A quantity from schrodinger

  • Thread starter Identity
  • Start date
  • #1
Identity
152
0
In a paper by schrodinger, he uses [tex]\Delta_p^\frac{1}{2}[/tex], and [tex]\Delta_p^{-\frac{1}{2}}[/tex] in a particular equation:

[tex]\Delta_p^\frac{1}{2} \sum_l \frac{\partial}{\partial q_l}\left(\Delta_p^{-\frac{1}{2}}\sum_k a_{lk} \frac{\partial \psi}{\partial q_k}\right)+\frac{8\pi^2}{h^2}(E-V)\psi = 0[/tex]

which he says is "well known from Gibbs' statistical mechanics". Could anyone tell me what particular quantities are, and where I could possibly read more about them (without having to start at the beginning of statistical mechanics)?

Thanks
 

Answers and Replies

  • #2
Physics Monkey
Science Advisor
Homework Helper
1,363
34
Hi Identity,

I believe Schrodinger is referring to the Laplacian in a curved space or in a non-Euclidean coordinate system. Although Gibbs may have considered this in the context of certain statistical mechanics systems (like rotors, etc), it doesn't have any necessary connection to statistical mechanics. You may read more about this here http://en.wikipedia.org/wiki/Laplace–Beltrami_operator

I believe [tex] \Delta_p = 1/|g| [/tex] (the determinant of the metric) and [tex] a_{i j} = g^{i j} [/tex] (the inverse metric) to convert between your notation and that of wikipedia. If you don't think this is what you want or if you're still confused give another shout.
 
  • #3
Identity
152
0
Thanks man :D

I don't really know much about metrics, but I kind of understand it as a Laplacian in curved space. Schrodinger was using generalised coordinates before, so I guess this would just be the laplacian in those coordinates
 

Suggested for: A quantity from schrodinger

Replies
1
Views
269
Replies
3
Views
494
Replies
5
Views
489
  • Last Post
Replies
1
Views
543
  • Last Post
Replies
4
Views
878
  • Last Post
Replies
2
Views
413
Replies
17
Views
868
Replies
1
Views
318
Replies
6
Views
514
Top