Quantum Fundamentals: NoTerm-2022
HW 8 : Due Day 20 F 3/4

1. Expectation Uncertainty S0 4371S

   Consider measuring the z-component of spin for each particle listed below.

   (i) List the possible outcomes of your experiment and determine the probability associated with each. Use a Different Representation: Draw a histogram of the probabilities.

   (ii) Find the expectation value and the uncertainty for your experiment. Compare: Does your result seem reasonable given your histogram?

   1. A spin-1/2 particle described by \(\left|{+}\right\rangle \).

   2. A spin-1 particle described by \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \).

   3. A spin-1/2 particle described by \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \).

2. Spin Uncertainty S0 4371S Consider the state \(\vert -1\rangle_y\) in a spin 1 system.
   1. Discuss the direction of the spin angular momentum for this quantum system.
   2. Calculate the expectation values and uncertainties for measurements of \(S_x\), \(S_y\), and \(S_z\).
3. Probabilities of Energy S0 4371S (adapted from McIntyre Problem # 3.2)

   1. Show that the probability of a measurement of the energy is time independent for a general state:

      \[\left|{\psi(t)}\right\rangle = \sum_n c_n(t) \left|{E_n}\right\rangle \]

      that evolves due to a time-independent Hamiltonian.

   2. Show that the probabilities of measurements of other observables that commute with the Hamiltonian are also time independent (neither operator has degeneracy).