Quantum Buzz
Variational Principle Recipe
1. Pick a trial function, psi that makes sense.
(a) The function should satisfy boundary conditions.
(b) The function should only have one maximum since that is the case for most groundstate wavefunctions.
(c) Don't forget, that the function must be square integrable within the defined region. For example, a sine-function within the region of an infinite square well of width can be integrated and normalized within that region.
2. Normalize the trial function.
3. Calculate (psi* H psi)this is the upper bound energy.
4. Minimize by calculating d(psi* H psi)/db = 0, where b is any undetermined constant in your trial function. Solve for b.
5. Substitute the normalization constant and b into your trial function and your done :-)
I hope this helps!
So, what's the big deal about perturbation theory and variational principle. Well, what if we knew exact solution to a potential well, i.e., we knew all the Eigenenergies and corresponding wavefunctions. Now let's introduce a small perturbation (this could be contamination in a material). Using perturbation theory, one can calculate the new energies and corresponding eigenfunctions (which are wavefunctions) without having to solve a complicated 2nd order differential equation. This method has proven to work very well and energies can be calculated to the desired accuracy. So, why waste time with the variational principle? What if one had a quantum well that was nothing like any of the wells that can easily be calculated using exact methods. The variational principle tells me that I can guess the wavefunction for ANY quantum well and calculate an upper bound for the ground state (lowest) energy. I can even minimize this upper bound by taking the first derivative and setting it equal to zero. This method also works quite well. Of course, it helps if the trial function is chosen sensibly. For example, the groundstate wavefuction will most likely only have only maximum. It is therefore advisable to choose something like a Gaussian or a half a cosine function as a trial function. Of course, if one is dealing with an infinite square well then it would be good to choose a trial function that goes to zero at the boundaries. In other words, your trial function must satisfy the boundary conditions (BC's) and be normalized.
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