How do you solve the time dependent Schrodinger wave equation?

Answer 1

If V is only a function of x, then the equation's general solution can be nearly solved using the method of separation of variables.

Starting with the time dependent Schrödinger equation:

(iℏ)∂Ψ/∂t = -(ℏ2/2m)∂2Ψ/∂x2 + VΨ, where iis the imaginary number, ℏ is Plank's constant/2π, Ψ is the wave function, m is mass, x is position, t is time, and V is the potential and is only a function of x in this case.

Now we find solutions of Ψ that are products of functions of either variable, i.e.

Ψ(x,t) = ψ(x) f(t)

Taking the first partial of Ψ in the above equation with respect to t gives:

∂Ψ/∂t = ψ df/dt

Taking the second partial of Ψ in that same equation above with respect to x gives:

∂2Ψ/∂x2 = d2ψ/dx2 f

Substituting these ordinary derivatives into the time dependent Schrödinger equation gives:

(iℏψ)df/dt = -(ℏ2/2m)d2ψ/dx2f + Vψf

Dividing through by ψf gives:

(iℏ)(1/f)df/dt = -(ℏ2/2m)(1/ψ)d2ψ/dx2 + V

This makes the left side of the equation a function of tonly, and the right a function of x only; meaning both sides have to be a constant for the equation to hold (I'm not going to prove why this is so, because if you've understood what I've written up to this point, you're probably familiar with the separation of variables technique).

That allows us to make two separate ordinary differential equations:

1) (1/f)df/dt = -(iE/ℏ)

2) -(ℏ2/2m)d2ψ/dx2 + Vψ = Eψ, where E is the constant that both separated differential equations must equal to (I did a little bit of algebra to these equations also). This is known as the time independent Schrödinger equation.

To solve 1) just multiply both sides of the equation by dt and integrate:

ln(f) = -(iEt/ℏ) + C. Exponentiating and, since C will be absorbed later, removing C:

f(t) = e-(iEt/ℏ)

Now, we have gone as far as we can until the potential, V(x), is specified. That leaves us with the general solution to the time dependent Schrödinger equation being:

Ψ(x,t) = ψ(x) e-(iEt/ℏ)

There are many solvable examples of ψ(x) for specific V(x). I've linked some below. Also, if V is a function of both x and t, there are methods to find solutions, such as perturbation theory and adiabatic approximation.