Phys 712 — Spring 2015 — Assignment 3

  1. Consider a single quantum particle in one dimension. Demonstrate that (I) \(\psi_p(x) = \langle p \vert x \rangle = (2\pi\hbar)^{-1/2}e^{ipx/\hbar}\) and (II) \([\hat{x},\hat{p}] = i\hbar\) are formally equivalent. Proceed as follows, proving that (I) implies (II) and then the reverse.

    (a) To start, assume (I). Show that \(\hat{T}(a) = e^{i\hat{p}a/\hbar}\) is an operator that translates the system by \(a\) (i.e. \(x \to x+a\)) and hence \(\hat{p}\) is a generator for translation.

    (b) Complete the proof of (I) implies (II) by establishing that

    \[\begin{split}\hat{T}(\Delta a)^\dagger \hat{x} \hat{T}(\Delta a) &= \hat{x} + \Delta a \hat{1} \\ &= \hat{x} + \frac{i\Delta a}{\hbar}(\hat{p}\hat{x} - \hat{x}{\hat{p}}) + O(\Delta a)^2 \end{split}\]

    for an infinitesmial translation \(\Delta a\).

    (c) Now assume (II). Show that \([\hat{x}^n,\hat{p}] = ni\hbar\hat{x}^{n-1}\). A recursive proof is easiest.

    (d) Use the result from part (c) to show that \([p,f(x)] = -i\hbar \partial_x f(x)\).

    (e) Complete the proof of (II) implies (I) by solving the differential equation

    \[[p,\psi_p(x)] = -i\hbar \partial_x \psi_p(x) = p\psi_p(x).\]

  2. A quantum particle lives on a collection of \(N\) sites arranged in a closed ring. (You can think of each individual site as a deep potential well that supports a highly localized ground state wavefunction.) Its dynamics include only tunnelling between adjacent sites. The Hamiltonian is

    \[\hat{H} = \sum_{i=1}^{N} \biggl[ -t\bigl( \lvert i \rangle \langle i+1 \rvert + \lvert i+1 \rangle \langle i \rvert \bigr) + \epsilon_i \lvert i \rangle \langle i \rvert \biggr],\]

    with the periodic boundary condition enforced by the equivalence \(\lvert N+1 \rangle \equiv \lvert 1 \rangle\).

    (a) Suppose that each site is identical, with \(\epsilon_i = \epsilon_0\) a uniform constant. Find a transformation that puts the Hamiltonian in diagonal form:

    \[\hat{H} = \sum_{k} (-2t\cos k + \epsilon_0) \lvert k \rangle \langle k \rvert.\]

    Be pricise about what values of \(k\) are summed over. You may want to invoke the concept of a Brillouin zone, following the discussion in Section 4.3 of Sakurai.

    (b) What is the ground state wave function?

    (c) What symmetry of the Hamiltonian is connected to \(k\) being a good quantum number?

    (d) Suppose that \(N\) is even and that there are two kinds of site alternating around the ring. Let’s use the notation \(\epsilon_i = \epsilon_{\text{even}}\) for \(i = 2, \ldots, N\), and \(\epsilon_i = \epsilon_{\text{odd}}\) for \(i = 1, 3, \ldots, N-1\) to denote the two different binding energies. Find a transformation that diagonalizes the Hamiltonian. Explain the significance of the new quantum number that arises. Account for the fact that there are now \(N/2\) inequivalent \(k\) values.

  3. Three spin-half objects arranged on the corners of an equilateral triangle interact via the Hamiltonian

    \[\hat{H} = J( \hat{\mathbf{S}}_1\cdot\hat{\mathbf{S}}_2 + \hat{\mathbf{S}}_2\cdot\hat{\mathbf{S}}_3 + \hat{\mathbf{S}}_3\cdot\hat{\mathbf{S}}_1).\]

    (a) We’ll span the Hilbert space with an orthonormal basis that is the product of the local spin-up and spin-down states at each site: e.g., \(\lvert \uparrow \rangle_1 \otimes \lvert \downarrow \rangle_2 \otimes \lvert \uparrow \rangle_3 \equiv \lvert \uparrow \downarrow \uparrow \rangle\) is a basis element. How many states are in the basis?

    (b) Relate the Hamiltonian to the total spin operator \(\hat{\mathbf{S}}^{\text{tot}} = \hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2 + \hat{\mathbf{S}}_3\). Argue that the Hamiltonian is invariant under a uniform rotation of the spins. Identify the three energy eigenvalues.

    (c) The angular addition rules for spin-half tell us that the spin sectors combine according to \(\tfrac{1}{2} \otimes \tfrac{1}{2} = 0 \oplus 1\) and \(s \otimes \tfrac{1}{2} = \bigl(s-\tfrac{1}{2}\bigl)\oplus \bigl(s+\tfrac{1}{2}\bigr)\). There are \(2s+1\) states in each sector \(s\). Determine the degeneracy of each energy eigenvalue. Since the energy eigenstates are also simultaneous eigenstates of the total spin operator, this amounts to counting how many states correspond to \(S^\text{tot} = 1/2\) and how many to \(S^\text{tot} = 3/2\).

    (d) Propose a term you could add to the Hamiltonian that would break the energy degeneracy.

    (e) Make a correspondence with binary number notation to assign a label to each of the basis states.

    \[\begin{align} \lvert 0 \rangle &= \lvert \downarrow \downarrow \downarrow \rangle\\ \lvert 1 \rangle &= \lvert \downarrow \downarrow \uparrow \rangle\\ \lvert 2 \rangle &= \lvert \downarrow \uparrow \downarrow \rangle\\ &\,\,\vdots \end{align}\]

    (f) Use the identity \(\hat{\mathbf{S}}\cdot\hat{\mathbf{S}}_j = \frac{1}{2}\bigl(\hat{S}^+_i\hat{S}^-_j + \hat{S}^-_i\hat{S}^+_j\bigr) + \hat{S}^z_i\hat{S}^z_j\) to determine how the Hamiltonian acts on each of the basis states. For example,

    \[\begin{align} \hat{H} \lvert 0 \rangle &= \frac{3}{4}\lvert 0 \rangle\\ \hat{H} \lvert 1 \rangle &= \frac{1}{2}\lvert 2 \rangle + \frac{1}{2}\lvert 4 \rangle - \frac{1}{4}\lvert 1 \rangle\\ &\,\,\vdots \end{align}\]

    Re-express the Hamiltonian as \(\hat{H} = \sum_{n,m} \lvert m \rangle H_{m,n} \langle n \rvert\). Complete the matrix of coefficients.

    \[H_{m,n} = \begin{pmatrix} \frac{3}{4} & 0 & 0 & 0 & \cdots \\ 0 & -\frac{1}{4} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} && \ddots\\ 0 & & \ddots\\ \vdots \end{pmatrix}\]

    (g) The basis \(\{ \lvert 0 \rangle, \lvert 1 \rangle, \ldots \}\) consists of states of well-defined \(S^{\text{tot}}_z\). Group them according to their \(S^{\text{tot}}_z\) eigenvalue. Rearrange the rows and columns of \(H_{m,n}\) to put the matrix in block diagonal form. There should be four blocks, corresponding to \(S^{\text{tot}}_z = -3/2, -1/2, 1/2, 3/2\).

    (h) Consider a permutation operation \(\hat{P}\) that relabels the spins \(1 \to 2 \to 3 \to 1\) (or equivalently, rotates them around the corners of the triangle). Argue that the Hamiltonian is invariant under such a relabelling.

    (i) Construct eigenstates of \(\hat{P}\) out of the states in the \(S^{\text{tot}}_z = -1/2, 1/2\) sectors. (Note that \(\hat{P}^3\) is an identity operation, so you should be looking for eigenvalues that are cube roots of unity.) Write down a basis of \(\hat{P}\) eigenstates that fully diagonalizes the Hamiltonian.

  4. “Time-reversal invariance” is more properly understood as the property that a system allows for a complete reversal of motion.

    (a) In class, we derived the propagator of a free particle in one spatial dimension:

    \[G(x_f,t_f; x_i,t_i) = \frac{m}{2\pi i \hbar (t_f - t_i)} \exp\biggl[ \frac{im(x_f-x_i)^2}{2\hbar(t_f-t_i)}\biggr].\]

    Argue that, in this case, reversal of motion manifests itself as

    \[G(x_f,t_f ; x_i,t_i)^* = G(x_i,t_i ; x_f,t_f).\]

    (b) A particle in a classical electromagnetic field is described by the path integral

    \[G(\mathbf{r}_f,t_f ; \mathbf{r}_i,t_i) = \int_{\mathbf{r}(t_i) = \mathbf{r}_i}^{\mathbf{r}(t_f) = \mathbf{r}_f} \mathcal{D}[\mathbf{r}(t)]e^{iS[\mathbf{r}(t)]/\hbar}\]

    with an action

    \[S = \frac{i}{\hbar}\int_{t_i}^{t_f} dt \Bigl[ \frac{1}{2}m\dot{r}^2 + \frac{e}{c}\dot{\mathbf{r}}\cdot\mathbf{A} - e\phi\Bigr].\]

    Deduce how \(\dot{\mathbf{r}}(t)\), \(\mathbf{A}(\mathbf{r}(t),t)\), and \(\phi(\mathbf{r}(t),t)\) must each transform under time reversal to ensure that \(G(\mathbf{r}_f,t_f ; \mathbf{r}_i,t_i) = G(\mathbf{r}_i,t_i; \mathbf{r}_f,t_f)^*\).

    (c) For an arbitrary, time-dependent Hamiltonian the evolution operator is

    \[\hat{U}(t_f,t_i) = T\exp\biggl(-\frac{i}{\hbar}\int_{t_i}^{t_f} \hat{H}(t) \biggr).\]

    Show that its hermitian conjugate can be written as

    \[\hat{U}(t_f,t_i)^\dagger = \tilde{T}\exp\biggl(\frac{i}{\hbar}\int_{t_i}^{t_f} \hat{H}(t) \biggr),\]

    where \(\tilde{T}\) is the (hopefully self-explanatory) reverse time ordering operator.

    (d) In a system that supports reversal of motion, the time reversal operator \(\Theta\) is an antiunitary operator that obeys

    \[\begin{align}\Theta \hat{H} \Theta^{-1} &= \hat{H}, \\ \Theta i\hat{H} \Theta^{-1} &= -i\hat{H}.\end{align}\]

    Prove that, for such a system, \(\hat{U}(t_f,t_i)^\dagger = \hat{U}(t_i,t_f)\) and \(G(\mathbf{r}_f,t_f ; \mathbf{r}_i,t_i) = G(\mathbf{r}_i,t_i; \mathbf{r}_f,t_f)^*\). You’ll need to think in terms of reversal over the particular time window, with \(t \to t_i + t_f -t\) and \(dt \to -dt\).