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## Homework Statement

Consider an electron that is constrained to be in a one dimensional box of size L, but is otherwise free to move inside the box.

i.) Write down the (time independent) Schrodinger equation for this particle, the boundary conditions for the wavefunction Ψ and find an expression for the energy levels.

ii.) Consider the process where the electron decays from the nth energy level to the ground state by emitting a photon. Find the wavelength of the emitted photon as function of L, n and m.

iii.) Consider now an electron that can freely move in a two dimensional square box. What are the energy levels in this case. Please motivate your answer.

## Homework Equations

I let, h' = h/2π

EΨ(x) = h'/2m * d²Ψ(x)/dx² + U(x)Ψ(x)

p = h/λ

k = nπ/L = 2π/λ

## The Attempt at a Solution

HI thanks for taking the time to help me. I have completed part i and ii and need them to be checked. As for part iii i dont have a clue any help is greatly appreciated. These are not h/w questions but past paper questions.

i.) EΨ(x) = -h'/2m * d²Ψ(x)/dx² + U(x)Ψ(x)

0<=x<=L

For the particle in the box U(x) = 0

EΨ(x) = -h'/2m * d²Ψ(x)/dx²

Ψ(x) = A1e^(ikx)+ A2e^(-ikx)

= (A1+A2)cos(kx) + i(A1-A2)sin(kx)

Ψ(0) = (A1+A2) = 0, therefore

A1 = -A2

Ψ(x) = i2A1sin(kx)

d²Ψ(x)/dx² = -i2A1k²sin(kx)

E[i2A1sin(kx)] = (-h'/2m)(-i2A1k²sin(kx))

E = (h')²k²/2m

ii.)

En = p²/2m

= h²/2mλ²

= h²k²/2m(2π)²

= (h')²n²π²/2mL²

= n²π²(h')²/2mL²

E = En - E1

= n²π²(h')²/2mL² - π²(h')²/2mL²

= π²(h')²/2mL²(n-1)

λ = hc/E

= hc/[π²(h')²/2mL²(n-1)]

= hc2mL²(n-1)/π²(h')²

= 8cmL²/h(n-1)

iii.) ???

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