Normalization of Quantum States

• group Superposition States for a Particle on a Ring

  group Small Group Activity

  30 min.

  Superposition States for a Particle on a Ring

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

  Quantum Ring Sequence

  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 Hydrogen Probabilities in Matrix Notation

  group Small Group Activity

  30 min.

  Hydrogen Probabilities in Matrix Notation

  Central Forces 2023 (2 years)
• group Time Dependence for a Quantum Particle on a Ring

  group Small Group Activity

  30 min.

  Time Dependence for a Quantum Particle on a Ring

  Theoretical Mechanics (6 years)

  central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

  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.
• group Energy and Angular Momentum for a Quantum Particle on a Ring

  group Small Group Activity

  30 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

  Quantum Ring Sequence

  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.
• group Wavefunctions on a Quantum Ring

  group Small Group Activity

  30 min.

  Wavefunctions on a Quantum Ring

  Central Forces 2023 (2 years)
• 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)

  central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence

  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.
• keyboard Position operator

  keyboard Computational Activity

  120 min.

  Position operator

  Computational Physics Lab II 2022

  quantum 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 Activity

  120 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 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 \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 Ring Function

  assignment Homework

  Ring Function

  Central Forces 2023 (3 years) Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.

  1. Find \(N\).

  2. Plot this wave function.
  3. Plot the probability density.
  4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
  5. What is the expectation value of \(L_z\) in this state?
• 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}