Solving the Mystery of Energy Eigenfunctions

[pivoxa15]

It seems the Schrödinger equation is written so that psi is an energy eigenfunction. So all psi are energy eigenfunctions? But how can it turn into other eigenfunctions like momentum? Or is it already a momentum eigenfunction as welll as the energy eigenfunction and so also position and so on...

 

[Manchot]

The time-dependent version of the Schroedinger equation makes no reference to energy eigenvalues. However, since the Hamiltonian is a Hermitian operator, any wavefunction can be expanded as a sum of its eigenvectors. If you crank out the resulting differential equation, the weighting coefficient of each eigenvector obeys a simple exponential relationship, which means that once the eigenvectors are found, it is a fairly simple matter to find the time-dependent behavior of the wavefunction. Thus, solving the Schroedinger equation is really just a problem of finding the Hamiltonian's eigenvalues.

 

[pivoxa15]

The great postulate of Schrödinger is that -i(hbar)d(psi)/dt = H(psi) where H is the hamiltonian.

So -i(hbar)d(psi)/dt = E(psi) = H(psi)

Which as you say nothing more than finding the Hamiltonian's eigenvalues.

So the Schrödinger equation is not that mysterious after all? psi as we know it is related to the eigenfunction of the hamiltonian. But has the intepretation as |psi|^2 is the probability density.

I've just realized that there are no momentum nor position eigenfunctions so there could have been some error in my OP.

 

[Gza]

The great postulate of Schrödinger is that -i(hbar)d(psi)/dt = H(psi) where H is the hamiltonian.

So -i(hbar)d(psi)/dt = E(psi) = H(psi)

Which as you say nothing more than finding the Hamiltonian's eigenvalues.

So the Schrödinger equation is not that mysterious after all? psi as we know it is related to the eigenfunction of the hamiltonian. But has the intepretation as |psi|^2 is the probability density.

I've just realized that there are no momentum nor position eigenfunctions so there could have been some error in my OP.

There is no real error in your op. Position eigenfunctions are dirac delta functions, and momentum eigenfunctions are plane waves (in the position basis at least.)

 

[CarlB]

It seems the Schrödinger equation is written so that psi is an energy eigenfunction. So all psi are energy eigenfunctions?

Let me amplify an answer you've already been given here. You can take any two solutions to the Schroedinger equation and combine them into another solution. For example:

$$\psi(x,t) = \psi_1(x,t) + \psi_2(x,t)$$

is also a solution if $$\psi_1, \psi_2$$ are. Now think about what happens if $$\psi_1$$ and $$\psi_2$$ are eigenfunctions of energy with different eigenvalues. In general, $$\psi$$ is not an eigenfunction of energy.