# The uncertainty relation and density of states

1. May 11, 2010

### paco_uk

1. The problem statement, all variables and given/known data

I am trying to understand how we go about calculating the density of states in situations where the available quantum states are continuous, e.g. electrons in a white dwarf.

I am happy to accept the uncertainty relation (we learned to derive it as the product of the standard deviations of the expectation values of two operators).

I am happy to accept the Pauli exclusion principle for fermions (we learnt that if you try and make an antisymmetric combination of two particles in identical states, it vanishes).

I don't understand how this is applied when we don't have discrete sates.

2. Relevant equations

For example, in calculating the degeneracy pressure in a white dwarf star, we say:

$$\Delta^{3} p \Delta^{3}x \geq \frac{h^{3}}{2}$$

But then we go on to say that this implies a maximum phase space density:

$$n_{ph}=\frac{2}{h}$$

3. The attempt at a solution

What I don't understand is how the uncertainty relation justifies this maximum density. The Pauli principal only creates a problem if we try and put two electrons in identical states so I don't understand what stops us putting them in arbitrarily similar states. Why does the uncertainty in one state affect how close we can pack the next state?

Many thanks for any help. If you think my best bet would be to go and read a book about it I would be grateful for any suggestions as to which book.

2. May 11, 2010

### paco_uk

P.S I'm sorry I couldn't get the maths to show. I'd be grateful if someone could tell me what I'm doing wrong there as well.

EDIT: Scratch that. I found it.