A quantity from schrodinger

[Identity]

In a paper by Schrödinger, he uses $$\Delta_p^\frac{1}{2}$$, and $$\Delta_p^{-\frac{1}{2}}$$ in a particular equation:

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

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

 

[Physics Monkey]

Hi Identity,

I believe Schrödinger 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 $$\Delta_p = 1/|g|$$ (the determinant of the metric) and $$a_{i j} = g^{i j}$$ (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.

 

[Identity]

Thanks man :D

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