Solutions to Schrodinger's Wave Equation

[Nothing000]

Homework Statement

Assume that $$\psi_{1}(x,t)$$ and $$\psi_{2}(x,t)$$ are solutions of the one-dimensional time-dependent Schrödinger's wave equations.
(a) Show that $$\psi_{1} + \psi_{2}$$ is a solution.

(b) Is $$\psi_{1} \cdot \psi_{2}$$ a solution of the Schrödinger's equation in general?

Homework Equations

Is this the "One-Dimensional Time-Dependent Schrödinger's Wave Equation":
$$\eta = \imath \hbar \cdot \frac{1}{\phi(t)} \cdot \frac{\partial \phi(t)}{ \partial t}$$

If so, it says in my book that the solution is $$\phi(t) = e^{- \imath (\frac{E}{\hbar})t$$

The Attempt at a Solution

I have a feeling that all I have to do is show that these solutions are linear, then use the superposition technique.

 

[Avodyne]

Your relevant equation is *half* of the time-dependent Schrödinger equation in the special case that there is no potential energy, and *after* separation of variables has been performed in x and t. (The other half involves x, and not t.)

Yes, linearity and superposition is the key point.

 

[Nothing000]

So I don't really even need to know what the solutions are? All I need to do is some sort of "proof" that the sum of the two solutions to the linear P.D.E. is also a solution?

If that is the case, do you think you could help me get started with working that out?

 

[Nothing000]

Could you please write the full time-dependent Schrödinger equation?

 

[Nothing000]

Anyone?

 

[Avodyne]

In one space dimension, the full time-dependent Schrödinger equation is

$$i\hbar{\partial\over\partial t}\psi(x,t) = \left[-{\hbar^2\over2m}{\partial^2\over\partial x^2}+V(x)\right]\psi(x,t)$$

Edit: the derivative on the right-hand side is wrt x, now fixed and correct.

 

[Nothing000]

So how do I show that $$\psi_{1}(x,t)$$ and $$\psi_{2}(x,t)$$ have linearity and superposition can be used to create a third solution?

 

[Avodyne]

You know that $$\psi_{1}(x,t)$$ and $$\psi_{2}(x,t)$$ obey this equation. You want to show that $$\psi_{1}(x,t) + \psi_{2}(x,t)$$ does as well. So, plug $$\psi_{1}(x,t) + \psi_{2}(x,t)$$ into the equation. Can you used what you know to show that the result is true?