1D Infinite Potential Well

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Homework Statement



Find the ground and first excited state eigenfunctions of for the 1D infinite square well with boundaries -L/2 and +L/2

Homework Equations


$$\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x) = E\psi(x)$$

The Attempt at a Solution


Okay so I know how to solve it and get that $$\psi_1(x) = \sqrt{\frac{2}{L}}\cos{\frac{\pi x}{L}}$$. Next, I also know that $$\psi_2(x) = \sqrt{\frac{2}{L}}\sin{\frac{2\pi x}{L}}$$ one could reason this by arguing that each eigenfunction should have ##n-1## nodes. However, what is a more mathematical reasoning to ##\psi## for a given excited state. I am sure it is quite simple, I just can't seem to see it.
 

Answers and Replies

  • #2
vela
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I'm not sure exactly what your question is. If you solve the Schrodinger equation with the appropriate boundary conditions — that is, solve the math problem — those are two of the solutions you get.
 

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