Quantum Fundamentals: NoTerm-2022
HW 9 : Due Day 23 W 3/9

1. Frequency S0 4372S 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 expectation value of \(M\) as a function of time? What is the frequency of oscillation of the expectation value of \(M\)?
2. Magnet S0 4372S

   Consider a spin-1/2 particle with a magnetic moment. At time \(t=0\), the state of the particle is \(\left|{\psi(t=0)}\right\rangle =\left|{+}\right\rangle \).

   1. If the observable \(S_x\) is measured at time \(t=0\), what are the possible results and the probabilities of those results?

   2. Instead of performing the above measurement, the system is allowed to evolve in a uniform magnetic field \(\vec{B}=B_0\, \hat y\). The Hamiltonian for a system in a uniform magnetic field \(\vec B=B_0\, \hat y\) is \(H=\omega_0\, S_y\). (You can treat \(\omega_0\) as a given parameter in your answers to the following two questions.)

      • Calculate the state of the system after a time \(t\) and represent this state using the \(S_z\) basis.
      • At time \(t\), the observable \(S_x\) is measured, what is the probability that a value \(\hbar\)/2 will be found?

3. Wavefunction Calculations S0 4372S

   Consider quantum particles in different 1-D potentials U(x).

   For each of the following wavefunctions:

   (i) Confirm that the wavefunction is normalizable by showing that the wavefunction goes to zero at \(\pm\infty\).

   (ii) Find the value of the overall constant that normalizes the state over all space.

   (iii) Calculate the probability of locating the particle in the region \(x = 0\) m to \(x = 1\) m.

   (iv) (For the first two states \(\left|{\psi_1}\right\rangle \) and \(\left|{\psi_2}\right\rangle \) only) Calculate the probability that the particle will be in a state

   \[\left|{\psi_{out}}\right\rangle \doteq \begin{cases} 0 & x < 0 \\ \sqrt{2}\sin \pi x & 0\leq x\leq 1 \\ 0 & x > 1 \end{cases}\]

   after an unspecified measurement process.

   (a) \(\left|{\psi_1}\right\rangle \doteq \begin{cases} 0 & x < -1 \\ A & -1\leq x\leq 1 \\ 0 & x > 1 \end{cases}\)

   (b) \(\left|{\psi_2}\right\rangle \doteq \begin{cases} 0 & x< 0 \\ Bx(x-3) & 0\leq x\leq 3 \\ 0 & x > 3 \end{cases}\)

   (c) \(\left|{\psi_3}\right\rangle \doteq \begin{cases} Ce^{k(x-1)} & x\leq 1 \\ Ce^{-k(x-1)} & x > 1 \end{cases}\)