In quantum mechanics, the behaviour of a charged oscillator in an electric field can be described using the Schrödinger equation
F = qe
F = - dV/ dX
dv = - Fdx = - qEdX
V = - qEX
When a charged oscillator in electric field E its Hamiltonian operator.
H' = - qEx = 𝛼 x
Energy eigen value of Ho are
ℏw(n+ 1/2)
Then the eigen values of H are
Direct approach
Using Perturbation theory
First order correction to energy
Second order correction
H'= 𝛾x³
Since the term containing equal power of a and a^+ contributes non zero value.
Solving these equations and understanding how the electric field affects the energy levels and wave functions of the system are all necessary for quantum mechanically analysing the charged oscillator in an electric field. The details of the potential and the nature of the electric field shape the solution.
This note is a part of the Physics Repository.