by SIMON JUDE BURGOS
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,
c. Finding involves a similar algebraic procedure
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.