarity with how to calculate measurement probabilities
• Solutions to the Schrodinger equation for a time independent Hamiltonian
• Dirac notation

## Activity: Introduction

Little introduction is needed, although you may want to review how to find a time-evolved state for a time-independent Hamiltonian.

Students work in groups to solve for the time dependence of two quantum particles under the influence of a Hamiltonian. Students find the time dependence of the particles' states and some measurement probabilities.

Time Evolution of a Spin-1/2 System

Two particles are under the influence of an interaction with a Hamiltonian that is proportional to $\hat{S}_{z}$:

$\hat{H} = \omega_0 \hat{S}_{z}$

At $t=0$:

1. one is in the state $\vert + \rangle _{x}$.
2. the other particle is in the state $\vert + \rangle$

For each particle:

1. What values of energy could you measure?
2. What are the energy eigenstates?
3. What state is each particle in at a later time t?

4. What are the probabilities for each energy measurement?
5. What is the probability that you would measure $S_x = {\hbar \over 2}$ state at time $t$? Does this probability change with time?

6. What is the probability that you would measure $S_z = {\hbar \over 2}$ at time $t$? Does this probability change with time?

7. Given a Hamiltonian, how would you determine which states are stationary states (states where no probabilities change with time)? Under what circumstances do measurement probabilities change with time?

## Activity: Wrap-up

The main points of this activity are addressed in the last question. Students should recognize:
• a pattern of how these calculations proceed,
• to recognize a stationary state
• that measurement probabilities of non-stationary states will be time-dependent UNLESS you measure a quantity that commutes with the Hamiltonian.
• face Time Evolution Refresher (Mini-Lecture)

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
Central Forces 2023 (3 years)

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• group Energy radiated from one oscillator

group Small Group Activity

30 min.

##### Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

This lecture is one step in motivating the form of the Planck distribution.
• group Expectation Values for a Particle on a Ring

group Small Group Activity

30 min.

##### Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• group Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals 2022 (3 years)

Warm-Up

• group Operators & Functions

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals 2022 (3 years) Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
• 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.
• assignment Ring Table

assignment Homework

##### Ring Table
Central Forces 2023 (3 years)

Attached, you will find a table showing different representations of physical quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be. pdf link for the Table or doc link for the Table

• 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 \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable $M$ is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} 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.
• 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. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} 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 Find Force Law: Logarithmic Spiral Orbit

assignment Homework

##### Find Force Law: Logarithmic Spiral Orbit
Central Forces 2023 (3 years)

In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a $1/r^2$ force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass $\mu$, under the influence of a central-force field, that moves in a logarithmic spiral orbit given by $r = ke^{\alpha \phi}$, where $k$ and $\alpha$ are constants.

Author Information
Elizabeth Gire
Learning Outcomes