Solving the Schrodinger Equation: WHAT DOES IT ALL MEAN?

Click For Summary

Discussion Overview

The discussion revolves around the interpretation of solutions to the time-independent Schrödinger equation, particularly in the context of stationary states and their implications for quantum systems. Participants explore the meaning of wave functions, energy eigenvalues, and potential applications in modeling molecular systems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the meaning of wave functions and energy eigenvalues after solving the Schrödinger equation for simple systems, seeking guidance on further research.
  • Another participant notes that stationary states are a method to solve the Schrödinger equation and that a general wave function can be expressed as a superposition of these eigenstates, which evolve over time.
  • A different participant describes the mathematical form of the solutions to the Schrödinger equation, relating it to ordinary differential equations and boundary value problems, while connecting it to de Broglie's hypothesis.
  • Another contribution clarifies that stationary states represent conditions where the system remains unchanged, likening them to stable electron orbits.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the interpretation of the solutions or the implications of stationary states, indicating that multiple views and uncertainties remain in the discussion.

Contextual Notes

Some participants mention specific systems and mathematical approaches, but there are unresolved aspects regarding the broader implications and interpretations of the wave functions and eigenvalues.

Who May Find This Useful

This discussion may be of interest to students and individuals exploring quantum mechanics, particularly those grappling with the conceptual underpinnings of the Schrödinger equation and its applications in various physical systems.

tomothy
Messages
20
Reaction score
0
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.
 
Physics news on Phys.org
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.
 
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.
 
The stationary state are the ones the system can stay in without changing or evolving to other states. Just like stable electron orbits.
 

Similar threads

Undergrad Schrödinger equation and classical wave equation
  • · Replies 39 ·
2
Replies
39
Views
4K
Undergrad Schrodinger equation for N particles in a box
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What is the Meaning of the Schrödinger Equation?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Derivation of Schrodinger equation (chicken and egg problem?)
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Why don't we bury Schrodinger's Cat?
  • · Replies 143 ·
5
Replies
143
Views
12K
Undergrad Why the linear combination of eigenfunctions is not a solution of the TISE
  • · Replies 24 ·
Replies
24
Views
3K
Graduate Schrodinger equation/Madelung equation phase transformation
  • · Replies 32 ·
2
Replies
32
Views
3K
Undergrad Solving the Schrodinger Equation for a particle being measured
  • · Replies 6 ·
Replies
6
Views
2K
High School Does the Schrödinger Equation Imply a Deterministic Universe?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Matrix Notation for potential in Schrodinger Equation
  • · Replies 6 ·
Replies
6
Views
2K