Allowed Eigenstates for a particle question

[Dindimin09]

 

[vanhees71]

So? What have you done? What are the relevant equations? You won't get help on this forum by just posting a problem!

 

[Dindimin09]

You will have to excuse my ignorance I am very new to Physics Forum and relatively so to QM. Please accept my apologies.

So inside the box V=0,
Thus ψ = Ae^ikx +B-ikx = Ccoskx + Dsinkx

Outside the box V = ∞ therefore ψ = 0
if ψ is continuous it must be zero at the edges of the box

→ ψ(0) = 0 and ψ(L) = 0
For ψ(0) = 0 then C = 0 therefore ψ(x) = Dsin(kx)

These boundary conditions lead to eigen functions in the form ψ(x) = Dsin(kx) and quantised values of k=n∏/L leading to quantised values of E given by:
E = hbar^2n^2/8mL^2

However if the eigenfunctions must be completely specified we normalize them:

→∫ψ*ψ dx = D^2sin^2(kx)dx=1 where D=√(2/L)= An and L=W

→ψn(z) = Ansin(n∏z/w)

Thanks for your help.