Deduce the formula giving the energy levels in terms of n for a free particle in a box of length l, using the fact that only stationary waves can be occupied by the particle.(adsbygoogle = window.adsbygoogle || []).push({});

- considering stationary waves, λ = 2l/n. (1)

1) Using E=hv/λ= hvn (2)

And

P = h/λ, v = h/m*λ, v = hn/2ml (3)

Subbing this into (2), E = h^2n^2/4ml^2

2) However using E=1/2mv^2, p=mv, and λ = h/mv,

1/2mv^2= p^2/2m = h^2/λ^2*2m

= h^2/2m * n^2/4l^2 ( applying (1) )

= h^2n^2/8ml^2

I’m confused as to why this is - is there something fundamentally wrong with trying to derive the energy levels for a particle in a box using 1/2mv^2=E and p=mv ?

Thanks a lot anyone - greatly appreciated :D.

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# Tiny part of derivation for the energy levels occupied by a free particle in a box.

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