Which Method Should Be Used to Calculate Energy in Standing Wave Formalism?

[Adhip]

Homework Statement

If you have seen the treatment of calculation of density of states using standing wave formalism in Modern Physics by Arthur Beiser,
their are these observations of which i am not completely convinced,

with wavelength fixed, he transforms wavelength into energy, using firstly debroglie's equation to transform lamda into momentum. And then momentum sq= 2mE, to get E.

The question is if we are using QM , why shouldn't we just transform lamda into frequency, and then directly using hv get energy.

The two give diverse results. Where am I missing the logic?

 

[zachzach]

Your post is not written very well, but I think I may understand you. You can transform wavelength into frequency. My question is, how did you do this? Matter waves and light waves are different. They have different dispersion relations.

 

[Adhip]

well, zachzach, thanks for the reply, i will reformulate it in clearer terms..

what we are doing is calculation of density of states in a box, which comes from the blackbody radiation and also can then be imagined to be particles in a box as quantum mechanically, particles in a box has standing wave solutions.

Now, same values of energy , momentum , or wavelength can be taken up by different combinations of quantum numbers describing the wave function.

Now,

n_x² + n_y² + n_z² = (2L/$$\lambda$$)²

where L is the dimension of the box.

Density of states is given by the number of unit n volume in a shell of radius given by 2L/$$\lambda$$.

Now lamda can be changed into frequency, which he does using dispersion relation for light, ie c=$$\lamda$$*$$\nu$$

Next we convert this into energy, for this their can be two possibilities

1) E= h$$\nu$$
2) straight from debroglie , get $$\lambda$$ changed to p
and then p²= 2mE to get E.

which and why?