Quantum harmonic oscillator

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The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics.

Hamiltonian[edit]

The Hamiltonian for the system is the following:

This Hamiltonian is a one dimensional Hamiltonian. Here is what each of the parts of the Hamiltonian mean:

  • m is the mass of the particle
  • The first term, is the usual kinetic energy term.
  • The second term, is the potential.

The potential term is very frequently written as . This is because the spring constant k is related to the oscillator frequency via the relationship . When this is done, the Hamiltonian reads

Time independent Schrödinger equation[edit]

The time independent Schrödinger equation is

and if we project onto the position basis, we get

Substituting our Hamiltonian into the equation, we get

The constants can be pulled out in front of the bra so the Schrödinger equation now reads

Now, consider the term Recall that so

For the other term , recall that so that Putting all the pieces together, the Schrödinger equation reads

Since we are working in the position basis, we have so we finally get

which is a differential equation which can be solved for . This is of course, the wavefunction of the system in the position basis.

Solutions to the quantum harmonic oscillator[edit]

There are different approaches to solving the quantum harmonic oscillator. One of them, involves directly solving the differential equation which was obtained in the previous section. We will do this first. Afterwards, we will solve this same system with the "operator factorization method" as a way to motivate the introduction of boson operators into our quantum mechanical theory.

First, let's define a characteristic length for the quantum harmonic oscillator. We can do this heuristically by looking at the units involved in our expression.

  • has units of
  • has units of
  • has units of

Hence, the quantity has the units of length. We will call this length the "characteristic length". If we substitute into the differential equation we will get

Note also, that the units of are also energy units. We can define as the characteristic energy of the system. (In fact, later, we will find that this energy happens to be the ground state zero point energy of the quantum harmonic oscillator.) So, putting we get

To solve this equation, first consider a simpler equation which describes the behaviour of the original wavefunction in some asymptotic limit. In the regime where the energy is very low, , the wavefunction should then satisfy the differential equation

The form of this equation suggests that . Substituting

or

This differential equation can be solved in many different ways. One approach is to take

The derivatives are (note carefully the summation limits)

and similarily

Substituting all of these terms into the above yields

and vanishes term by term provided that

The most straightforward way to enforce these relationships is to set the numerator to zero. This leads to

It's a very interesting result since the energy is now constrained to take on certain discrete values.


Feedback[edit]

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