Harmonic oscillator derivation of wave functions

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Homework Help Overview

The discussion revolves around deriving wave functions for a harmonic oscillator, specifically focusing on the explicit expressions required in part (iv) of the problem. Participants are exploring the mathematical framework of quantum mechanics related to this topic.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about where to begin and mentions a concern about making assumptions. They seek guidance on deriving wave functions. Another participant provides a mathematical expression related to the Hamiltonian and suggests that expressing momentum and position operators in the x-basis leads to a differential equation for the wave function. The original poster then confirms they have derived a second-order differential equation but is unsure how to proceed with solving it.

Discussion Status

The discussion is active, with participants engaging in mathematical reasoning and exploring the derivation process. Some guidance has been offered regarding the formulation of the differential equation, but there is no explicit consensus on the next steps for solving it.

Contextual Notes

The original poster references a PDF with their question and previous answers, indicating that there may be additional context or constraints not fully detailed in the thread. There is also an implication that the topic is commonly covered in quantum mechanics literature, suggesting a reliance on external resources for foundational understanding.

jtaa
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here is a link to the pdf file with my question and answershttp://dl.dropbox.com/u/2399196/harmonic%20osc.pdf

i'm not sure where to start, because i don't want to assume anything that i haven't been given.
i'm stuck on part (iv) where i have to derive explicit expressions for 2 wave functions.

thanks!
 
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Well, from your 2 expressions for [itex]H[/itex] you have

[tex]\left( \frac{1}{2}\hat{p}^2+\frac{1}{2}\hat{x}^2\right) \Psi_0(x)=\hbar\left(\hat{N}+\frac{1}{2} \right)\Psi_0(x) = \frac{\hbar}{2}\Psi_0(x)[/tex]

So, if you express [itex]\hat{p}^2[/itex] and [itex]\hat{x}^2[/itex] in the [itex]x[/itex]-basis, you will have a differential equation you can solve for [itex]\Psi_0(x)[/itex]
 
ok i get a second order diff equation that looks like this:

http://dl.dropbox.com/u/2399196/2orderdiff.png

but how do i solve that?
 
Last edited:
This is a topic covered in many quantum mechanics and mathematical methods texts. I'd suggest you start there.
 

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