I Time-dependent to time-independent Schrödinger equation

LagrangeEuler
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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?
 
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LagrangeEuler said:
and when in general is that possible?
When it works!
 
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And when it works?
 
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.
 
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LagrangeEuler said:
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 Schrodinger equation takes this form when the Hamiltonian is a function of position but not of time.
 
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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..
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
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