Why we cannot solve quantum sho directly by fro envious series.

In summary, when solving the quantum harmonic oscillator by using the Schodinger equation, we first look at the large x behavior and use a substitution to approximate it. Then, we use the series method to solve the equation in terms of u(x). There is a question about whether we can use power series or Frobenius series at the beginning, but ultimately, the only way to find out is to try it ourselves.
  • #1
wowowo2006
13
0
I have just learned the quantum harmonic oscillator
When we start to Solve the schodinger equation. y"= (K - x^2) y
We look at large x behaviour and use a substitution y = u(x)* exp(-x^2 /2)
to approximate the large x behaviour first,
then we use series method to solve the equation which is in terms of u(x)

I wonder if we can use power series or froenvious series to solve the equation at the very beginning
 
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  • #2
If you don't believe your teachers or the textbook authors, there is no reason for you to believe us.
You only have one option remaining: try it and see ;)
 

1. Why is solving quantum sho directly by frozen series difficult?

Solving quantum sho directly by frozen series is difficult because it involves a complex mathematical problem known as the "many-body problem." This problem arises when trying to describe the behavior of a system with multiple interacting particles, which is the case in quantum sho.

2. What is the significance of using frozen series in solving quantum sho?

Frozen series, also known as perturbation theory, is a mathematical approach used to approximate solutions to complex problems. In the case of quantum sho, it allows us to solve for the energy levels of the system by breaking it down into simpler parts and then adding these solutions together.

3. Can we not solve quantum sho directly without using frozen series?

While it is possible to solve quantum sho without using frozen series, it becomes increasingly difficult as the number of particles in the system increases. This is because the calculations become more and more complex, making it impractical to solve directly.

4. Is there a limit to how accurate frozen series can be in solving quantum sho?

Yes, there is a limit to the accuracy of frozen series in solving quantum sho. This is due to the fact that it is an approximation method and does not take into account all possible interactions between particles. Therefore, the more terms we include in the series, the more accurate our solution will be.

5. Are there other methods for solving quantum sho besides frozen series?

Yes, there are other methods for solving quantum sho such as numerical techniques and variational methods. These methods may be more accurate and efficient in certain cases, but frozen series remains a commonly used method due to its simplicity and effectiveness for many systems.

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