Why Does My Physics Calculation Differ from the Markscheme?

[thomas49th]

The question I am struggling is number 12 (c)(ii) which starts on page 14, but my part is on 16, the very last question of the paper

http://www.hinchingbrookeschool.net/science/documents/2864June2003.pdf

I want to calculate the length of a box where n = 3

$$E_{n} = n^{2}E_{1}$$

Therefore for the n = 3 quantum state, E = 1.215x10^-21

But the markscheme goes into somthing completely different, so I must be taking the wrong route. If I work it through I get the wrong answer

Page 32 ON Mark Scheme:
http://www.hinchingbrookeschool.net/science/documents/2003JuneMS.pdf

Can someone explain my misunderstanding of the physics involved here?

Thanks :)

 

[cepheid]

You are right that the general conceptual outline is to calculate E in the n=3 energy level, to use this to find the de Broglie wavelength, and then to use that, combined with the assumption that the box supports standing waves with 1.5 wavelengths as shown, to calculate the length of the box.

Your problem arises with the calculation of the energy in the n=3 level. The relation you have quoted, E_n = n^2 (E_1), is only true for the hydrogen atom. This is not a hydrogen atom. It should be obvious upon a closer inspection of the problem that this relation does not apply. (Hint: can you see that if the energy did vary as n^2, that the levels would NOT be evenly-spaced as stated?)

What this is is a simplistic "particle in a box" model that does not necessarily correspond to a physically realistic potential. Because the levels are evenly spaced, you calculate E_3 as follows:

E(n =3 ) = E(ground state) + two spacings

The ground state energy is 1.35e-21 J (given), and the spacing is supposedly then 2.70e-21 J (based on what's in the marking scheme). That's not quite what I get for the spacing, but it is close. Here is what I got as an answer for the energy level spacing:

http://www.google.com/search?client=safari&rls=en-us&q=4e12+Hz+*+planck's+constant&ie=UTF-8&oe=UTF-8

I hope that helps.