Phys 451 — Fall 2014 — Assignment 1

  1. Consider a quantum particle whose Hilbert space is discrete, consisting of just three states. We’ll label them \(\lvert 1 \rangle, \lvert 2 \rangle,\) and \(\lvert 3 \rangle\). Suppose that these states form an orthonormal basis and that the particle is prepared at time \(t=0\) in a quantum superposition

    \[\lvert \psi \rangle = 2\lvert 1 \rangle + i\lvert 2 \rangle + (2-i)\lvert 3 \rangle.\]

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

    (b) What is the normalized state with the same physical properties as \(\lvert \psi \rangle\)?

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

    (d) Determine explict values for \(P_1, P_2,\) and \(P_3\).

    (e) Show that if the basis states are also energy eigenstates—i.e., \(\hat{H}\lvert n \rangle = E_n\lvert n \rangle\)—then the time evolution of the wave function is described by

    \[\lvert \psi(t) \rangle = 2e^{-iE_1t/\hbar}\lvert 1 \rangle + ie^{-iE_2t/\hbar}\lvert 2 \rangle + (2-i)e^{-iE_3t/\hbar}\lvert 3 \rangle.\]

  2. A particle confined in one spatial dimension is prepared in a state \(\lvert \psi(t) \rangle = \lvert \psi(0) \rangle\) with 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 \(\Delta p\).

  3. States \(\lvert \psi \rangle\) and \(\lvert \phi \rangle\) have wave functions \(\psi(\mathbf{r}) = \langle \mathbf{r} \vert \psi \rangle\) and \(\tilde{\phi}(\mathbf{k}) = \langle \mathbf{k} \vert \phi \rangle\). Here we’ll use a tilde to distinguish between the real-space and wave-vector-space representations.

    (a) Show that \(\langle \psi \vert \phi \rangle \propto \int\!d^3r\,d^3k\,e^{i\mathbf{k}\cdot\mathbf{r}} \psi(\mathbf{r})^* \tilde{\phi}(\mathbf{k})\).

    (b) In the case where \(\tilde{\phi}(\mathbf{k})\) is strongly peaked in the vicinity of a particular value \(\mathbf{k}_0\), argue that \(\langle \psi \vert \phi \rangle \propto \int\!d^3r\,e^{i\mathbf{k}_0\cdot\mathbf{r}}\psi(\mathbf{r})^* \propto \tilde{\psi}(\mathbf{k}_0)^*\).

    (c) In the case where \(\tilde{\phi}(\mathbf{k})\) is an almost perfectly flat distribution, argue that \(\langle \psi \vert \phi \rangle \propto \psi(\mathbf{0})^*\).

  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?