The perturbation theory is best applied in the determination of the approximate correction to the energy levels and eigenstates after a certain perturbation is introduced to a real quantum system. To understand this deeply, let us look at this example.

which can be written further as

.

We recall that it was shown in the properties of quantum oscillators that

and

and substituting these to our equation , we then get

.

We also have the relation that

.

Since m = n+1 (not equal to n), then we now have

so,

Finally,

.

Thus, the first-order correction is indeed equal to 0.

For the second-order correction, it is found using the fundamental equation of the second order perturbation theory which is

where

.

Following the same procedure as in getting the first-order correction in simplifying the numerator of the equation, that is, using the raising and lowering operators, we get

and simplifying, we now have

.

With the delta function, it is important to note that

,

and the above equation becomes

.

Substituting this to our fundamental equation, it becomes

and for a harmonic oscillator,

and

.

Then, our second-order equation becomes

.

Simplifying the numerator, we now have

.

It is important to note that

So, now we have the equation,

.

Finally,

.

This is the second-order correction to the energy levels.

b) The Schrödinger equation (SE) can be solved exactly in this case by a change of variables. Find the exact energies and show that they are consistent with the perturbation theory approximation.

Solutions:

The Schrödinger equation for this potential is:

By change of variables, we let

.

Considering first the potential part of the SE and changing the variables, we have

.

Thus, substituting this to our SE, it becomes,

and rearranging terms, we get

which is the SE for simple harmonic oscillator in the variable x’.
We know that,

and finally

.

In the above equation, the second term is the second order correction to the energy level and since we found that the first order correction is zero, thus this solution is consistent with the perturbation theory approximation.

About the author:

Ann finished her BS Physics degree at MSU main campus in Marawi City and is pursuing now a graduate degree at MSU-IIT, Iligan City. She is into performing experiments in Material Science and hopes to become one of the experimental physicists of the country someday.

In this post we investigate the properties of a quantum oscillator by using an algebraic tool in quantum mechanics called ‘ladder operators’. Using the ladder operator it becomes easy to find the following properties for a quantum oscillator in a given energy level:  the average position and momentum and the square of these values as well as the average kinetic energy of a simple harmonic oscillator. In formal notation, we are looking for the following respective quantities: , , , and .

Some discussion about ladder operators

We begin by introducing the so-called ladder operators. There are two types: the raising operator, symbolized by , and the lowering operator, symbolized by .  For reasons that will be evident later, the two are also called creation and annihilation operators respectively.

The ladder operators come from the roots of the Hamiltonian for a simple harmonic oscillator. The Hamiltonian is given by

which can be rewritten as

We then take the roots or factors of the expression inside the brackets. We should note however that we are dealing here with operators which do not commute. Simple algebraic factoring yields two roots:

To be clear, we rewrite the two roots separately below as

where the momentum operator is given by

To be able to find the expectation values of (position operator) , (momentum operator) and (kinetic energy),  we express the position operator and momentum operator in terms of the ladder operators and . We add the two roots in order to get the expression for the position operator in terms of the ladder operators as

and then by subtracting the lowering from the raising operator gives the expression for the momentum operator as

Now we consider the product of the two ladder operators. Since operators do not commute there are different results when we change the order when multiplying both operators:

from which we derive the expression for the Hamiltonian as
.
The term in the braces is just the dimensionless Hamiltonian operator which is more convenient for our purposes:

This Hamiltonian operator can be expressed differently by multiplying the ladder operators in a different order. Then we get

and its dimensionless counterpart is just

The Schroedinger eigenvalue equation for a simple harmonic oscillator will then yield

hence it follows that

Now we can operate these ladder operators to and see how the eigenvalues behave. We write down the action of the lowering operator as
.
Its adjoint is given by

Multiplying the latter 2 equations gives us

since is the eigenfunction is normalized and is given, then

we finally arrive at the result that for the raising operator we have

And also for lowering operator the result is
.

When using ladder operators it is imporatnt to note that orthogonality condition must be satisfied. The orthogonality condition  is given by,

Finding the properties of a quantum oscillator

Using the preceding results, we can now find the desired solutions to the problem initially given at the top of this post; which are
a. In finding , we proceed as follows using the derived expression for the position operator in terms of the ladder operators. We note that  where <n| is any eigenvector. So we write,

b. we can find  in the same manner

c. Finding  involves a similar algebraic procedure

d. We repeat the same algebraic procedure in finding for .

e. Finally we can derive the expectation value for the kinetic energy, <T> in a straightforward way as

.

Relation to Heisenberg’s Uncertainty Principle

The quantum oscillator we have described above obeys the Heisenberg uncertainty principle.

We use the results from a) to d) above in proving these statements.

Using the above results, it is easy to see that

We thus have seen that the quantum harmonic oscillator satisfies the Heisenberg uncertainty principle.

About the Author:

SIMON JUDE BURGOS is a graduate student in Physics at the Mindanao State University-Iligan Institute of Technology (MSU-IIT) in Mindanao, Philippines. He goals to work in research facilities in the field of medical physics. He will be finishing his masters degree soon and hope to go on to Ph.D. physics research in the near future.