## Activity: Quantum Expectation Values

Quantum Fundamentals Winter 2021
• 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 Winter 2021

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.
• group Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals Winter 2021
• group Operators & Functions

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals Winter 2021 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 Matrix Elements and Completeness Relations

assignment Homework

##### Matrix Elements and Completeness Relations
Quantum Fundamentals Winter 2021

Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

What if I want to calculate the matrix elements using a different basis??

The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: $\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y$

In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

where $I$ is the identity operator: $I=\left|{+}\right\rangle _y {}_y\left\langle {+}\right|+\left|{-}\right\rangle _y {}_y\left\langle {-}\right|$. This effectively rewrite the $\left|{+}\right\rangle$ in the $\left|{\pm}\right\rangle _y$ basis.

Find the top row matrix elements of the operator $\hat{S}_y$ in the $S_z$ basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals Winter 2021
• face Quantum Reference Sheet

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces Spring 2021 Central Forces Spring 2021
• group Changing Spin Bases with a Completeness Relation

group Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Quantum Fundamentals Winter 2021

Students work in small groups to use completeness relations to change the basis of quantum states.
• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho=A\pi\sin(\pi z/h).$$
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)$$
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

AIMS Maxwell AIMS 21 Static Fields Winter 2021

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

Expectation Values and Uncertainty

You have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the $z$-direction.

Consider one of the different initial spin states described below:

A spin 1/2 particle described by:

1. $\left|{+}\right\rangle$
2. $\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle$
3. $\left|{+}\right\rangle _x$

A spin 1 particle described by:

4. $\left|{0}\right\rangle$
5. $\left|{-1}\right\rangle _x$
6. $\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle$
• List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.

• Plot a histogram of the probabilities.

• Find the expectation value and uncertainty of the z-component of spin.

Learning Outcomes