Time-dependent to time-independent Schrödinger equation

[LagrangeEuler]

Why you can do separation of variables in time-dependent

Schrödinger equation​

$$i \hbar \frac{\partial \psi(\vec{r},t)}{\partial t}=-\frac{\hbar^2}{2m}\Delta \psi(\vec{r},t)+V(\vec{r})\psi(\vec{r},t)$$
with
$$\psi(\vec{r},t)=\varphi(\vec{r})T(t)$$
and when in general is that possible?

 

[PeroK]

and when in general is that possible?

When it works!

 

[LagrangeEuler]

And when it works?

 

[dextercioby]

The separability of time dependence is a simplifying assumption in the context of the explicit time independence of the potential. But a strict condition is valid in the opposite way: If the potential is time-dependent, the assumption of splitting (separation of) variables is untenable.

 

[Nugatory]

And when it works?

Separation of variables is a general technique for solving multivariable differential equations, when we can algebraically manipulate the equation to get all of one variable on one side and all of the other variable on the other side.

The Schrödinger equation takes this form when the Hamiltonian is a function of position but not of time.

 

[VVS2000]

I think it would be better to later look into how the wave function can also be represented as a product of it's "spatial" part and "spin" part
This is a widely used method to simplify multivariable differential equations..