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).