Quantum Harmonic Oscillator & Schrodinger's Equation
Let F be some operator acting on a vector x. If F does not change the orientation of the vector x, x is an eigenvector of the operator, satisfying the equation
F(x) = λx
where λ is a real or complex number, the eigenvalue corresponding to the non-zero eigenvector x.. Thus the operator F will only change the length of the vector (not the orientation) by a factor given by the eigenvalue. Find the details here.
In terms of basic physical interpretation, the Hamiltonian represents the total kinetic and potential energy of the system traditionally denoted by T and V respectively. Hamiltonian mechanics was formulated starting from Lagrangian mechanics.
H = T + V
Let's apply this on the Harmonic Oscillator.
From the Fig. 1,
V = W = kx2/2
Fig. 1: Spring Potential Energy [1]
T = 1/2 mv2 , where v is the velocity
In terms of momentum, it becomes
T = 1/2 (mv)2/m = 1/2p2/m
Therefore,
Hence, p = mdx/dt & F = dp/dt = -kx (Hooke's Law)
The ground state energy for the quantum harmonic oscillator can be the minimum energy allowed by uncertainty principle (from [1]).
In the transition to wave equation, the physical variables take the form of operators i.e.
The energy becomes the Hamiltonian operator which acts upon the wave function to generate the evolution of the wavefunction in time and space.
or, H
(x) = E (x)
This is the Schrodinger equation which gives the quantized energies of the system. The permissible solutions of the equation are restricted by certain conditions expressed by mathematical equations called eigenfunctions. So, En are the only possible energy values or the eigenvalues associated with the quantum state or eigenvector (energy levels) that satisfied the equation.
Refer to the solution of the Schrodinger equation for the hydrogen atom.
References:
[1] HyperPhysics