The infinite potential well isnt silly at all. In fact its probably the most basic bound state problem and its extremely illustrative. The basic ideas that come up here can then be used in lots of other situations...
Anyway here's how to solve the infinite potential well problem.
Method 1:Using De Broglie's Principle:This is probably the simplest and most intuitive way to tackle the problem. We just try and look for the standing wave wavelengths that can exist inside a box of length L. So if the potential well has a lenght L then we want
n*w/2 = L
Where w is the wavelength and n is an integer. This immediately gives us the energy eigenvalues
w = h/p
So p = nh/2L
So E = p2/2m = n2h2/(8L2m)
where m is the mass of the particle confined to the well.
Method 2:Solving the Schrodinger equation
Lets look for stationary solutions of the schrodinger equation which satisfy:
H(psi) = E*psi
H has the form
p2/2m + V(x)
where V is the potential function which is 0 in the interval (0,L) and infinite everywhere else.
This tells us that psi must be zero outside the well and we're left with the equation:
-(h-bar)2/2m d2(psi)/dx2 = E(psi)
where continuity of psi gives the homogenous boundary conditions
psi(0)=psi(L)=0
Now we get the same answer. The eigenvalues for this boundary value problem are
2m*E/(h-bar)2 = (n*pi/L)2
Which gives us again for E
E=n2h2/(8L2m)
the normalised wave function has the form
sqrt(2/L) sin(n*pi*x/L)
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