Quantum harmonic oscillator

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




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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