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Problem 3 Identical Particles (10 Marks) a The Hamiltonian $H$ for two noninteracting particles in an infinite one- dimensional square well of width $a$ is egin{equation} -frac{hbar^2}{2m}frac{partial^2 psi}{partial x_1^2} - frac{hbar^2}{2m}frac{partial^2 psi}{partial x_2^2} = Epsi end{equation} if $0 le x_1, x_2 le a$, otherwise $psi = 0$ If the particles are distinguishable, then the solution can be written as a simple product of the wavefunctions representing each particle. Show that this simple product composite wavefunction formed from the two one-particle states $psi_{n_1}(x) = sqrt{2/a}sin(n_1pi x_1/a)$ and $psi_{n_2}(x) = sqrt{2/a}sin(n_2pi x_1/a)$ is an eigenfunction of $H$ with energy $E_{n_1, n_2} = (n_1^2 + n_2^2)pi^2hbar^2/2ma^2$. b Give the eigenfunctions, energies and degeneracies of (i) the ground state and (ii) the first excited state of this system of two distinguishable parti- cles. c Repeat (b) when the two particles are (i) bosons. (Hint: Remember that for bosons, the wavefunctions are given by $(sqrt{2/a})(psi_{n_1}(x) + psi_{n_2}(x))$ d What is the ground state wavefunction and energy if the two particles are fermions? e Show that the average energy per free electron ($E_{tot}/N_q$) as a fraction of the Fermi energy is given by $(3/5)E_F$.

          Problem 3 Identical Particles (10 Marks)
a The Hamiltonian $H$ for two noninteracting particles in an infinite one-
dimensional square well of width $a$ is
egin{equation*}
-frac{hbar^2}{2m}frac{partial^2 psi}{partial x_1^2} - frac{hbar^2}{2m}frac{partial^2 psi}{partial x_2^2} = Epsi
end{equation*} if $0 le x_1, x_2 le a$, otherwise $psi = 0$
If the particles are distinguishable, then the solution can be written as a
simple product of the wavefunctions representing each particle.
Show that this simple product composite wavefunction formed from the
two one-particle states $psi_{n_1}(x) = sqrt{2/a}sin(n_1pi x_1/a)$ and $psi_{n_2}(x) = sqrt{2/a}sin(n_2pi x_1/a)$
is an eigenfunction of $H$ with energy $E_{n_1, n_2} = (n_1^2 + n_2^2)pi^2hbar^2/2ma^2$.
b Give the eigenfunctions, energies and degeneracies of (i) the ground state
and (ii) the first excited state of this system of two distinguishable parti-
cles.
c Repeat (b) when the two particles are (i) bosons. (Hint: Remember that
for bosons, the wavefunctions are given by $(sqrt{2/a})(psi_{n_1}(x) + psi_{n_2}(x))$
d What is the ground state wavefunction and energy if the two particles are
fermions?
e Show that the average energy per free electron ($E_{tot}/N_q$) as a fraction of
the Fermi energy is given by $(3/5)E_F$.

$Problem 3 Identical Particles (10 Marks) a The Hamiltonian H for two noninteracting particles in an infinite one- dimensional square well of width a is eginequation* -frachbar^22mfracpartial^2 psipartial x1^2 - frachbar^22mfracpartial^2 psipartial x2^2 = Epsi endequation* if 0 le x1, x2 le a, otherwise psi = 0 If the particles are distinguishable, then the solution can be written as a simple product of the wavefunctions representing each particle. Show that this simple product composite wavefunction formed from the two one-particle states psin1(x) = sqrt2/asin(n1pi x1/a) and psin2(x) = sqrt2/asin(n2pi x1/a) is an eigenfunction of H with energy En1, n2 = (n1^2 + n2^2)pi^2hbar^2/2ma^2. b Give the eigenfunctions, energies and degeneracies of (i) the ground state and (ii) the first excited state of this system of two distinguishable parti- cles. c Repeat (b) when the two particles are (i) bosons. (Hint: Remember that for bosons, the wavefunctions are given by (sqrt2/a)(psin1(x) + psin2(x)) d What is the ground state wavefunction and energy if the two particles are fermions? e Show that the average energy per free electron (Etot/Nq) as a fraction of the Fermi energy is given by (3/5)EF.$

Added by Albert F.

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