Introduction to Quantum Mechanics
Phys 451 — Fall 2015 — Assignment 1
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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 \(E_0 \neq E_1\).
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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.
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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 \(\Delta p\).
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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?