by JESSICA IRISH LOPEZ
The use of ladder operators in simple quantum oscillators was discussed by Simon Jude Burgos in an earlier post. We can further look at other quantum oscillators properties using again the ladder operator concept.
For example, we can derive the nth eigenstate from the ground state by applying the creation or raising operator n times, i.e.
To prove the above statement, we consider an eigenvector where . We operate the raising operator, , to , -times. By applying the operator this way, we will always get . To clear things out, we check the behavior of the eigenvector when we operate to it, considering several values of ,
By doing the above steps as alpha goes to n we will get,
Further Use of Ladder Operators
Using the ladder operator concept allows us to calculate, in a few algebraic steps, the expectation values of more complicated quantities such as:
From the definition of ladder operators as given earlier, it allows us to express the position operator in terms of the raising operator and the lowering operator and this is given by
As the problem requires us the cube of the position operator so we can then simply write,
Substituting the latter equation to the given problem:
The left hand side can also be written as
The next step we need to do is to operate each term to the state |n=2>. We also remember the result of the ladder operators when operated to an eigenvector yields,
With these we can easily obtain the following results:
It will just be a simple step to get the inner product of each of these terms with the state |n=3>. However, before applying the results obtained above, we can directly check if the terms will satisfy the orthogonality condition
It is easy to see that for the orthogonality condition to be satisfied the only terms that should remain are those with eigenvector |3>. Therefore,
So that finally we obtain,
which is the desired answer.
About the Author:
JESSICA IRISH LOPEZ is a graduate student in Physics at the Mindanao State University-Iligan Institute of Technology (MSU-IIT) in Mindanao, Philippines. She will be finishing his masters degree soon and hope to go on to Ph.D. physics research in the near future.