E5 E4 E3 E2 Ei

En kT

Fig 10.11.

Energy levels in a degenerate gas.The energies of the levels are indicated on the right. In each level an upward arrow represents an electron with its spin in one sense, and a downward arrow represents an electron with its spin in the opposite sense.The dashed line indicates the average thermal energy per particle.The total energy is the sum of the energies of the individual electrons.

being in higher states than one would expect just based on temperature.)

We can also describe this pressure in terms of the uncertainty principle. In Chapter 3, we saw that we must think of electrons as having wave properties. We can only talk about the probability of finding an electron in a given place, or moving with a given speed. As a result of this wave property, we cannot simultaneously describe the position and momentum of the electron. If we can determine the momentum with an uncertainty Ap, and the position with an uncertainty Ax, the uncertainty principle tells us that

For a given Ax, the uncertainty in the momentum

Fig 10.12.

Pressure in a degenerate gas.We consider the force on the section of area A of the right-hand wall of the box, due to the x-component of the motions of the particles.

x-component of the momentum. The momentum per second per unit surface area is just the pressure exerted by the gas on the wall:

If we have ne electrons per unit volume, then there is one electron per box with volume 1/ne. The side of such a box is (1/ne)1/3, so the average spacing between electrons is

If we say that the average momentum is of the order of the momentum uncertainty, then px = h/2v Ax = (h/2w ) ne 1/3

The speed of each electron is its momentum divided by its mass, so

When the density becomes very high, we are trying to force the electrons close together. This means that we are trying to confine them to a small Ax. Therefore, the uncertainty principle tells us that the uncertainty in the momentum is large. This means that large momenta are possible. These high momentum electrons are responsible for the increased pressure. Fig. 10.12 shows a container with density n and particles moving with speed vx in the x-direction. The number of particles hitting a wall per second per unit surface area is n vx. The momentum per second per unit surface area delivered to the wall is then n vx px, where px is the

Px/me

This gives a pressure, using equation (10.3), of

This is just an estimate of the pressure. A more detailed calculation yields a pressure that is a factor of about two higher than that given in equation (10.7).

Equation (10.7) gives the pressure in terms of electron density. We would like to express it in terms of the total mass density p. If the density of positive ions with charge Ze is nZ, then in a neutral gas the density of electrons must be ne = Zn

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