*face*Time Evolution Refresher (Mini-Lecture)*face*Lecture30 min.

##### Time Evolution Refresher (Mini-Lecture)

Central Forces 2023 (3 years) The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.*group*Working with Representations on the Ring*group*Small Group Activity30 min.

##### Working with Representations on the Ring

Central Forces 2023 (3 years)*assignment*Normalization of Quantum States*assignment*Homework##### Normalization of Quantum States

Central Forces 2023 (3 years) Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}*group*Superposition States for a Particle on a Ring*group*Small Group Activity30 min.

##### Superposition States for a Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.*group*Energy and Angular Momentum for a Quantum Particle on a Ring*group*Small Group Activity30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.*keyboard*Position operator*keyboard*Computational Activity120 min.

##### Position operator

Computational Physics Lab II 2022quantum mechanics operator matrix element particle in a box eigenfunction

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.*keyboard*Sinusoidal basis set*keyboard*Computational Activity120 min.

##### Sinusoidal basis set

Computational Physics Lab II 2023 (2 years)inner product wave function quantum mechanics particle in a box

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*ISW Position Measurement*assignment*Homework##### ISW Position Measurement

time evoluation infinite square well Quantum Fundamentals 2023A particle in an infinite square well potential has an initial state vector \[\left|{\Psi(0)}\right\rangle = A\big(\left|{\phi_1}\right\rangle -\left|{\phi_2}\right\rangle +i\left|{\phi_3}\right\rangle \big)\]

where \(|\phi_n\rangle\) are the energy eigenstates. You have previously found \(\left|{\Psi(t)}\right\rangle \) for this state.

Use a computer to graph the wave function \(\Psi(x,t)\) and probability density \(\rho(x,t)\). Choose a few interesting values of \(t\) to include in your submission.

Use a computer to calculate the probability of measuring the particle to be near the middle of the well (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.

Choose another location in the well, different from the location above. Use a computer to calculate the probability of measuring the particle to be near your chosen location (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.

Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.

- The time dependence for a wave function like this is complicated. Write a lengthy description in words about the major features of this wave function and its probability density, how they change in time, and why they change the way they do. Comment on any interesting features you noticed that you have not already discussed in the questions above and describe any additional things you learned from the process of solving this problem.

*format_list_numbered*Quantum Ring Sequence*format_list_numbered*Sequence##### Quantum Ring Sequence

Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a*Mathematica*notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.*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): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.-
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.)- 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.
- 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.
- 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.
- 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.