What are typical initial conditions for the Schrödinger eq?

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Discussion Overview

The discussion revolves around the initial conditions for solving the Schrödinger equation, particularly in the context of one-dimensional free electrons. Participants explore both time-dependent and time-independent analyses, discussing various approaches and assumptions related to initial conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about general initial conditions for solving the Schrödinger equation for 1D free electrons.
  • Another participant explains that initial conditions depend on whether the analysis is time-dependent or time-independent, noting that time-dependent analyses often involve wave-packets localized in position and momentum.
  • A participant describes their approach to a time-independent analysis, proposing to use zero-point energy and suggesting two initial conditions: the wavefunction being 1 at position zero and its slope being zero at the same point.
  • Another participant acknowledges the proposed initial conditions but points out that while they can lead to a solution, they do not assist in finding the energy eigenstates.
  • A subsequent reply clarifies that knowing the eigenstates inherently includes knowledge of the ground state, which is the eigenstate with the lowest energy.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the proposed initial conditions and their relationship to finding energy eigenstates. The discussion remains unresolved regarding the effectiveness of the initial conditions in aiding the determination of eigenstates.

Contextual Notes

There is uncertainty regarding the completeness of the proposed initial conditions and their dependence on the specific context of the problem. The discussion also highlights the distinction between time-dependent and time-independent analyses without resolving the implications of this distinction.

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

Thanks!
 
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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.
 
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stevendaryl said:
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
 
SeM said:
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.
 
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stevendaryl said:
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.
 
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.
 

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