Griffiths quantum harmonic oscillator derivation

[Syrus]

Homework Statement

I am unsure as to a step in Griffiths's derivation of the quantum harmonic oscillator. In particular, I am wondering how he arrived at the equations at the top of the second attached photo, from the last equation (at the bottom) of the first photo (which is the recursion relation approximated for large j).

Any help?

Homework Equations

The Attempt at a Solution

 

[voko]

Substitute the approx. solution into the approx. equation. Is it satisfied?

 

[Syrus]

Is there a way to derive the solutions besides via a fortiori means?

 

[voko]

The could be some clever way, but since the exact solution is known, the approx. solution could equally have been obtained from it.

EDIT: it actually seems very simple: since for every "next" term we divide the previous by j/2, one should expect (j/2)! in the denominator, and c/(j/2)! should be fairly obvious as a possible solution.

 

[paranormal]

I am not familiar with the specific derivation of the quantum harmonic oscillator by Griffiths. However, I can offer some general insights and suggestions that may help you understand the derivation better.

Firstly, it is important to note that the quantum harmonic oscillator is a widely studied and well-understood system in quantum mechanics. Therefore, there are many resources available that may provide a more detailed explanation of the derivation, such as textbooks, online lectures or discussions with colleagues.

In terms of the specific step you are struggling with, it may be helpful to review the concept of recursion relations and how they are used in physics. Additionally, it may be beneficial to break down the equations and examine each term individually to understand their meaning and how they contribute to the overall derivation.

It is also important to keep in mind that derivations can often be complex and require a strong understanding of the underlying principles and mathematical techniques. If you are still struggling, I suggest seeking guidance from your instructor or a more experienced colleague who may be able to provide a more in-depth explanation.

Overall, understanding derivations in physics can take time and effort, so don't be discouraged if you are struggling. Keep persevering and seeking help when needed, and eventually, you will have a better understanding of the derivation.