Solving the Schrodinger equation

  • #1
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Main Question or Discussion Point

This extract is from my college notes.

"Because of the inherent difficulty of obtaining even grossly approximate solutions of the complete Schrodinger equation, one typically focuses on reduced formulations that are believed to capture the essential features of the problem of interest. This has resulted in a number of parallel strands in the study of the electronic properties of solids and is, in large part, responsible for the richness of the subject as a whole. Experiments are constantly discovering new phenomena that either must be accommodated within the existing theoretical framework, or provide the basis for expanding this framework by introducing new fundamental concepts. Theory, in turn, makes predictions that challenge existing concepts and which must be tested by experiments."


inherent: Why has this word been used?

complete: Wat are the incomplete and the complete versions?

reduced formulations: Reduced? How?

This has resulted in a number of parallel strands in the study of the electronic properties of solids and is, in large part, responsible for the richness of the subject as a whole.:
I don't understand.

for expanding this framework by introducing new fundamental concepts: And why should the new fundamntal concepts fit into (be consistent with) the old theroetical framework?

Theory makes predictions that challenge existing concepts: How can a self-consistent theory make predictions that challenge the theory?

Finally, I don't see any connection between the first two and the last two sentences.

Anyone that helps I thank.
 

Answers and Replies

  • #2
jtbell
Mentor
15,551
3,493
The complete Schrödinger equation for a system of merely two particles, in Cartesian coordinates, is

[tex]- \frac{\hbar^2}{2m_1}
\left( {\frac{\partial^2 \Psi}{\partial x_1^2} + \frac{\partial^2 \Psi}{\partial y_1^2} + \frac{\partial^2 \Psi}{\partial z_1^2} \right)
- \frac{\hbar^2}{2m_2}
\left( {\frac{\partial^2 \Psi}{\partial x_2^2} + \frac{\partial^2 \Psi}{\partial y_2^2} + \frac{\partial^2 \Psi}{\partial z_2^2} \right)
+ V(x_1, y_1, z_1, x_2, y_2, z_2, t) \Psi = -i \hbar \frac{\partial \Psi}{\partial t}[/tex]

where

[tex]\Psi = \Psi(x_1, y_1, z_1, x_2, y_2, z_2, t)[/tex]

For systems with more particles, extend the equation correspondingly with more variables and terms. The key is that psi is a function of 3N+1 variables, where N is the number of particles in the system. To make any headway in solving this, you need to make simplifying assumptions that reduce the number of variables that you have to deal with.
 
  • #3
Bill_K
Science Advisor
Insights Author
4,155
195
failexam, I agree. Whoever wrote that passage must be getting paid by the word.
 
  • #4
359
4
This is probably oversimplifying what you asked, but I think part of the problem is that there are not a lot of exactly solvable problems in Quantum Mechanics. In other words, we don't know how to solve the equations describing the physics, without resorting to approximate methods.

In most QM texts you solve unrealistic problems like the particle in the box, that are solvable. Then you might do the free particle (which in purest terms is also unrealistic) and the hydrogen atom. But, very quickly you go into time-dependent and time-independent perturbation theory, which are "approximation schemes".

Theory makes predictions that challenge existing concepts: How can a self-consistent theory make predictions that challenge the theory?
I think the trick is self-consistent within its domain of validity.

GR is a good example. It, more or less, predicts singularities. In other words, it predicts a phenomenon it offers no explanation for.

This happens because it is going outside the domain where the theory is valid.

A self-consistent theory can make its boundaries of validity very evident in this way.
 
  • #5
1,344
32
Thnaks for helping.

failexam, I agree. Whoever wrote that passage must be getting paid by the word.
Why should he be getting paid by the word?
 

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