- 1.
- (a)
- The probability of finding an electron, of a given state
of the hydrogen atom, at some distance between and is given by

How much more likely is a electron to be at a distance of from the nucleus than a distance of ?

(Use the attached table in figure 2 below of the normalised wave functions for the hydrogen atom.) (8)

- (b)
- Find the ground state energy of the hydrogen atom, and
,

by substituting the appropriate radial wave function into the radial wave equation

As an intermediate result, also find an expression for , the Bohr radius,

(Note that for the ground state and , and use the attached table in figure 2 below of the normalised wave functions for the hydrogen atom.) (12)

- 2.
- (a)
- Starting with the wave-functions for two identical non-interacting
quantum particles,
show that the probability for two fermions to occupy the same quantum
state is zero, and
that the probability for two bosons to occupy the same quantum state is
twice that for two classical particles to occupy the same state.

(Note : .) (6)

- (b)
- Use this result to state and explain the relative pressure
exerted by similar gases of classical molecules, bosons, or
fermions for the same temperature.
(6)

- (c)
- Sketch the Fermi-Dirac distribution (see question 5a) at low but finite temperature
and explain why only electrons near the Fermi-level
are expected to participate in transport properties.
(5)

- (d)
- Show for this distribution (see question 5a) that if the average occupancy of a state of energy
is at any temperature, then the average occupancy of a state of energy
is .

(7)

[25]

- 3.
- (a)
- Classify the materials (a) to (f) below as metals, insulators or semi-conductors
specifying the dopant type as well where necessary.

- (b)
- In a semi-conductor, the energy of a band is given by .
At what value of and will the electron have infinite mass ?
(9)