Particle in the 1D infinite square well

Question

I'm trying to draw the phase space for a particle moving freely between 0 and $L$. I guess $H=E$(total energy, constant)$=frac{p_{x}^2}{2m}$ so $p_{x}=pmsqrt{2mE}$ for every $x$ between $0$ and $L$, and taking the positive sign when going from $0$ to $L$ and the negative sign when going from $L$ to $0$. If I draw the phase space for this particle, with energy between $E$ and $delta E$ I think I should get what i drew on the image. However, it seems a bit odd, is there something wrong?

InertialObserver · Answer

Your situation is a particle in a box, and as such the energy is quantized to be

$$ E= E_n = E_1 n^2$$.

If you plot $E$ vs. $n^2$ for $n>0$ you get one half of a discrete parabola. What you want is to know the number of states that lie between energies $E$ and $E + dE$. That is,

$$dn = frac{dn}{dE} dE$$

We call $frac{dn}{dE}$ the density of states. To find this we solve for $n$ obtaining

$$n = sqrt{frac{E}{E_1}}$$

Therefore, the density of states is

$$
frac{dn}{dE} = frac{1}{2sqrt{E_1}} cdot frac{1}{sqrt{E}}
$$

Particle in the 1D infinite square well

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