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 *ξ*^{2} 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.

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Tags: Friday Physics, Ground State, Ground State Energy, Harmonic Oscillator, Hermite Polynomials, physics, Quantum Harmonic Oscillator, Quantum Mechanics, Schrödinger Equation, Zero-Point Energy

This entry was posted on January 16, 2009 at 5:21 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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