Schrødinger equation deriving diff equation

[fisher garry]

In the attachment I have used the derivation of the wave equation with a slight modification to derive the diff equation that when solved gives the time independent schrødinger equation. Since I have dome this myself I am worried that this might be totally wrong or a bit wrong. Could someone look through and see if they seem to believe that the derivation is ok or point out what is wrong. Thanks!

 

[bhobba]

I haven't gone through it in detail, but from a quick glance it looks ok - except for your charge density thing.

However a string has got nothing to do with Schroedinger's equation. Your equating to charge density is simply pulled out of nowhere with no physical basis whatsoever.

The most elegant derivation however is from the Lagrangian:
http://faculty.cua.edu/sober/612/Lagrangian_wave_eq.pdf

If you haven't encountered Lagrangian's yet it would really help your studies in QM. At the beginner level Susskind's book is good:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20

The true basis of Schrödinger's equation is symmetry, and you will find its correct derivation in Chapter 3 of Ballentine:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

It is however advanced so will likely be a challenge to follow at the beginner level - but persevere - you will get the gist.

Also understanding that symmetry is the rock bottom essence of much of physics is one of the truly great insights and, once understood, is quite breathtaking in its beauty. Enough said - you must experience it for yourself. Just to set the stage:
http://www.pnas.org/content/93/25/14256.full

Thanks
Bill

 

[fisher garry]

Thanks for the reply and the links. I see that the string can have nothing to do with the schrødinger equation I am just making an attempt of a derivation.

In hindsight I found out that I did not get my derivation. My problem is in the attachment. Can anyone get me out of it?

 

[bhobba]

Note Schroedinger's equation is complex - and your added bit from Maxwell's equations are real.

To see a correct derivation, as mentioned previously, read chapter 3 - Ballentine.

But if you want to see how its derived from classical analogies check out:
http://arxiv.org/pdf/1204.0653.pdf

There you will see the key assumption is complex numbers - that's the magic - and mystery of QM and why your attempt will fail.

Actually its not a mystery - Feynman sorted it out years ago - but that is another story - the link above explains it.

Thanks
Bill

 

[fisher garry]

Thanks for the reply! I guess the derivation from your link or the maxwell equation derivation will get me the answer. And that my attempt seems a bit out of bounds.