Quantum Harmonic Oscillator necessary DE

[mjlist16]

I was reading through my Principles of Quantum Mechanics textbook and arrived at the section that discusses the quantum harmonic oscillator. In this discussion the equation ψ"-(y^2)ψ=0 presents itself and a solution is given as ψ=(y^m)*e^((-y^2)/2), similar to a gaussian function i assume. My book has given no derivation that yielded this answer it only supplied it, so my question is if an answer of this sort can be algebraically solved for, or what kind of logic would lead to such an answer. Thank you in advance for any responses.

 

[mfb]

Guessing the general shape of the solution is the easiest way here: The second derivative is the function, multiplied by y. This leads to the approach ##e^{-y^2}## (with the minus to let it get smaller for large y).
If you calculate derivatives of that, you get a lot of factors of 2y. To cancel the 2, you can modify the function to ##e^{-y^2/2}##, and then you can multiply it with a sum of y^m to get a solution.

 

[mjlist16]

Thank you, it's just hard for me to accept this guessing method as I go further into physics, i just can't help but believe that there exists a way to explicitly solve this equation. Would a series solution work for this type?

 

[Bill_K]

A power series does work, but after you factor out the exp(-y²/2). Not just any power series but a finite power series, i.e. a polynomial. Which is what you need in order to find discrete eigensolutions.

This is not guesswork - you can justify the factorization by thinking about the DE's behavior near infinity. But in practical terms the reason you don't just start chugging away on a power series is that you'll get a three-term recurrence formula.

 

[mjlist16]

thank you for your guidance I think i may have worked out something that satisfies my ocd.

 

[Nugatory]

Thank you, it's just hard for me to accept this guessing method as I go further into physics, i just can't help but believe that there exists a way to explicitly solve this equation.

It's not physics that requires this sort of guessing, it's solving differential equations that does. However, you'll come across <understatement>a lot</understatement> of differential equations in physics... so you might as well get used to it.

Some people find the term "guess" to be somewhat pejorative, and even below the dignity of any serious scientist. You can always use the phrase "thoughtful selection of a suitable ansatz" instead, and vary it by substituting "brilliant", "inspired", "clever" for "thoughtful" as needed.

I've never seen a derivation of the Schwarzschild solution that used the word "guess", but I've also never seen one that didn't start with an ansatz.

 

[Bill_K]

Some people find the term "guess" to be somewhat pejorative, and even below the dignity of any serious scientist. You can always use the phrase "thoughtful selection of a suitable ansatz" instead, and vary it by substituting "brilliant", "inspired", "clever" for "thoughtful" as needed.

If it works once, it's a trick. If it works twice, it's a method.