## Activity: Spin-1 Time Evolution

Quantum Fundamentals 2023
Students do calculations for time evolution for spin-1.
None
Spin-1 Time Evolution
1. Use a completeness relation to rewrite each of the states below in the $z$-direction.

• $|\psi_A\rangle = |-1\rangle_x$
• $|\psi_B\rangle = \frac{1}{\sqrt{2}}|+1\rangle_z - \frac{1}{\sqrt{2}}|-1\rangle_z$
• $|\psi_C\rangle = |-1\rangle_z$

2. Suppose the Hamiltonian is given by $\hat{H} = \omega_o \hat{S}_x$.

1. Identify the energy eigenstates and their corresponding energy eigenvalues.
2. Use them to write each state at a later instant in time.
3. Represent each state using Arms as complex numbers.

3. For $|\psi_C\rangle$:

1. List the possible values of a measurement of $S_z$ and calculate the corresponding probabilities.

2. List the possible values of a measurement of $S_x$ and calculate the corresponding probabilities.

3. Graph any probabilities that depend on time.

• 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 2023 (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.
• 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 2023 (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.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
• 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.
• assignment Wavefunctions

assignment Homework

##### Wavefunctions
Quantum Fundamentals 2023 (3 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 $0<x<1$.

• 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.
• assignment Frequency

assignment Homework

##### Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2023 (3 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 Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals 2023 (3 years)

Warm-Up

• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

##### Sinusoidal basis set
Computational Physics Lab II 2023 (2 years)

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.
• group Time Dependence for a Quantum Particle on a Ring Part 1

group Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring Part 1
Theoretical Mechanics (6 years)

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

Learning Outcomes