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Particles and Dimensions

1. We saw in class that a 3-dimensional ideal gas obeys pV = 3/2E, where p is the pressure, V is the volume, and E is the internal energy (average kinetic energy). Derive the corresponding formula for a two-dimensional ideal gas, i.e. a collection of noninteracting particles that moves on a plane. (Hint: suppose a two-dimensional ideal gas of N particles, with internal energy E, is confined by one-dimensional "walls" to an area A; the pressure is defined as the force per unit length).
2. Each r seconds a particle, which was initially x = 0, jumps either left or right (each with probability 1/2) a distance a. At time tn= nr, the particle is at location xk = ka with probability P(n, k). Calculate P(n, k), <k> and <(delta(k))^2>.

Solution Preview

Hi!

As promised, here are the solutions to this two problems.

Let's recall Schrodinger's equation solutions in square two dimensional infinite well with area :

With boundary conditions:

Separating the equation we obtain:

Since the equation is now separated, the terms on the left hand side are independent of each other and therefore for the equation to be correct they must be constants.

Hence we get two equations:

The solutions of the first equations are:

Thus:

The solutions for the other equation (in the y-direction) are the same (after all, it is practically the same equation with the same boundary conditions) hence:

Each pair combination is a microstate in state space.

In state space, it looks like this:

We need to calculate the density of states, that is, how many states we have between energy E and energy E+dE as a function of the energy.

Therefore, the density of states is the number of states that occupy quarter of the ring of radius n and thickness dn, which is proportional to the area of the ring ...

Solution Summary

This solution provides calculations for various questions involving particles and dimensions.

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