Solving Schroedinger Eqn: Interpreting Separation Constant G as Energy

[DaTario]

Hi All,

When solving the Schroedinger equation for the zero potential situation, what is the argument for interpreting the separation constant G as the energy of the system. The dimensional aspect I understand but it seems not to be suficient. I guess I am missing some detail.

Best wishes,

DaTario

 

[ShayanJ]

The Schrödinger equation is $\tilde{H} \psi=i\hbar\frac{\partial \psi}{\partial t}$ where $\tilde{H}$ is the Hamiltonian of the system.If Hamiltonian is independent of time,then a separation is possible which gives,as the spatial part, $\tilde{H}\psi=G\psi$.
One reason to interpret G as the energy of the system,is that in classical Hamiltonian mechanics,if only conservative forces are present,Hamiltonian equals the total energy of the system.
But the main reason is that,it works!
Schrödinger derived his equation but you can't call such derivations proofs because they're all based on things not so much accepted.People just postulate things and if it turns out to work,so its OK.Schrödinger postulated G to be energy and it turned out to work.