Ladder operators and a 1-D QHO

  • Thread starter gronke
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  • #1
gronke
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Homework Statement


ZPryy.jpg



Homework Equations


sSbrM.png



The Attempt at a Solution


I solved part a) correctly, I believe, giving me

ψ = e[itex]^{-(√(km)/\hbar)x^{2}}[/itex]

and a normalization constant A = ((π[itex]\hbar[/itex])/(km))[itex]^{-1/4}[/itex]

I'm having difficulty with part b. I'm not exactly sure how I create a linear combination. I think I'm supposed to write the linear combination like so:

a + a[itex]^{+}[/itex] = 2(k/2)[itex]^{1/2}[/itex]x

(a + a[itex]^{+}[/itex])(2k[itex]^{-1/2}[/itex]) = x

However, I am not sure how I use the operator methods to show that xψ0 is the first excited state.

If anyone could nudge me in the right direction, I'd be eternally grateful! Thank you.
 

Answers and Replies

  • #2
vela
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What do you get when you calculate ##x\psi_0## with the expression for x you have in terms of the raising and lowering operators?
 

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