Perturbation and Variational Theory

[Rachael_Victoria]

Ok so I am trying to expand my understanding of these two concepts. Here is what I understand so far.
There are very few Schrödinger Equations that are exactly solvable
Both Theories are used to approximate a solution
Perturbation Theory utilizes a similar function with a known solution and adds the small perturbation to bring the answer closer to the actual value of the solution for the unsolvable wave function.
Variational Theory takes a trial wave function and sets up a set of parameters, a1, a2, etc and results in a value Ev. Ev is always larger than E, the actual ground state energy. Adjusting the parameters to minimize Ev will bring you closer and closer to the actual value of the ground state energy, for your actual wave function, E.
Is this correct and can anyone expand on what I have here? I understand how to do the actual mathematics involved but am a little shaky on why I am doing them. Also if anyone has a clear definition of how the variational principal relates to Huckle theory I would really appreciate it.
Thanks

 

[dextercioby]

Well,apparently u got the "big picture" right.And if u say that "details" (mathematical difficulties) are not a problem to u,then u got yourself a "touch down".U say u don't understand the point in making all those painful calculations??I believe your post has the answer inside.Exact methods in quantum theory are usable only in the case of isolated systems for which the Hamiltonian has a particular form,involving simple kinetical pieces (usually squares of momenta) and that the potential has a simple form which allows exact integrations.When all these don' t occur,i.e.the Hamiltonian has a sh**** part that spoils the beauty behind it all,then,in order to obtain by theoretical methods experimentally verifiable results,u need to make use of approximate methods.Whether they're perturbative or not,it's less relevant.Mathematics will be the key.

Are u sure it's Huckle??I haven't read too much QM,but the only guy with a name close to his is Hueckel (with an umlaut,actually) and he formulated the molecular orbital method (MOM) to describe chemical bonds in molecules.If it isn't him,then I'm sorry i cannot be of any assistance anymore.

Good luck!
Daniel.

 

[R0man]

Your understanding of perturbation and variational theory is mostly correct. Perturbation theory is a method used to approximate the solution of a problem by adding a small perturbation to a known solution. This is useful for problems that cannot be solved exactly, such as the Schrödinger equation for many-particle systems. The perturbation is assumed to be small enough that it does not significantly change the behavior of the system, and the solution can be found by using the known solution as a starting point and then adding the perturbation term.

Variational theory, on the other hand, is based on the variational principle, which states that the exact solution to a problem is always less than or equal to any trial solution. This means that by choosing an appropriate trial wave function and varying its parameters, we can get closer and closer to the exact solution. In other words, we are minimizing the energy of the trial wave function to get the best possible approximation of the true ground state energy.

As for the relation to Hückel theory, Hückel theory is a simplified approach to solving the Schrödinger equation for conjugated systems, such as molecules with delocalized electrons. It uses a variational approach to approximate the molecular orbitals and energies. So, variational theory is a more general concept that can be applied to various problems, while Hückel theory is a specific application of variational theory to a particular type of problem.

In summary, perturbation and variational theory are both methods used to approximate solutions to problems that cannot be solved exactly. Perturbation theory adds a small perturbation to a known solution, while variational theory uses the variational principle to minimize the energy of a trial solution. Both are important tools in theoretical physics and chemistry, and understanding their principles can greatly enhance your understanding of these fields.