Chapter 7, Identical Particles Video Solutions, Modern Quantum Mechanics

Problem 1

Liquid helium makes a transition to a macroscopic quantum fluid, called superfluid helium, when cooled below a phase transition temperature $T=2.17 \mathrm{~K}$. Calculate the de Broglie wavelength $\lambda=h / p$ for helium atoms with average energy at this temperature, and compare it to the size of the atom itself. Use this to predict the superfluid transition temperature for other noble gases, and explain why none of them can form superfluids. (You will need to look up some empirical data for these elements.)

Penny Riley

Numerade Educator

02:11

Problem 2

Three identical particles are in a one-dimensional harmonic oscillator potential well with classical angular frequency $\omega$.
a. Write the complete time-independent Hamiltonian for this system, and express it in coordinate space as a differential equation whose solution is the three-body wave function $\Psi\left(x_{1}, x_{2}, x_{3}\right)$.
b. Assume the particles have zero spin. Use the single-particle wave functions to construct the ground-state wave function $\Psi_{0}\left(x_{1}, x_{2}, x_{3}\right)$, and show that it satisfies the differential equation in (a), and find the ground-state energy.
c. Assume the particles have spin $\frac{1}{2}$. Repeat (b), and also construct the ground state spin state from single-particle spin eigenstates.

Zhuxi Luo

Numerade Educator

02:57

Problem 3

a. $N$ identical spin $\frac{1}{2}$ particles are subjected to a one-dimensional simple harmonic oscillator potential. Ignore any mutual interactions between the particles. What is the ground-state energy? What is the Fermi energy?
b. What are the ground-state and Fermi energies if we ignore the mutual interactions and assume $\mathrm{N}$ to be very large?

Mahnoor Amin

Numerade Educator

03:55

Problem 4

It is obvious that two nonidentical spin 1 particles with no orbital angular momenta (that is, $s$-states for both) can form $j=0, j=1$, and $j=2$. Suppose, however, that the two particles are identical. What restrictions do we get?

Sam Stansfield

Numerade Educator

00:37

Problem 5

Discuss what would happen to the energy levels of a helium atom if the electron were a spinless boson. Be as quantitative as you can.

Devon Burke

Numerade Educator

05:50

Problem 6

Three spin 0 particles are situated at the corners of an equilateral triangle. Let us define the $z$-axis to go through the center and in the direction normal to the plane of the triangle. The whole system is free to rotate about the $z$-axis. Using statistics considerations, obtain restrictions on the magnetic quantum numbers corresponding to $J_{z}$.

Mahnoor Amin

Numerade Educator

08:57

Problem 7

Consider three weakly interacting, identical spin 1 particles.
a. Suppose the space part of the state vector is known to be symmetric under interchange of any pair. Using notation $|+\rangle|0\rangle|+\rangle$ for particle 1 in $m_{s}=+1$,
particle 2 in $m_{s}=0$, particle 3 in $m_{s}=+1$, and so on, construct the normalized spin states in the following three cases.
(i) All three of them in $|+\rangle$.
(ii) Two of them in $|+\rangle$, one in $|0\rangle$.
(iii) All three in different spin states.
What is the total spin in each case?
b. Attempt to do the same problem when the space part is antisymmetric under interchange of any pair.

Mahnoor Amin

Numerade Educator

08:38

Problem 8

A porphyrin ring is a molecule which is present in chlorophyll, hemoglobin, and other biological compounds. It can be modeled as 18 electrons moving freely along a one-dimensional circular path of radius $R=0.4 \mathrm{~nm}$.
a. Using a polar angular coordinate $\theta$, write down the appropriately normalized single-particle wave functions $\psi(\theta)$, including periodic boundary conditions. Find an expression for the single-particle energy eigenvalues.
b. Find the electron configurations and energies for the ground state and first excited state of porphyrin.
c. Find a numerical value for the wavelength of electromagnetic radiation that would excite the ground state into the first excited state. This is a very simple model, and porphyrin comes in many varieties, but compare your result to an experimental result.

Ronald Prasad

Numerade Educator

01:44

Problem 9

Show that for an operator $a$ which, with its adjoint, obeys the anticommutation relation $\left\{a, a^{\dagger}\right\}=a a^{\dagger}+a^{\dagger} a=1$, then the operator $N=a^{\dagger} a$ has eigenstates with the eigenvalues 0 and 1 .

Nick Johnson

Numerade Educator

03:55

Problem 10

Suppose the electron were a spin $\frac{3}{2}$ particle obeying Fermi-Dirac statistics. Write the configuration of a hypothetical Ne $(Z=10)$ atom made up of such "electrons" [that is, the analogue of $(1 s)^{2}(2 s)^{2}(2 p)^{6}$ ]. Show that the configuration is highly degenerate. What is the ground state (the lowest term) of the hypothetical Ne atom in spectroscopic notation $\left({ }^{2 S+1} L_{J}\right.$, where $S, L$, and $J$ stand for the total spin, the total orbital angular momentum, and the total angular momentum, respectively) when exchange splitting and spin-orbit splitting are taken into account?

Suzanne W.

Numerade Educator

08:57

Problem 11

Two identical spin $\frac{1}{2}$ fermions move in one dimension under the influence of the infinite-wall potential $V=\infty$ for $x<0, x>L$, and $V=0$ for $0 \leq x \leq L .$
a. Write the ground-state wave function and the ground-state energy when the two particles are constrained to a triplet spin state (ortho state).
b. Repeat (a) when they are in a singlet spin state (para state).
c. Let us now suppose that the two particles interact mutually via a very short range attractive potential that can be approximated by
$$
V=-\lambda \delta\left(x_{1}-x_{2}\right) \quad(\lambda>0) .
$$
Assuming that perturbation theory is valid even with such a singular potential, discuss semiquantitatively what happens to the energy levels obtained in (a) and (b).

Mahnoor Amin

Numerade Educator

09:46

Problem 12

Consider the case of a single particle of mass $m$ in a one-dimensional simple harmonic oscillator potential $V(x)=m \omega^{2} x^{2} / 2$.
a. Using a form for $\tilde{\rho}(x)$ proportional to $\exp \left(-a x^{2}\right)$, calculate the energy functional (7.69) in terms of $a$. Minimize the result with respect to $a$ and show that you get the correct ground-state energy.
b. Now show that a form proportional to $\exp \left(-a x^{2}\right)$ is a solution to the differential equation (7.76), for the appropriate value of $a$. Check that (7.69) gives you the correct ground-state energy, and show that it equals the value of $\mu$.

Coleman Green

Numerade Educator

View

Problem 13

Show by explicit construction for $N=2$ that the normalization condition $(7.80)$ is met for $n(\mathbf{x})$ as defined by $(7.89)$ with the multiparticle wave function $(7.88)$ for the cases when
a. $\phi_{1}(\mathbf{x})$ and $\phi_{2}(\mathbf{x})$ are distinct orthonormal functions,
b. $\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x})$.

Joseph David

Numerade Educator

08:28

Problem 14

Using whatever code or application you prefer, fill in the details of the calculation on the helium atom in density functional theory.
a. Confirm the dimensionless forms (7.102) and (7.103).
b. Show that (7.104) is properly normalized.
c. Verify (7.105).
d. Obtain (7.108), or something very close, based on your own numerical approach.
e. Complete the calculation and reproduce Table $7.1$ and Figure $7.7 .$
f. Try repeating the calculation using an initial density $n^{(0)}(r)$ that is smarter than (7.104). In fact, (7.108) shows that the first Kohn-Sham ground-state wave function is very close to the $1 s$ state for $Z=Z_{\text {eff }}$. What starting density does this imply?
There are many variations to this problem you might consider, including trying different forms for the exchange-correlation energy, a larger or smaller set of basis states, or more iterations or different starting densities.

Amit Srivastava

Numerade Educator

02:53

Problem 15

$$
\text { Prove the relations }(7.170) \text {, and then carry through the calculation to derive }(7.176) \text {. }
$$

Amany Waheeb

Numerade Educator

09:59

Problem 16

A Hamiltonian for a system of bosons has the form
$$
\mathscr{H}=\sum_{\mathbf{k}} T(\mathbf{k}) a_{\mathbf{k}}^{\dagger} a_{\mathbf{k}}+\lambda \sum_{\mathbf{l}} \sum_{\mathbf{m}} V(\mathbf{l}+\mathbf{m}) a_{\mathbf{1}}^{\dagger} a_{-\mathbf{1}}^{\dagger} a_{\mathbf{m}} a_{-\mathbf{m}}
$$
where $\lambda$ is a constant. Prove that the number operator
$$
\mathscr{N}=\sum_{\mathbf{k}} a_{\mathbf{k}}^{\dagger} a_{\mathbf{k}}
$$
is a constant of the motion.

Isaac Huidobro

Numerade Educator