Chapter 10 Systems with $N$ Degrees of Freedom

10.1 $N$ Particles in One Dimension

Exercise 10.1.1 Show the following:

(1)

(2)

(3)If

then

and similarly with $1\to 2$.

(4)

Exercise 10.1.2 Imagine a fictitious world in which the single-particle Hilbert space is two-dimensional. Let us denote the basis vectors by $|+\rangle$ and $|-\rangle$. Let

be operators in $\mathbb{V}_1$ and $\mathbb{V}_2$, respectively (the $\pm$ signs label the basis vectors. Thus $b=\langle+| \sigma_1^{(1)}|-\rangle$ etc.) The space $\mathbb{V}_1 \otimes \mathbb{V}_2$ is spanned by four vectors $|+\rangle \otimes|+\rangle$, $|+\rangle \otimes|-\rangle$, $|-\rangle\otimes|+\rangle$, $|-\rangle\otimes|-\rangle$. Show (using the method of images or otherwise) that

(1)

(Recall that $\langle\alpha|\otimes\langle\beta|$ is the bra corresponding to $|\alpha\rangle\otimes|\beta\rangle$.)

(2) $\sigma_{2}^{(1)\otimes(2)}=\begin{pmatrix} e & f & 0 & 0\\ g & h & 0 & 0\\ 0 & 0 & e & f\\ 0 & 0 & g & h \end{pmatrix}$

(3) $(\sigma_{1}\sigma_{2})^{(1)\otimes(2)}=\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}=\begin{pmatrix} ae & af & be & bf\\ ag & ah & bg & bh\\ ce & cf & de & df\\ cg & ch & dg & dh \end{pmatrix}$

Do part (3) in two ways, by taking the matrix product of $\sigma_{1}^{(1)\otimes (2)}$ and $\sigma_{2}^{(1)\otimes (2)}$ and by directly computing the matrix elements of $\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}$.

Exercise 10.1.3 Consider the Hamiltonian of the coupled mass system:

We know from Example 1.8.6 that $\mathscr{H}$ can be decoupled if we use normal coordinates

and the corresponding momenta

(1) Rewrite $\mathscr{H}$ in terms of normal coordinates. Verify that the normal coordinates are also canonical, i.e., that

Now quantize the system, promoting these variables to operators obeying

Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{\text{I}}$ and $X_{\text{II}}$.

(2) Quantize the system directly, by promoting $x_{1}$, $x_{2}$, $p_{1}$, and $p_{2}$ to quantum operators. Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{1}$ and $X_{2}$. Now change from $x_{1}$, $x_{2}$ (and of course $\partial/\partial x_{1}$, $\partial/\partial x_{2}$) to $x_{\text{I}}$, $x_{\text{II}}$ (and $\partial/\partial x_{\text{I}}$, $\partial/\partial x_{\text{II}}$) in the differential equation. You should end up with the result form part (1).

10.2 More Particles in More Dimensions

Exercise 10.2.1 (Particle in a Three-Dimensional Box) Recall that a particle in a one-dimensional box extending from $x=0$ to $L$ is confined to the region $0 \leqslant x \leqslant L$; its wave function vanishes at the edges $x=0$ and $L$ and beyond (Exercise 5.2.5). Consider now a particle confined in a three-dimensional cubic box of volume $L^3$. Choosing as the origin one of its corners, and the $x, y$, and $z$ axes along the three edges meeting there, show that the normalized energy eigenfunctions are

where

and $n_i$ are positive integers.

Exercise 10.2.2 Quantize the two-dimensional oscillator for which

(1) Show that the allowed energies are

(2) Write down the corresponding wave functions in terms of single oscillator wave functions. Verify that they have definite parity (even/odd) number $x \rightarrow-x, y \rightarrow-y$ and that the parity depends only on $n=n_x+n_y$.

(3) Consider next the isotropic oscillator $\left(\omega_x=\omega_y\right)$. Write explicit, normalized eigenfunctions of the first three states (that is, for the cases $n=0$ and 1). Reexpress your results in terms of polar coordinates $\rho$ and $\phi$ (for later use). Show that the degeneracy of a level with $E=(n+1) \hbar \omega$ is $(n+1)$.

Exercise 10.2.3 Quantize the three-dimensional isotropic oscillator for which

(1) Show that $E=(n+3 / 2) \hbar \omega$; $n=n_x+n_y+n_z$; $n_x, n_y, n_z=0,1,2, \ldots$.

(2) Write the corresponding eigenfunctions in terms of single-oscillator wave functions and verify that the parity of the level with a given $n$ is $(-1)^n$. Reexpress the first four states in terms of spherical coordinates. Show that the degeneracy of a level with energy $E=$ $(n+3 / 2) \hbar \omega$ is $(n+1)(n+2) / 2$.

10.3 Identical Particles

Exercise 10.3.1 Two identical bosons are found to be in states $|\phi\rangle$ and $|\psi\rangle$. Write down the normalized state vector describing the system when $\langle\phi \mid \psi\rangle \neq 0$.

Exercise 10.3.2 When an energy measurement is made on a system of three bosons in a box, the $n$ values obtained were $3$, $3$, and $4$. Write down a symmetrized, normalized state vector.

Exercise 10.3.3 Imagine a situation in which there are three particles and only three states $a$, $b$, and $c$ available to them. Show that the total number of allowed, distinct configurations for this system is

(1) 27 if they are labeled

(2) 10 if they are bosons

(3) 1 if they are fermions

Exercise 10.3.4 Two identical particles of mass $m$ are in a one-dimensional box of length $L$. Energy measurement of the system yields the value $E_{\text{sys}}=\hbar^2 \pi^2/mL^2$. Write down the state vector of the system. Repeat for $E_{\text{sys}}=5 \hbar^2 \pi^2 / 2 m L^2$. (There are two possible vectors in this case.) You are not told if they are bosons or fermions. You may assume that the only degrees of freedom are orbital.

Exercise 10.3.5 Consider the exchange operator $P_{12}$ whose action on the $X$ basis is

(1) Show that $P_{12}$ has eigenvalues $\pm 1$. (It is Hermitian and unitary.)

(2) Show that its action on the basis ket $\left|\omega_1, \omega_2\right\rangle$ is also to exchange the labels $1$ and $2$, and hence that $\mathbb{V}_{S/A}$ are its eigenspaces with eigenvalues $\pm 1$.

(3) Show that $P_{12} X_1 P_{12}=X_2$, $P_{12} X_2 P_{12}=X_1$ and similarly for $P_1$ and $P_2$. Then show that $P_{12} \Omega\left(X_1, P_1 ; X_2, P_2\right) P_{12}=\Omega\left(X_2, P_2 ; X_1, P_1\right)$. [Consider the action on $\left|x_1, x_2\right\rangle$ or $\left|p_1, p_2\right\rangle$. As for the functions of $X$ and $P$, assume they are given by power series and consider any term in the series. If you need help, peek into the discussion leading to Eq. (11.2.22).]

(4) Show that the Hamiltonian and propagator for two identical particles are left unaffected under $H \rightarrow P_{12} H P_{12}$ and $U \rightarrow P_{12} U P_{12}$. Given this, show that any eigenstate of $P_{12}$ continues to remain an eigenstate with the same eigenvalue as time passes, i.e., elements of $\mathbb{V}_{S/A}$ never leave the symmetric or antisymmetric subspaces they start in.

Exercise 10.3.6 Consider a composite object such as the hydrogen atom. Will it behave as a boson or fermion? Argue in general that objects containing an even/odd number of fermions will behave as bosons/fermions.