Quantum harmonic oscillator Definition and 103 Threads

Can someone explain to me how they went from U(x)=(1/2)kx^2 to E=(n+1/2)(h/2pi)w? http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

Hi, Is there an explicit formula for the eigenfunctions of the harmonic oscillator? By explicit, I mean "not written as the nth power of the operator (ax-d/dx) acting on the ground state". Thanks.

Hi, I am wondering how i would go about calculating the canonical partition function for a system of N quantum harmonic oscillators. The idea of the question is that we are treating photons as oscillators with a discrete energy spectrum. I'm confused as whether to use Maxwell-Boltmann...