Quantum Mechanics, Schrodinger equations and energy eigenvalues

[Badger01]

How do you find an expression for the energy eigenvalues from the TISE (Time Indipendant Schrödinger Equation) for a given potential.

e.g. why is:
E = (N + 1) hbar*omega
an expression for the energy eigenvalues for a potential of:
V = 1/2*m*omega²x²
??

I really have no idea where to start with this.

thanks for any help
(sorry about the lack of simbols, i couldn't get them to work)
for a better setup see
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html
the section on energy eigenvalues

 

[Steely Dan]

How do you find an expression for the energy eigenvalues from the TISE (Time Indipendant Schrödinger Equation) for a given potential.

e.g. why is:
E = (N + 1) hbar*omega
an expression for the energy eigenvalues for a potential of:
V = 1/2*m*omega²x²
??

I really have no idea where to start with this.

thanks for any help
(sorry about the lack of simbols, i couldn't get them to work)
for a better setup see
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html
the section on energy eigenvalues

There is not a straightforward method for cleanly determining the energy eigenvalues for an arbitrary potential, because not every arbitrary potential results in a TISE that has a closed-form solution. The most general way to go about it, though, is simply to solve the differential equation (if you can). Once you have the wavefunction(s) that solves the TISE, you then substitute it back into the TISE and evaluate the Hamiltonian acting on the wavefunction. If it is an eigenfunction of the Hamiltonian, the left side of $$H \psi = E \psi$$ turns into some number times ψ, so the wave-function cancels out and you're left with the energy.

But there are often other methods for determining the energy levels without explicitly finding the wave-functions. In the case of the harmonic oscillator potential, that would be the ladder operator method. It's in pretty much any introductory level QM book (e.g. Griffiths).