Separation of variables and the Schrodinger equation 

Neil Volk Our professor gave this as a homework .. this video was my savior

Yeah. That point really is a lovely idea.

It's because the Schrödinger's equation has been derived by taking the formula for the wave function of a "free particle" (plane wave) and replacing the ω and k with E/ℏ and p/ℏ, and then looking for the time and space derivatives, which are needed for the Hamiltonian. The wave equation is not a Hamiltonian - it expresses the dependence between the curvature in space and the curvature in time of the wave function, which are both second-order derivatives. The Schrödinger's equation, on the other hand, expresses the dependence between the kinetic energy (related to changes in time) and the potential energy (related to changes in space), which are first-order derivatives. There's a nice analogy you can take a look at, regarding harmonic motion: The equation for the Simple Harmonic Oscillator (SHO) i s d²x/dt² = -ω²·x, so it is has a second-order derivative, and the solution is that nice sinusoidal oscillation you're probably well familiar with. But the harmonic oscillator can be also expressed in the Hamiltonian way, in terms of energies: the total energy E consists of kinetic energy m·v²/2 and potential energy k·x²/2, so the Hamiltonian is E = m·v²/2 + k·x²/2. Now you can replace the `v` in this formula by the first derivative of position with respect to time, dx/dt, obtaining: E = m·(dx/dt)²/2 + k·x²/2 It is just a different way to express the conditions on the function which describes the motion of the oscillator: one is in terms of curvatures, the other is in terms of energies. But the solution is the same for both, because they both describe the same harmonic oscillator.

That relationship came from simplifying the original differential equation, so if that equality is false then X(x)•T(t) would not be a solution to the differential equation (Six years ago, hopefully this is still relevant for you 😅)

@@faielgila7375 hopefully he hasnt bene stuck on this for 6 years lol

@@pedramnoohi2715 or dead

No! The ODE has been solved by rearranging the variables and integrating, it is not a seond order ODE with the general solution of Aexp[-wt] + Bexp[wt]

Sci Twi I've been wondering myself. From what I can gather (also from the book), if there is another function in the PDE that makes it so that the variables can't be seperated easily, solving the PDE will become a whole lot harder. The example they gave was if V was V(x,t) instead of just V(x), then it will not be possible to solve with SOV if V's variables aren't seperable​ themselves...? I don't know. Maybe there exists some weird PDE that has has the exact opposite property.

*\0(*_*)0/* ᕦ(ಠ_ಠ)ᕤ😡🥴😤

fuck yeah baby