Question 2.

Ideal and Bose Gases

a)    Helium has two stable isotopes. Name them, and state for each whether it is a boson or a fermion. Justify your answer. (1 point)

b)   For each isotope, provide the function which describes the thermal average occupancy of an orbital of energy E, in contact with a reservoir of chemical potential u. (1 point)

c)    In which regime do these two distribution functions coincide? What can be said about the likelihood of multiple occupancy of an orbital in this regime? Justify your answer. (1 point)

d)   State the equipartition theorem and use it to provide the internal energy per particle of an ideal gas of non-interacting mono-atomic particles confined in a 3D box of volume V, at     temperature T. (2 points)

e)    Using the previous result, work out an approximate value of the quantum temperature TQ, (for which the thermal de Broglie wavelength 入T = becomes comparable to the mean     interparticle separation l), and show that it is equivalent to within a constant to the critical temperature TC =mkB(2几ℏ2) ()2/3 for an ideal Bose gas. (2 points)

f)    We consider an ideal Bose gas with total number of particles N, occupancy of the ground state (of energy 0) N0 and occupancy of all excited states,

Ne = dED(E)fB (E).

What relationship exists between these values at TC? (1 point)

g)    For the same ideal Bose gas discussed above (Q 2f), what relationship exists between these values at T = 0 ? Use this relationship and fB (0) to derive the value of the chemical            potential as T → 0. Relate this result to the concept of Bose enhancement (or bosonic stimulation).  (2 points)

Question 3.

Fermi Gas

Consider an ideal 2D gas of N nonrelativistic fermions of spin S = 3/2 confined to a square of area

A = L2.

a)    Find the density of states D(E) as a function of energy E. (3 points)

b)   Find the Fermi energy EF of the gas. (3 points)

c)    Find the ground state energy of the gas U0 and express the final result in terms of N and EF.

(4 points)

Question 4.

Ising Model

a)   The 1D Ising model is sufficiently simple that there are analytical solutions for key                 thermodynamic quantities such as the thermal average energy, U = −NE tanh(). What does the smoothness of this function physically imply for a chain of spins? Additionally,        comment on the length scale of the interactions in the basic model. (2 points)

In the figure below, filled squares represent a spin up configuration while an empty box represents a spin down configuration.

b)   How many spin configurations are possible for this 9 × 9 lattice? (1 point)

c)    Assuming the number of up and down spins is conserved, describe what can qualitatively happen if spins are allowed to change place in the 9 × 9 lattice. (2 points)

d)   The block spin transformation is used to simplify the analysis of large or complex lattices and is related to a theoretical method called the renormalization group. In this transformation,    we replace a 3 × 3 block of dipoles (see heavy outline) with a single dipole. The state of this  single dipole is determined by which state (up or down) is in the majority in the 3 × 3 block.

Reduce the lattice above using the block spin transformation and determine the energy of this new lattice configuration using the mean field approximation. Start by drawing a         schematic figure indicating the new lattice configuration. Additionally, comment on the     number of configurations for this new lattice. (3 points)

e)   A common modification to the Ising model involves the introduction of an external magnetic field which adds a potential energy term −uBsi, that is:

E = −e ∑ sisj − uB∑ si

             i

Using this energy statement, suggest a form for the partition function when considering N total sites. Hint: You are not required to find a closed form of the partition function, rather set up the initial step by correctly identifying the summations required . (2 points)