Come One, Come All, and Cast Your Vote

[bluestar]

Vote on which university has the correct normalized quantum harmonic oscillator equation!

Is it University #1
Georgia State University and the Hyperphysics site presented by the physics department.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html#c1

$$\psi_n \left( y \right) = \left( \frac {\alpha}{{\pi}} \right) ^ \frac{1}{{4}} \frac{1}{{\sqrt{2^nn!}}}H_n\left(y\right)e^{\frac{-y^2}{{2}}}$$
Where $$y = \sqrt{\alpha} x$$
and $$\alpha = \frac{m\omega}{{\hbar}}$$


Is it University #2
Oxford University with a paper from the Physical Chemistry department
http://physchem.ox.ac.uk/~hill/tutorials/qm2_tutorial/sho_series.pdf

$$\psi_v\left(y\right) = \frac{1}{{\sqrt{\pi}{{2^vv!}}}}H_v\left(y\right)e^{\frac{-y^2}{{2}}}$$
Where $$y = \sqrt{\frac{m\omega}{{\hbar}}}x$$


Or is it University #3
University of Glasgow with a paper from the Physics department
http://www.physics.gla.ac.uk/~dmiller/lectures/P409M_Miller_2008_section4.pdf

$$\psi_n \left( x \right) = \sqrt{\frac{\alpha}{{\sqrt{\pi}2^nn!}}}H_n\left(ax\right)e^\frac{-\alpha^2x^2}{{2}}$$
Where y = $$\alpha x$$
and $$\alpha = \sqrt{\frac{m\omega}{{\hbar}}}$$


The normalizing constant is different for each equation so they all can not be the same.
When you cast your vote it would be helpful to others if you would justify your selection.
Thanks for participating.

 

[jtbell]

Check it yourself! Evaluate the normalization integral for each one and see which one(s) give you 1 for the result.

Of course, you'll have to pick a particular value of $n$ (or $\nu$ as the case may be) and use the corresponding particular Hermite polynomial.

 

[bluestar]

Are you saying that I should evaluate each function for a specific n (or v) and the one that results in a value of 1 has the valid normalization constant? If so, I can do that.

Or

Are you suggesting evaluate the normalization integral. That I won’t be able to do for a couple of them do not provide that information. That is why I am asking for opinions.

 

[bluestar]

I did some more checking around and found the answer to my question.

Shown below is the normalized harmonic equation from Michael Fowler at the University of Virginia. As can be seen the structure of the normalization constant is very similar to Georgia States Hyperphysics equation. After closer examination of the Hyperphysics equation I found the two equations to be equal. I also reexamined the Glasgow equation and found it was also equal to the other two. The confusion arose because the alphas were defined differently between Hyperphysics and Glasgow.

University of Virginia - Michael Fowler page

$$\psi_n \left( y \right) = \left( \frac {m\omega}{{\hbar\pi}} \right) ^ \frac{1}{{4}} \frac{1}{{\sqrt{2^nn!}}}H_n\left(y\right)e^{\frac{-y^2}{{2}}}$$
Where $$y = \frac{x}{{\alpha}}$$
and $$\alpha = \sqrt{\frac{\hbar}{{m\omega}}}$$

I am still puzzled as to why the prestigious Oxford equation does not match the others.
If you have any thoughts on the puzzle chime in.

I am also puzzled why 91 readers didn’t catch my initial error including JTBELL mentor. :-)