Chapter 21, Identical Particles Video Solutions, Quantum mechanics

Problem 1

(a) Show that if $V(r)$ is a two-particle interaction that depends only on the distance $r$ between the particles, the matrix element of the interaction in the k-representation may be reduced to
$$
\left\langle\mathbf{k}_3 \mathbf{k}_4|V| \mathbf{k}_1 \mathbf{k}_2\right\rangle=\delta\left(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3-\mathbf{k}_4\right) \frac{1}{(2 \pi)^3} \int V(r) e^{-t q \cdot \mathbf{r}} d^3 r
$$
where $\hbar \mathbf{q}$ is the momentum transfer $\hbar\left(\mathbf{k}_3-\mathbf{k}_1\right)$.
(b) For this interaction, show that the mutual potential energy operator is
$$
\mathscr{V}=\frac{1}{2} \iiint d^3 k_1 d^3 k_2 d^3 q \boldsymbol{\phi}^{\dagger}\left(\mathbf{k}_1+\mathbf{q}\right) \boldsymbol{\phi}^{\dagger}\left(\mathbf{k}_2-\mathbf{q}\right) \boldsymbol{\phi}\left(\mathbf{k}_2\right) \boldsymbol{\phi}\left(\mathbf{k}_1\right) \boldsymbol{F}(\mathbf{q})
$$
where $F(\mathbf{q})$ is the Fourier transform of the displacement-invariant interaction.

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Problem 2

Show that the diagonal part of the interaction operator $\mathscr{V}$, found in Problem 1 in the $\mathbf{k}$-representation, arises from momentum transfers $\mathbf{q}=0$ and $\mathbf{q}=\mathbf{k}_2-\mathbf{k}_1$, respectively. Write down the two interaction terms and identify them as $\operatorname{direct}(\mathbf{q}=0)$ and exchange $\left(\mathbf{q}=\mathbf{k}_2-\mathbf{k}_1\right.$ ) interactions. Draw the corresponding diagrams (Figure 21.1).

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Problem 3

In the $\mathbf{k}$-representation, calculate the matrix element of the interaction in Problem 1 for the screened Coulomb potential $V_0 e^{-\alpha r} / \alpha r$ and plot it as a function of $q$. For bosons and fermions, construct the corresponding two-particle interaction operator $\mathscr{V}$ for identical particles in terms of the creation and annihilation operators in $\mathbf{k}$-space.

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Problem 4

Defining the momentum space annihilation operator ${ }^3$
$$
\boldsymbol{\phi}(\mathbf{p})=\int\langle\mathbf{p} \mid \mathbf{r}\rangle \boldsymbol{\psi}(\mathbf{r}) d^3 r
$$
derive the commutation (or anticommutation) relations for $\boldsymbol{\phi}(\mathbf{p})$ and $\boldsymbol{\phi}^{\dagger}(\mathbf{p})$. For the Bose-Einstein case, show that the mixed commutator of field operators in coordinate and momentum space is
$$
\left[\boldsymbol{\phi}(\mathbf{p}), \boldsymbol{\psi}^{\dagger}(\mathbf{r})\right]=\langle\mathbf{p} \mid \mathbf{r}\rangle
$$

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20:23

Problem 5

In the second-quantization formalism, define the additive position and total momentum operators
$$
\mathbf{r}=\int \boldsymbol{\psi}^{\dagger}(\mathbf{r}) \mathbf{r} \boldsymbol{\psi}(\mathbf{r}) d^3 r \quad \text { and } \quad \mathbf{p}=\int \boldsymbol{\phi}^{\dagger}(\mathbf{p}) \mathbf{p} \boldsymbol{\phi}(\mathbf{p}) d^3 p
$$
and prove that for bosons their commutator is
$$
[\mathbf{r}, \mathbf{p}]=i \hbar N \mathbf{1}
$$
where $N$ is the operator representing the total number of particles. Derive the Heisenberg uncertainty relation for position and momentum of a system of bosons, and interpret the result.

Khoobchandra Agrawal

Numerade Educator

16:39

Problem 6

Local particle and current density operators at position $\mathbf{r}$ are defined in the secondquantization formalism as
$$
\rho(\mathbf{r})=\int \boldsymbol{\psi}^{\dagger}\left(\mathbf{r}^{\prime}\right) \delta\left(\mathbf{r}^{\prime}-\mathbf{r}\right) \boldsymbol{\psi}\left(\mathbf{r}^{\prime}\right) d^3 r^{\prime}
$$
and
$$
j(\mathbf{r})=\frac{\hbar}{2 m i} \int d^3 r^{\prime} \boldsymbol{\psi}^{\dagger}\left(\mathbf{r}^{\prime}\right)\left[\nabla^{\prime} \delta\left(\mathbf{r}^{\prime}-\mathbf{r}\right)+\delta\left(\mathbf{r}^{\prime}-\mathbf{r}\right) \nabla^{\prime}\right] \boldsymbol{\psi}\left(\mathbf{r}^{\prime}\right)
$$
(a) Show that the expectation values of these operators for one-particle states are the usual expressions.
(b) Derive the formulas for the operators $\rho(\mathbf{r})$ and $j(\mathbf{r})$ in the momentum representation.

Sheh Lit Chang

University of Washington

01:57

Problem 7

Two identical bosons or fermions in a state
$$
\Psi^{(2)}=A \sum_{i j} c_i d_j a_j^{\dagger} a_i^{\dagger} \Psi^{(0)}=A \sum_j d_j a_j^{\dagger} \sum_i c_i a_i^{\dagger}|0\rangle
$$
are said to be uncorrelated (except for the effect of statistics). If $\sum\left|c_l\right|^2=$ $\sum\left|d_i\right|^2=1$, determine the normalization constant $A$ in terms of the sum $S=\sum c_i^* d_i$.
(a) In this state, work out the expectation value of an additive one-particle operator in terms of the one-particle amplitudes $c_i$ and $d_i$ and the matrix elements $\langle i|K| j\rangle$.
(b) Show that if $S=0$, the expectation value is the same as if the two particles with amplitudes $c_i$ and $d_i$ were distinguishable.
(c) Work out the expectation value of a diagonal interaction operator in terms of $c_i, d_i$, and the matrix elements $\langle i j|K| k \ell\rangle=V_{i j} \delta_{i k} \delta_{j e}$. Show that the result is the same as for distinguishable particles if the states of the two particles do not overlap, i.e., if $c_i d_i=0$ for all $i$.

Mahnoor Amin

Numerade Educator

08:57

Problem 8

A state of $n$ identical particles (bosons or fermions) is denoted by $\left|\Psi^{(n)}\right\rangle$.
For $n=1$, the probability of finding the particle in the one-particle basis state $i$ is the expectation value $\left\langle\Psi^{(1)}\left|N_i\right| \Psi^{(1)}\right\rangle$. (See Exercise 21.1.)
(a) For $n=2$, prove that the probability of finding both particles in the oneparticle basis state $i$ is the expectation value of $N_i\left(N_i-1\right) / 2$.
(b) For $n=3$, obtain the function of $N_l$ whose expectation value is the probability of finding all three particles in the same basis state $i$.
(c) For $n=2$, show that the expectation value of $N_i N_j$ is the probability of finding the two particles in two different basis states, $i \neq j$. Prove that the probability of finding one particle in basis state $i$ and the other particle not in basis state $i$ is the expectation value of $N_i\left(2-N_i\right)$.

Mahnoor Amin

Numerade Educator