(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a particle of mass m moving in a one-dimensional potential,

[tex]V(x)=\infty[/tex] for [tex]x\leq0[/tex]

[tex]V(x)=\frac{1}{2}m{\omega^2}{x^2}[/tex] for [tex]x>0[/tex]

This potential describes an elastic spring (with spring constant K = m[tex]\omega^2[/tex]) that can be extended but not compressed.

By reference to the solution of the 1-D harmonic oscillator potential sketch and state the form of the valid eigenfunctions, and state the corresponding eigenvalues, for the ground and first excited state.

2. Relevant equations

[tex]E=(n+\frac{1}{2})\hbar\omega[/tex]

3. The attempt at a solution

https://www.physicsforums.com/attachment.php?attachmentid=23025&stc=1&d=1263339354

Here is the potential sketch

Since the potential V(x)=[tex]\infty[/tex] for [tex]x\leq0[/tex], The potential at x=0 is infinity. So the wave function at x=0 must equal 0. If it were anything else the particle would have infinite energy. Therefore eigenfunctions must pass through the origin.

http://131.104.156.23/Lectures/CHEM_207/CHEM_207_Pictures/p75a_72gif

From the graph, n=1 and n=3 are the lowest eigenfunctions to pass through the origin.

Ground state eigenfunction n=1 [tex]{E_1}=(1+\frac{1}{2})\hbar\omega=\frac{3}{2}\hbar\omega[/tex]

1st excited state eigenfunction n=3 [tex]{E_2}=(3+\frac{1}{2})\hbar\omega=\frac{7}{2}\hbar\omega[/tex]

Can someone tell me if this is right?

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# 1-D harmonic oscillator problem

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