Density of States Derivation -

[JackStrong]

Density of States Derivation -urgent

Homework Statement

Homework Equations

λ=h/Px (Px = x momentum)

L/λ=nx

The Attempt at a Solution

A summarized derivation from the lecturer has proven to be problematic when revising:

as same in xyz direction just cube one direction

L.Px = H.nx [if nx =1

L.dpx=H

[xyz direction

L^3.dpx.dpy.dpz=h^3

it then jumps to sphere range of momentum v=4/3.∏.R^3

note _p = bar p ( a p with a line ontop)

_p to _p + d_p

r= _p

N(p)dp = (2(4∏._p^2.d_p))/h^3


Could someone please help.

Thanks. Jack

 

[mark726]

Dear Jack,

I understand your urgency in trying to understand the derivation of the density of states. It is an important concept in many fields of science, and it is crucial to have a clear understanding of it.

Let's start by defining some terms. The density of states, denoted by N, is a measure of how many quantum states are available per unit energy range at a given energy. It is usually expressed as a function of the energy, denoted by E. The equation for the density of states is:

N(E) = (2V / h^3) * √(E/m)

where V is the volume of the system and m is the mass of the particle.

Now, let's go through the derivation step by step.

1. The first equation you have written, λ = h/Px, is the de Broglie wavelength, which relates the momentum (Px) of a particle to its wavelength (λ). This equation is valid for all three directions (x, y, z).

2. Next, we have the equation L/λ = nx, where L is the length of the system and nx is the number of wavelengths that fit in the system. This equation is also valid for all three directions.

3. The next step is where the lecturer mentions "cube one direction." This means that we are considering a cube-shaped system, where the length (L) is the same in all three directions. So, instead of writing L/λ = nx for all three directions, we can just write L/λ = n for one direction. This is because nx = ny = nz = n.

4. Now, we can combine the two equations we have so far to get L.Px = h.n. This equation is also valid for all three directions.

5. Next, we introduce the concept of "differential momentum," denoted by dPx. This means that we are considering a small range of momentum, from Px to Px + dPx. This is similar to considering a small interval of x values, from x to x + dx. So, we can rewrite the previous equation as L.dPx = h.

6. Now, we need to consider all three directions together. Since we are dealing with a cube-shaped system, we can write L^3.dPx.dPy.dPz = h^3. This is because dPx = dPy = dPz = dP.

7. Next, we