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with regards to, it follows:

$\psi \cdot \chi ={\frac {i}{\hbar }}\psi W\cdot \chi ,$

$\psi _{,k}^{,k}={\frac {-1}{\hbar ^{2}}}\psi W_{,k}W^{'k},$

(6)$\psi _{,k}^{,k}-2{\frac {e}{c}}{\frac {i}{\hbar }}\varphi ^{k}\psi _{,k}-{\frac {1}{\hbar ^{2}}}\left(m^{2}c^{2}+{\frac {e^{2}}{c^{2}}}\varphi _{k}\varphi ^{k}\right)\psi =0.$^[1]

This equation coincides with that of Klein-Fock,^[2] which also without the introduction of somewhat artificial fifth coordinate can be obtained.^[3]

Leningrad, Physikalisches Institut der Universität.

1. It should be noted that in the field-free case from (6) the speed of the de Broglie's phase waves can easily be obtained. This case yields $\psi =\psi 'e^{{\frac {i}{\hbar }}Et}$, where $\psi '$ is independent of $t$, because

   $\nabla ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}-{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0$

   and

   ${\frac {\partial ^{2}\psi }{\partial t^{2}}}=-{\frac {E^{2}}{\hbar ^{2}}}\psi ,$

   $\nabla ^{2}\psi -\left({\frac {1}{c^{2}}}-{\frac {m^{2}c^{2}}{E^{2}}}\right){\frac {\partial ^{2}\psi }{\partial t^{2}}}=0.$

   Let's set

   $E={\frac {mc^{2}}{\sqrt {1-{\frac {{\boldsymbol {b}}^{2}}{c^{2}}}}}},$

   so that

   $\nabla ^{2}\psi -{\frac {{\boldsymbol {b}}^{2}}{c^{4}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}=0,$

   or, in other words, the required speed is $V={\frac {c^{2}}{\boldsymbol {b}}}$, which could also be deduced directly from the relativistic Hamiltonian equation (5).

2. O. Klein, ZS. f.Phys.37,895,1926; V. Fock, ZS. f. Phys.39, 226, 1926.
3. Anmerknng bei der Korrektur. Meanwhile E. Schrödinger (Ann. d. Phys. 81, 132, 1926) has obtained the same equation successfully with the help of the operator method.