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On the derivation of Klein–Fock equation

From D. Iwanenko and L. Landau in Leningrad

Received on 8 October 1926

 

It is shown that the generalized Schrödinger equation can be obtained by a transition from the relativistic analog of Hamilton's problem.

 

From the theory of relativity, the following expression is known for the differential of the action function:

(1)$dW=-mc\,ds+{\frac {e}{c}}\varphi _{k}dx^{k},$

where

$ds^{2}=-dx_{k}dx^{k}.$

The generalized momenta are given by

(2)$p_{k}={\frac {\partial L}{\partial {\dot {x}}^{k}}}={\frac {\partial (dW)}{\partial (dx^{k})}}=+mc\,{\frac {dx_{k}}{ds}}+{\frac {e}{c}}\varphi _{k},$

$\left(L={\frac {dW}{dt}}\,\,\,{\text{the Lagrangian function}}\right).$

It follows:

$p_{k}-{\frac {e}{c}}\varphi _{k}=mc\,{\frac {dx_{k}}{ds}},$

(3)$\left(p_{k}-{\frac {e}{c}}\varphi _{k}\right)\left(p^{k}-{\frac {e}{c}}\varphi ^{k}\right)+m^{2}c^{2}=0.$^[1]

On the other hand:

(4)$p_{k}={\frac {\partial W}{\partial x^{k}}}=W\cdot \chi ,$

also

(5)$W_{,k}W^{,k}-2{\frac {e}{c}}\varphi ^{k}W_{,k}+{\frac {e^{2}}{c^{2}}}\varphi _{k}\varphi ^{k}+m^{2}c^{2}=0.$

Let's make an assumption similar to Schrödinger's^[2] that equation (5) is the limit of a linear equation for

$\psi =e^{{\frac {i}{\hbar }}W}$^[3] as $\hbar \rightarrow 0$

1. See also P. A. M. Dirac, Proc. Roy. Soc. (A) 111, 405, 1926.
2. E. Schrödinger, Ann. d. Phys. 79, 489, 1926.
3. $\hbar$ denotes Planck's constant divided by $2\pi$. Zeitschrift für Physik. Bd. XL.