What is a quantum harmonic oscillator?

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):

Since there is no dampening factor, the energy levels are evenly spaced, separated by ℏomega, or $h \nu$ (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:

$\textcolor{b l u e}{{\hat{H}}_{\text{SHO}}} = \hat{K} + \hat{V}$

= -ℏ^2/(2mu)d^2/(dx^2) + 1/2kx^2

= [-iℏd/(dx)]^2/(2mu) + 1/2kx^2

$= \textcolor{b l u e}{{\hat{p}}^{2} / \left(2 \mu\right) + \frac{1}{2} k {x}^{2}}$

where $\mu = \frac{{m}_{1} {m}_{2}}{{m}_{1} + {m}_{2}}$ is the reduced mass, $\hat{p}$ is the momentum operator, $k$ is the force constant, and $x$ is the relative displacement from equilibrium. $\hat{K}$ and $\hat{V}$ 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.

$\textcolor{b l u e}{{\psi}_{\upsilon} \left(x\right) = {N}_{\upsilon} {H}_{\upsilon} \left(\sqrt{\alpha} x\right) {e}^{- \alpha {x}^{2} \text{/} 2}}$

where:

• N_(upsilon) = [1/(2^(upsilon) upsilon!)(alpha/(pi))^"1/2"]^"1/2" is the normalization constant.
• ${H}_{\upsilon} \left(\sqrt{\alpha} x\right) = {\left(- 1\right)}^{\upsilon} {e}^{- \alpha {x}^{2}} {d}^{\upsilon} / \left(d {\left(\sqrt{\alpha} x\right)}^{\upsilon}\right) \left[{e}^{- \alpha {x}^{2}}\right]$ 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, ${\psi}_{0} \left(x\right) = N {e}^{- c {x}^{2}}$, where $N = {\left(\frac{2 c}{\pi}\right)}^{\text{1/4}}$ 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.