Perturbation Theory: Quantum Oscillator Problem | Quantum Science Philippines

## Perturbation Theory: Quantum Oscillator Problem

by ANCELIE C. ROSALES

In quantum mechanics, the perturbation theory is a very important mathematical tool which is used to approximate physical quantities that describe complicated quantum systems based on our knowledge on the simpler ones. It tells us how to correct the solutions to the unperturbed or undisturbed problem to approximately account for the influence of the perturbation, as long as the perturbation is small compared to the unperturbed Hamiltonian.

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.

Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field E so that the potential energy is shifted by an amount H’ = – qEx.

a) Show that there is no first-order change in the energy levels and calculate the second-order correction.

Solutions:

The first-order change in the energy levels with this given perturbation, H’ = -qEx , is found using the fundamental result of the first-order perturbation theory which states that the change in energy is just the average value of the perturbation Hamiltonian in the unperturbed states:

.

Substituting the given perturbation into the equation, we get

where n is the nth eigenfunction. Employing the ladder operators (raising and lowering operators, a+ & a, respectively) on x as in the equation,

and we get the inner product

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.

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.

### 26 Responses to “Perturbation Theory: Quantum Oscillator Problem”

1. Engr. Ronald Castillon Says:

I heard about this Perturbation theory before but it was not quite interested for me. As i read in your article this time, i didn’t expect that the nature and equations of the theory will goes like that. It’s great. You did very well. You can be one of the leading physicist of the country.
Congratulations!

2. deby Says:

the illustration is clear giving it a nice illustration for people not very knowlegdeable in the filed of quantum mechanics but but the part of the first correction, there may arouse some questions pertaining to where such equation was derived.

3. Normie Jean Sajor Says:

Congratulations!You explained very well the perturbation theory, brief but understandable. You gave all the details clearly in order for us to understand. You did a great job!

4. John Paul Aseniero Says:

I really like the way you explain the problem regarding the perturbation theory. As a physics enthusiast, it is very important also to know the details and applications of relevant physics concepts especially in quantum mechanics. Based on your article even a none physics major could understand its general concept. Congratulations for that nice achievement.

5. Hananish Joy Odarve Says:

You have presented the perturbation theory well which is nice because it helps those who are new to the field to grasp the idea of the theory and you have shown how to use the perturbation theory in determining the correction in a perturbed system.

keep it up.

6. Sandra L. Manulat Says:

The discussion on the perturbation theory is very detailed and it gave a great deal of information. I think, though that the equation be numbered or labeled for reference. Thank you!

7. Majvell Kay Odarve Says:

Hello Ms. Ann,

Your presentation of the perturbation theory is simple and clear. A beginner to the idea would be easily guided through your presentation of the topic. However, I would like to comment on the presentation of the equations. I think it would be better to place equation numbers for easy reference. And also there is just a very minor error when the second order correction is derived. When the lowering operator was acted on |n>, the answer should have |n-1> and not |n>… this was corrected on the next lie though, but it might be confusing for others. Thank you for sharing this and God bless…

8. Gibson Tipdas Maglasang Says:

Congratulations Ms. Rosales for a job well done. You have contributed a lot to the readers specially those people seeking the correctness, application and uses of the perturbation theory. I am also very well enlightened in this topic.

However, it should also be noted and emphasized the usage of perturbation theory on degenerate and non-degenerate states.

Congratulations once again and God bless you.

9. Lutchie Dyan S. Mendoza Says:

Good day Ma’am Ann=)
The over all presentation is understandable, needed information(i.e equations, steps, mathematical properties regarding the topic) were sufficiently provided, and i congratulate you for that(“,). The article also tells us the importance of perturbation theory in quantum mechanics as well as its application. Thumb’s up for you ma’am!;-)

10. Michael Jabines Says:

This article is very simple and understandable that even a non-major physics can understand and even I can also understand the concept. Its solution is well organized and you can easily grasp the idea and topic even if without prior knowledge. One thumbs-up for this article.

11. Edmar G. Pantohan Says:

This is a very good post. Using schrodinger equation we can solve the energy 2nd order correction and it turns out that it is consistent with the perturbation theory.I learned another approach from this post. Nice.

12. carlo paul p. morente Says:

A well organized presentation on the basic of perturbation theory. I hope to see more complicated examples of perturbed system posted in this site. Nevertheless this post is very helpful upon my understanding of the topic.

13. karl patrick casas Says:

Though we have not encountered some of these methods, they are introduced to the readers very well. I would like to suggest equation numbers in case of citations for comments. Also, it’s amazing to think that we could also use the ladder operators here for harmonic oscillator that we have learned in undergraduate. Thanks anyway!

14. Rommel J. Jagus Says:

The illustration of the idea is simple and elegant. But, I would like to comment on the details of the solutions especially on how Ms. Ann derived the second-order correction. This might confuse the beginners of the topic.

15. Eric M. Alcantara Says:

I’m really impressed on the way you present the topic. I just like to comment on the way you present your solution. It is quite confusing for the beginners of the topic. I think it would be better if you explain the need of employing such tricks like changing the variables for better understanding.

16. Christine Marie Ceblano Says:

Hello ma’am! You forgot to include “m=n-1” in getting the 1st-order correction you only put “since m=n+1”. It’s also better if you put number in every equation so it would be easy to refer and substitute equations to equations. You should at least explain why you choose m=n+1 in simplifying your answer for second-order correction to the energy levels. Overall, I like your solution ma’am. Nice work.

17. Liza Dangkulos Says:

The solution to each example is clearly explained.
The examples truly showed how perturbation theory
can be applied. The article is articulately written.
Congrats ma’am ann!

18. Lotis Racines Says:

Going through with your presentation, I think you accidentally forgot to put |n-1> in the second equation of your article, 3rd page. You put |n> in the 2nd term instead of |n-1>. What’s confusing is on the 4th page of the article. I admit that not all cases on perturbation is easy to “sink in” in our minds. I’m just confused of the way terms are changed. Why a factor of 2 isn’t included in changing the variables?
(But I believe your solution is right, and same with your whole presentation.) Nevertheless, your article is but interesting. I admire your patience on solving this case of perturbation. Quite difficult but you made it. Cheers!

19. Edwin B. Fabillar Says:

Thanks Ma”am Ann for making your article simple and understandable. Though Perturbation Theory is just like a pacific ocean to me but I really appreciate the detailed solution in your example. Challenging approach! Keep up the good work and GOD bless.

20. catherine therese quinones Says:

I find this article very interesting. The solutions were written in an orderly manner though it is highly mathematical. However I have this question, Why did you change variables in part b of your solution? What is your basis of doing such? Nevertheless, you got the right answer in the end. Very amazing indeed!

21. Marichu T. Miscala Says:

oscillator problem is one of the few good examples to study in order for someone to understand how a perturbed system actually behaves. It’s good to know that the process of solving such problem is presented here and thank you for posting this online. People can now have an easy access and use it for reference for their works in perturbed system-related topics and quantum mechanics in general. Although there are some clerical errors and missed equations, overall the article is very informative and useful.

22. ancelie Says:

hi all,

thanks a lot for your comments..i was flattered by all of your compliments and remarks…its elating that you learned something from my post…yah..i notice some clerical errors ..am sorry about that…but i will try to correct them and make the presentation of my solutions more detailed and more comprehensible…again, thank you very much.

23. Thu Vien So Says:

I find this article very interesting. The solutions were written in an orderly manner though it is highly mathematical. However I have this question,

24. Tol Pagna Says:

oh it is so good for perturbation theory. you explained in detailed .It is useful for theoretical physicist

25. Misho Says:

How to slove 1st and 2nd perturbation of 1st excited state of LHO?

26. William M Says:

This is a wonderful and concise treatment of perturbation theory with ladder operators, and the first such treatment I have been able to find. Thank you for the explanation!