Chapter 22, Applications to Many-Body Systems Video Solutions, Quantum mechanics

Problem 1

Consider a system of identical bosons with only two one-particle basis states, $a_{1 / 2}^{\dagger} \Psi^{(0)}$ and $a_{-1 / 2}^{\dagger} \Psi^{(0)}$. Define the Hermitian operators $x, p_x, y, p_y$ by the relations
$$
a_{1 / 2}=\frac{1}{\sqrt{2 \hbar}}\left(c x+i \frac{p_x}{c}\right), \quad a_{-1 / 2}=\frac{1}{\sqrt{2 \hbar}}\left(c y+i \frac{p_y}{c}\right)
$$
where $c$ is an arbitrary real constant, and derive the commutation relations for these Hermitian operators. Express the angular momentum operator (22.6) in terms of these "coordinates" and "momenta," and also evaluate $\mathscr{F}^2$. Relate $\mathscr{F}^2$ to the square of the Hamiltonian of an isotropic two-dimensional harmonic oscillator by making the identification $c=\sqrt{m \omega}$, and show the connection between the eigenvalues of these operators.

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02:19

Problem 2

(a) Using the fermion creation operators $a_{j m}^{\dagger}$, appropriate to particles with angular momentum $j$, form the closed-shell state in which all one-particle states $m=-j$ to $+j$ are occupied.
(b) Prove that the closed shell has zero total angular momentum.
(c) If a fermion with magnetic quantum number $m$ is missing from a closed shell of particles with angular momentum $j$, show that, for coupling angular momenta, the hole state may be treated like a one-particle state with magnetic quantum number $-m$ and an effective creation operator $(-1)^{j-m} a_{j m}$.

Dominador Tan

Numerade Educator

Problem 3

Consider the unperturbed states $a_{n m_n}^{\dagger} \cdots a_{k m_k}^{\dagger} \cdots a_{1 m_1}^{\dagger}|\mathbf{0}\rangle$ of $n$ spin one-half particles, each occupying one of $n$ equivalent, degenerate orthogonal orbitals labeled by the quantum number $k$, and with $m_k= \pm 1 / 2$ denoting the spin quantum number associated with the orbital $k$. Show that in the space of the $2^n$ unperturbed states a spin-independent two-body interaction may, in first-order perturbation theory, be replaced by the effective exchange (or Heisenberg) Hamiltonian
$$
\mathscr{H}_{\text {eff }}=-\frac{1}{\hbar^2} \sum_{k \ell}\langle k \ell|V| \ell k\rangle \mathbf{S}_k \cdot \mathbf{S}_{\ell}+\text { const. }
$$
where $\mathbf{S}_k$ is the localized spin operator
$$
\mathbf{S}_k=\frac{\hbar}{2} \sum_{m_k m_k^{\prime}} a_{k m_k}^{\dagger} a_{k m_k^{\prime}}\left\langle m_k|\boldsymbol{\sigma}| m_k^{\prime}\right\rangle
$$

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Problem 4

For a Fermi gas of free particles with Fermi momentum $p_F$, calculate the ground state expectation value of the pair density operator
$$
\sum_{\sigma^{\prime}, \sigma^{\prime \prime}} \boldsymbol{\psi}_{\sigma^{\prime}}^{\dagger}\left(\mathbf{r}^{\prime}\right) \boldsymbol{\psi}_{\sigma^{\prime}\left(\mathbf{r}^{\prime \prime}\right)}^{\dagger} \boldsymbol{\psi}_{\sigma^{\prime}}\left(\mathbf{r}^{\prime \prime}\right) \boldsymbol{\psi}_{\sigma^{\prime}}\left(\mathbf{r}^{\prime}\right)
$$
in coordinate space and show that there is a repulsive interaction that would be absent if the particles were not identical. Show that there is no spatial correlation between particles of opposite spin.

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Problem 5

Calculate in first order the energies of the ${ }^1 S,{ }^3 P$, and ${ }^1 D$ states arising from the atomic configuration $p^2$ (two electrons with $\ell=1$ in the same shell). Use the multipole expansion
$$
\frac{e^2}{\left|\mathbf{r}^{\prime}-\mathbf{r}^{\prime \prime}\right|}=e^2 \sum_{k=0}^{\infty} \frac{4 \pi}{2 k+1} \gamma_k\left(r^{\prime}, r^{\prime \prime}\right) \sum_{q=-k}^k(-1)^q Y_k^q\left(\hat{\mathbf{r}}^{\prime}\right) Y_k^{-q}\left(\hat{\mathbf{r}}^{\prime \prime}\right)
$$
for the interaction energy between the electrons, and show that the term energies may be expressed as
$$
\begin{aligned}
& E\left({ }^1 S\right)=E_0+\left\langle\gamma_0\right\rangle+\frac{10}{25}\left\langle\gamma_2\right\rangle \\
& E\left({ }^3 P\right)=E_0+\left\langle\gamma_0\right\rangle-\frac{5}{25}\left\langle\gamma_2\right\rangle \\
& E\left({ }^1 D\right)=E_0+\left\langle\gamma_0\right\rangle+\frac{1}{25}\left\langle\gamma_2\right\rangle
\end{aligned}
$$
where $\left\langle\gamma_k\right\rangle$ is the radial integral
$$
\left\langle\gamma_k\right\rangle=e^2 \iint \gamma_k\left(r^{\prime}, r^{\prime \prime}\right)\left[R\left(r^{\prime}\right)\right]^2\left[R\left(r^{\prime \prime}\right)\right]^2 r^{\prime 2} r^{\prime 2} d r^{\prime} d r^{\prime \prime}
$$

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Problem 6

Apply the Hartree-Fock method to a system of two "electrons" which are attracted to the coordinate origin by an isotropic harmonic oscillator potential $m \omega^2 r^2 / 2$ and which interact with each other through a potential $V=C\left(\mathbf{r}^{\prime}-\mathbf{r}^{\prime \prime}\right)^2$. Solve the HartreeFock equations for the ground state and compare with the exact result and with first-order perturbation theory.

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