What are typical initial conditions for the Schrödinger eq?

[SeM]

Hi, I am wondering if there exists some general initial conditions for solving the Schödinger eqn. for 1D free electrons ?

Thanks!

 

[stevendaryl]

Typically, in solving Schrödinger's equation, you are either doing a time-dependent analysis, in which case the initial conditions are part of the problem statement, or you are doing a time-independent analysis, in which case, people are usually looking for energy eigenstates.

If you are doing a time-dependent analysis, the initial condition might be that the electron is a wave-packet of some sort localized at some particular position and momentum.

 

[SeM]

Typically, in solving Schrödinger's equation, you are either doing a time-dependent analysis, in which case the initial conditions are part of the problem statement, or you are doing a time-independent analysis, in which case, people are usually looking for energy eigenstates.
.

Hi, I am doing a time-independent analysis, and thought of using the zero-point energy for E, in the Schrödinger eqn, and then solve it with the two initial conditions:

1) assume that the wavefunction is 1 at position zero (x=0), as it would be normalized to.

2) its slope, Psi', is zero at the same point, x=0. Does this sound reasonable?

Thanks

 

[stevendaryl]

Hi, I am doing a time-independent analysis, and thought of using the zero-point energy for E, in the Schrödinger eqn, and then solve it with the two initial conditions:

1) assume that the wavefunction is 1 at position zero (x=0), as it would be normalized to.

2) its slope, Psi', is zero at the same point, x=0. Does this sound reasonable?

Thanks

If you already know a complete set of energy eigenstates then you can find a solution having those properties. Those properties don't help you find the eigenstates, though.

 

[SeM]

If you already know a complete set of energy eigenstates then you can find a solution having those properties. Those properties don't help you find the eigenstates, though.

I know, there is another procedure to find the eigenstates. This is just to find the analytical solution to the ground state.

 

[stevendaryl]

I'm not sure what you mean. The ground is the eigenstate with the lowest energy. So if you know the eigenstates, you already know the ground state.