Physics Friday 55

Classically, a one-dimensional harmonic oscillator is a system with a mass under a restoring force proportional to displacement from the equilibrium position: F=-kx. The energy is , and the equation of motion  has solution , where.

Analogously, a one-dimensional quantum harmonic oscillator is a particle with Hamiltonian , where here p is the quantum momentum operator, and x the position operator. In the position basis, this is then
.
A cursory examination of the expectation for energy

shows that we can expect our energies to be non-negative.

Most quantum mechanics books and courses I have encountered address the energy eigenstates by means of the raising and lowering operators. Instead, here we will use an analytical method of finding the eigenvalues to the time-independent Schrödinger equation
.
Let us replace x and E with corresponding dimensionless variables: we define the dimensionless variables , . Then , 
the Schrödinger equation becomes
.

To find a solution to this differential equation, we consider asymptotic behavior. Namely, when the energy is small, the ξ2 dominates over ε, and our equation approaches , which hints at a solution that behaves like a Gaussian. Adopting, then, a test form
, and plugging this into our differential equation , we get
,
or, eliminating the common Gaussian factor,

Applying the power series method, we plug in  to find

Which gives recursion relation .

Now, we return to the physics of the problem, namely that the wavefunctions have to be normalized. This means that ψ→0 as x→±∞. Now, considering that
, and comparing to the series expansion of the gaussian, we see our series diverges too fast for ψ to be normalized, unless the series terminates. This requires  for some n, which gives , and as E≥0, we see that the ground state has energy ; this is known as the ground state energy, or zero-point energy.

Solving for the wavefunctions themselves requires more mathematics, and gives solutions involving the Hermite polynomials.