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 $p_{x}=\pm\sqrt{2mE}$ 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?

enter image description here

  • $\begingroup$ If the particle is confined to move from 0,L then that's just the infinte square well, right? $\endgroup$ – InertialObserver May 4 '19 at 17:41
  • $\begingroup$ Yes it is... but i had never tried to draw its phase space... I used to solve Schrodingers equation and graph a couple of wavefunctions, that was it $\endgroup$ – Juan Pablo Arcila May 4 '19 at 17:44
  • $\begingroup$ From your comment, it looks like you are discussing a quantum particle, for which there are ferocious "uncertainty-principle"-type constraints... You are probably graphing an aspirational Wigner function distribution, but then the cartoon you sketched is not accurate... See Exercise 0.2 , p 29 here. $\endgroup$ – Cosmas Zachos May 4 '19 at 18:38
  • $\begingroup$ If you were really serious about this, see Belloni et al, AmJPhys 72 (2004) 1183. $\endgroup$ – Cosmas Zachos May 4 '19 at 19:27

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

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


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