A beam of spin-\(\frac{1}{2}\) particles is prepared in the initial state \[ \left\vert \psi\right\rangle = \sqrt{\frac{2}{5}}\; |+\rangle_x - \sqrt{\frac{3}{5}}\; |-\rangle_x \](Note: this state is written in the \(S_x\) basis!)

1. What are the possible results of a measurement of \(S_x\), with what probabilities?

2. Repeat part a for measurements of \(S_z\).

3. Suppose you start with a particle in the state given above, measure \(S_x\), and happen to get \(+\hbar /2\). You then take that same particle and measure \(S_z\). What are the possible results and with what probability would you measure each possible result?

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

None

None

• How to translate a complicated wavefunction into eigenstates.
• Refresher on how to find expectation values and probabilities in a region.
• How to use the symmetry of the wavefunction to tell you something about measurements.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.

A 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.

1. 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.

2. 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.

3. 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.

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

5. 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.

1. Set-Up a Sequential Measurement

   1. Add an analyzer to the experiment by:

      1. Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
      2. Click and drag a new connection from the analyzer to empty space to create a new element. A new analyzer is one of the options.

   2. Measure \(S_z\) twice in succession.

      

      What is the probability that a particle leaving the first analyzer with \(S_z=\frac{+\hbar}{2}\) will be measured by the second analyzer to have \(S_z=\frac{-\hbar}{2}\)?

   3. Try all four possible combinations of input/outputs for the second analyzer.

      

      What have you learned from these experiments?

   
   

2. Try All Combinations of Sequential Measurements

   In the table, enter the probability of a particle exiting the 2nd analyzer with the spin indicated in row if the particle enters the 2nd analyzer with the spin indicated in each column.

   

   
   

3. You can rotate the Stern-Gerlach analyzers to any direction you want (using spherical coordinates).

   Choose an arbitrary direction (not along one of the coordinate axes) for the 1st analyzer and measure the spin along the coordinate directions for the 2nd analyzer.

   

Students become acquainted with the Spins Simulations of Stern-Gerlach Experiments and record measurement probabilities of spin components for a spin-1/2 system. Students start developing intuitions for the results of quantum measurements for this system.

• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.

Students consider the dimensions of spin-state kets and position-basis kets.

Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

1. Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

Note

In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

1. What is the energy of the ground state and the first and second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.

2. At room temperature, what is the relative probability of finding a hydrogen molecule in the \(\ell=0\) state versus finding it in any one of the \(\ell=1\) states?
   i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)

3. At what temperature is the value of this ratio 1?

4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the \(\ell=2\) states versus that of finding it in the ground state?
   i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.

These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.

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.

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.

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.

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.

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.

Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

Students find a wavefunction that corresponds to a Gaussian probability density.

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.