Wave Function from ground to excited

[terp.asessed]

I now understand how the wavefunction graphs look from the hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html.

However, what I fail to understand, is how the wavefunction equation was derived for the 1st excited state, from:

ground_wavelength₀ = (a/pi)^1/4 e^(-ax2/2), where a = alpha

to

excited_wavelength₁ = (a/pi)^1/4 2^1/2e^(-ax2/2), where a = alpha

If someone could explain, merci beaucoup!

 

[jtbell]

For an outline of the procedure, see here:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html
http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_18/node4.html

For more mathematical details, see here:

http://quantummechanics.ucsd.edu/ph130a/130_notes/node162.html#derive:HOdiffeq

 

[terp.asessed]

Awesome, thank you!