What is a quantum harmonic oscillator?

1 Answer
Nov 5, 2016

The quantum harmonic oscillator is essentially a two-body problem consisting of two solid spheres connected by a spring.

A basic version of it is called the simple harmonic oscillator (SHO), in which the spring has no dampening factor (no anharmonicity constant):

http://hyperphysics.phy-astr.gsu.edu/

Since there is no dampening factor, the energy levels are evenly spaced, separated by #ℏomega#, or #hnu# (the anharmonic oscillator would have quadratic convergence of the energy levels). Furthermore, the lowest energy level is #1/2ℏomega#, and not #0#.

The Hamiltonian operator for the SHO system in one dimension is:

#color(blue)(hatH_"SHO") = hatK + hatV#

#= -ℏ^2/(2mu)d^2/(dx^2) + 1/2kx^2#

#= [-iℏd/(dx)]^2/(2mu) + 1/2kx^2#

#= color(blue)(hatp^2/(2mu) + 1/2kx^2)#

where #mu = (m_1m_2)/(m_1 + m_2)# is the reduced mass, #hatp# is the momentum operator, #k# is the force constant, and #x# is the relative displacement from equilibrium. #hatK# and #hatV# were the kinetic and potential energy operators.

The normalized wave function for the #upsilon#th energy level in general is the product of a Hermite polynomial and a decaying exponential.

#color(blue)(psi_(upsilon)(x) = N_(upsilon)H_(upsilon)(sqrtalphax)e^(-alphax^2"/"2))#

where:

  • #N_(upsilon) = [1/(2^(upsilon) upsilon!)(alpha/(pi))^"1/2"]^"1/2"# is the normalization constant.
  • #H_(upsilon)(sqrtalphax) = (-1)^(upsilon)e^(-alphax^2)d^(upsilon)/(d (sqrtalphax)^(upsilon))[e^(-alphax^2)]# is the Hermite polynomial.
  • #alpha = sqrt((kmu)/(ℏ^2))# is a variable defined for convenience of expressing the function.

Applying the variational method on some trial wave function, #psi_0(x) = Ne^(-cx^2)#, where #N = ((2c)/pi)^"1/4"# upon normalization, would give you:

#color(blue)(E_0 = 1/2ℏomega)#

which is in general,

#bb(E_(upsilon) = ℏomega(upsilon + 1/2))#,

as shown in the first image.