how do you solve quantum equation for example prove that for any operator A, E|<i|A|j>|^=Tr(AA^T), where Tr denotes the trace help would be appreciated

- Mar 26th 2008, 03:37 PMjanembahow do you solve quantum physics equations example is included
how do you solve quantum equation for example prove that for any operator A, E|<i|A|j>|^=Tr(AA^T), where Tr denotes the trace help would be appreciated

- Mar 26th 2008, 04:36 PMtopsquark
$\displaystyle \sum_{ij} < i | A | j > = Tr(A A^{\dagger})$

Where did you get this relation from? It isn't correct. Say, for example, the spectrum of orthonormal eigenstates of A is used:

$\displaystyle \sum_{ij} < i | A | j > = \sum_{ij} < i | a_j | j > = \sum_{ij} a_j \delta _{ij} = \sum_i a_i$

and

$\displaystyle Tr(A A^{\dagger}) = \sum_i < i | A A^{\dagger} | i > = \sum a_i a_i^*$

These two expressions are clearly not equal to each other in general.

-Dan - Mar 26th 2008, 04:54 PMjanemba
yeah but how can you solve it

- Mar 26th 2008, 05:04 PMtopsquark
- Mar 26th 2008, 06:05 PMjanemba
you know when see quantum physics in college and it has really really long equations how can you solve that and whats the use of equations like the Schroedinger equations and all of the rest?

- Mar 26th 2008, 06:09 PMmr fantastic
- Mar 26th 2008, 06:13 PMjanemba
oK

- Mar 26th 2008, 06:22 PMtopsquark
Well the S-equation is the non-Relativisitic wave equation. That is to say the solutions that come out of it are "scalar" wavefunctions that represent the probability distribution of a particle being measureable in a specific place. Being non-Relativistic, it is not the most general possible equation, but if you want to do Relativistic equations then you need to know the spin of the particle, etc. and things get very messy very quickly.

As far as how to solve the equations that is going to depend heavily on what the equation is. For example, the Navier-Stokes equations are miserably difficult to solve even under simple circumstances, whereas it can be hard not to find solutions to the Poisson equation. (At least solutions that might be likely for a charge distribution, for example.)

Your question is really too broad to even try to answer. Do you have any specific interests?

-Dan - Mar 26th 2008, 06:53 PMjanemba
but did you ever solve the Navier-Stokes equations and what rule did you use to solve the the Navier-Stokes equation

- Mar 26th 2008, 07:03 PMtopsquark
Pffl. See this page. Let me put it to you this way: I decided I wanted to study Quantum theory because 1) I stink at experimentation and 2) I didn't EVER want to try to solve something as nasty as Navier-Stokes. (Oddly enough fluid mechanics is more complicated, at least Mathematically, than QM.) (Then of course, I ran headlong into General Relativity and found out that there are certain similarities between an incompressible fluid flow and quantum probability densities. Ah well. :( )

Despite my interests I try to steer clear of the truly nasty stuff.

-Dan - Mar 26th 2008, 07:10 PMjanemba
im 13 And studying quantum mechanics i know most of calculus

- Mar 26th 2008, 07:30 PMtopsquark
I had a 12 year old in on of my Physics I classes that I TA's at Purdue. I liked him: he was the only one that laughed at all of my jokes.

Keep at it. You're young enough to have lots of time to run into all of this stuff. Give it a few more years and you'll hit the more advanced techniques that your are learning the basics of now.

-Dan - Mar 27th 2008, 07:05 PMjanemba
wouldnt the navier stokes equation would be much easier by using the finite volume method ?