Solving the Schrodinger Equation: WHAT DOES IT ALL MEAN?

[tomothy]

I'm an A-level student (I don't know what the US equivalent is sorry, I'm not an undergraduate is what I'm saying), and I've independently done a project on wave functions for a few simple stationary systems; particle in a box and quantum harmonic oscillator are the ones I focused on in the end.

However something's been troubling me lately. I've solved the time-independent Schrödinger equation, and I have a series of energy eigenvalues and corresponding eigenfunctions. But what do these actually mean? I know the wave function can be interpreted as a probability amplitude and it can be used to find 'expectation values' for position and momentum.

One problem I'm considering is modelling a conjugated pi system in a molecule like butadiene using the particle in an (infinite walled) box.

Honestly, I'm really very lost. So lost in fact I don't even really know what I'm asking. Even just pointing me towards something specific to research would be a great help for me. Thank you all in advance.

 

[Matterwave]

The stationary states are just a good method to solve the Schroedinger equation. A general wave function will be in a superposition of eigenstates. Since we know how each of those eigenstates evolve in time, it will be easy for us to figure out how a general wave function evolves in time.

 

[Jim_G]

I'll leave out vectors / eigenvalues and just use 'n' for this explanation if it's OK.

To me it looks like it starts out as an ordinary differential equation of the second order.
(The second derivative of ψ(x) = k^2*ψ(x) basically).

The solutions should be of the form ψ(x) = A sin (ωx)

Then the quantum part.

de Broglie's hypothesis λ = h/p is combined with k = 2∏/λ

and one winds up with a solution of the form ψ(x) = A sin (nωx)
where n represents integer quantum values. Reminds one of a
Fourier series.

The system it is solved for determines the boundary values of ψ(x)
Example: ψ(x) = 0 at the walls of a potential well.

So it becomes a boundary value problem (for the constants etc.).

Hope this helps.

 

[Antiphon]

The stationary state are the ones the system can stay in without changing or evolving to other states. Just like stable electron orbits.