4.5: Exercises

Exercises

Exercise $\PageIndex{1}$

Consider a system of two identical particles. Each single-particle Hilbert space $\mathscr{H}^{(1)}$ is spanned by a basis $\{|\mu_i\}$. The exchange operator is defined on $\mathscr{H}^{(2)} = \mathscr{H}^{(1)} \otimes \mathscr{H}^{(1)}$ by

$$P \Big (\sum_{ij} \psi_{ij} |\mu_i\rangle|\mu_j\rangle \Big) \;\equiv\; \sum_{ij} \psi_{ij} |\mu_j\rangle|\mu_i\rangle.$$

Prove that $\hat{P}$ is linear, unitary, and Hermitian. Moreover, prove that the operation is basis-independent: i.e., given any other basis $\{\nu_j\}$ that spans $\mathscr{H}^{(1)}$,

$$P \Big (\sum_{ij} \varphi_{ij} |\nu_i\rangle|\nu_j\rangle \Big) \;=\; \sum_{ij} \varphi_{ij} |\nu_j\rangle|\nu_i\rangle.$$

Exercise $\PageIndex{2}$

Prove that the exchange operator commutes with the Hamiltonian

$$\hat{H} = - \frac{\hbar^2}{2m_e} \Big(\nabla_1^2 + \nabla^2_2\Big) + \frac{e^2}{4\pi\varepsilon_0|\mathbf{r}_1 - \mathbf{r}_2|}.$$

Exercise $\PageIndex{3}$

An $N$-boson state can be written as

$$|\phi_1,\phi_2,\dots,\phi_N\rangle = \mathcal{N} \sum_p \Big(|\phi_{p(1)}\rangle |\phi_{p(2)}\rangle |\phi_{p(3)}\rangle \cdots |\phi_{p(N)}\rangle\Big).$$

Prove that the normalization constant is

$$\mathcal{N} = \sqrt{\frac{1}{N!\prod_\mu n_\mu!}},$$

where $n_\mu$ denotes the number of particles occupying the single-particle state $\mu$.

Exercise $\PageIndex{4}$

$\mathscr{H}_{S}^{(N)}$ and $\mathscr{H}_{A}^{(N)}$ denote the Hilbert spaces of $N$-particle states that are totally symmetric and totally antisymmetric under exchange, respectively. Prove that

$$\begin{align}\begin{aligned} \mathrm{dim}\left(\mathscr{H}_{S}^{(N)}\right) &= \frac{(d+N-1)!}{N!(d-1)!}, \\ \mathrm{dim}\left(\mathscr{H}_{A}^{(N)}\right) &= \frac{d!}{N!(d-N)!}. \end{aligned}\end{align}$$

Exercise $\PageIndex{5}$

Prove that for boson creation and annihilation operators, $[\hat{a}_\mu,\hat{a}_\nu] = [\hat{a}_\mu^\dagger,\hat{a}_\nu^\dagger] = 0$.

Exercise $\PageIndex{6}$

Let $\hat{A}_1$ be an observable (Hermitian operator) for single-particle states. Given a single-particle basis $\{|\varphi_1\rangle,|\varphi_2\rangle,\dots\}$, define the bosonic multi-particle observable

$$\hat{A} = \sum_{\mu\nu} \,a^\dagger_\mu \; \langle\varphi_\mu|\hat{A}_1|\varphi_\nu\rangle \; a_\nu,$$

where $a_\mu^\dagger$ and $a_\mu$ are creation and annihilation operators satisfying the usual bosonic commutation relations, $[a_\mu,a_\nu] = 0$ and $[a_\mu,a_\nu^\dagger] = \delta_{\mu\nu}$. Prove that $\hat{A}$ commutes with the total number operator:

$$\Big[\hat{A}, \sum_\mu a^\dagger_\mu a_\mu \Big] = 0.$$

Next, repeat the proof for a fermionic multi-particle observable

$$\hat{A} = \sum_{\mu\nu} \,c^\dagger_\mu \; \langle\varphi_\mu|\hat{A}_1|\varphi_\nu\rangle \; c_\nu,$$

where $c_\mu^\dagger$ and $c_\mu$ are creation and annihilation operators satisfying the fermionic anticommutation relations, $\{c_\mu,c_\nu\} = 0$ and $\{c_\mu,c_\nu^\dagger\} = \delta_{\mu\nu}$. In this case, prove that

$$\Big[\hat{A}, \sum_\mu c^\dagger_\mu c_\mu \Big] = 0.$$

Further Reading

[1] Bransden & Joachain, §10.1–10.5

[2] Sakurai, §6

[3] J. M. Leinaas and J. Myrheim, On the Theory of Identical Particles, Nuovo Cimento B 37, 1 (1977).

[4] F. Wilczek, The Persistence of Ether, Physics Today 52, 11 (1999).