Phys 451 — Fall 2015 — Assignment 1

  1. Consider a quantum system whose Hilbert space is discrete, consisting of just two states. We’ll label them \(\lvert 0 \rangle\) and \(\lvert 1 \rangle\). Suppose that these states form an orthonormal basis and that the system is prepared at time \(t=0\) in a quantum superposition

    \[\lvert \psi \rangle = \lvert \psi(t=0) \rangle = \lvert 0 \rangle + (1+\sqrt{2}i)\lvert 1 \rangle.\]

    (a) Write down the bra state dual to this ket.

    (b) Compute the value of the overlap \(\langle \psi \vert \psi \rangle\).

    (c) Construct the normalized state \(\lvert \tilde{\psi} \rangle\) with the same physical properties as \(\lvert \psi \rangle\).

    (d) Argue that the probability of finding the particle in state \(\lvert n \rangle\) is \(P_n = \lvert \langle n \vert \tilde{\psi} \rangle\rvert^2 = \lvert \langle n \vert \psi \rangle\rvert^2 / \langle\psi \vert \psi \rangle\).

    (e) Determine explict values for \(P_0\) and \(P_1\). Show that \(P_0 + P_1=1\).

    (f) Assume that the basis states are also energy eigenstates; i.e., \(\hat{H}\lvert n \rangle = E_n\lvert n \rangle\). Determine the explcit time evolution of the state

    \[\lvert \psi(t) \rangle = \exp(-i\hat{H} t/ \hbar) \lvert \psi(0) \rangle.\]

    (g) Comment on the time evolution in that cases where \(E_0 = E_1\) and \(E_0 \neq E_1\).

  2. Using the same basis \(\{ \lvert 0 \rangle, \lvert 1 \rangle\}\) from question 1, we present an operator in ket-bra outer product form:

    \[\hat{O} = \lvert 0 \rangle \langle 1 \rvert + \lvert 1 \rangle \langle 0 \rvert.\]

    (a) Compute its four matrix elements \(\langle m \rvert \hat{O} \lvert n \rangle\).

    (b) Show explicitly that \(\langle m \rvert \hat{O}^2 \lvert n \rangle = \sum_{l=0,1} \langle m \rvert \hat{O} \lvert l \rangle \langle l \rvert \hat{O} \lvert n \rangle\), and explain why this is so.

    (c) Show that there is an orthonormal basis \(\{ \lvert + \rangle, \lvert - \rangle\}\) built from \(\lvert + \rangle \sim \lvert 0 \rangle + \lvert 1 \rangle\) and \(\lvert - \rangle \sim \lvert 0 \rangle - \lvert 1 \rangle\).

    (d) Express \(\hat{O}\) in this rotated basis.

    (e) Solve the eigenproblem for \(\hat{O}\) each of the two choices of basis.

  3. A particle confined in one spatial dimension is prepared in a state \(\lvert \psi(t) \rangle = \lvert \psi(0) \rangle\) that has no time dependence whatsoever. The wave function has the form \(\langle x \vert \psi \rangle = \psi(x) = Ce^{-ax^2}\).

    (a) Determine the normalization constant \(C\).

    (b) What is the probability of measuring the particle in the interval \([-1/\sqrt{a},1/\sqrt{a}]\)?

    (c) Apply the Schrödinger equation \(i \hbar \partial \psi/\partial t = -(\hbar^2/2m)\partial^2 \psi/\partial x^2 + V(x)\psi(x)\) to find an explicit expression for the external potential \(V(x)\).

    (d) Compute these four expectation values: \(\langle \hat{x} \rangle, \langle \hat{x}^2 \rangle, \langle \hat{p} \rangle, \langle \hat{p}^2 \rangle\). Evaluate the product \(\Delta x \Delta p\), where the \(\Delta\) implies \(\Delta x = (\langle \hat{x}^2 \rangle - \langle \hat{x} \rangle^2)^{1/2}\) and the analogous definition for \(\Delta p\).

  4. A particle confined to an \(L \times L \times L\) box has a wave function \(\psi(\mathbf{r}) = (2/L)^{3/2}\prod_{a=x,y,z}\sin(k_aa)\), where we have used the notation \(\mathbf{r} = (x,y,z)\) and \(\mathbf{k} = (k_x,k_y,k_z)\).

    (a) Compute its probability current \(\mathbf{j}(\mathbf{r})\).

    (b) Decompose \(\psi(\mathbf{r})\) into a superposition of free-particle wave functions of definite momentum.

    (c) What is a strict precondition for obtaining a nonzero current?