Question

Write a comparative essay, on the perturbation theory, the variational principle and the WKB approximation, including their applications in quantum mechanics.

Answer 1

Perturbation Theory: It is a mathematical procedure that helps in finding the approximate solutions to the perturbed (or slightly changed) state of the system.

There are two types of perturbation theory, degenerate (only changed by one step) and non-degenerate (changed by several steps or values) perturbation theory.

The mathematical equation for the perturbation theory is as follows:

Thus, perturbation theory uses Hamiltonian and Schrodinger waves equation (second-order differential equation), to approximate the slightly changed system by .

Variational Principle: When we cannot achieve the solution using Schrodinger wave equation, we use Variational Principle, because it gives the upper bound energy state of the system.

Upper bound energy state means the maximum energy state possible for that system, therefore, it helps us to determine the wavefunctions and nature of the system.

the mathematical equation is:

  

WKB Approximation: The WKB approximation is the method which approximates the solution of time-independent Schrodinger II order Differential wave equation. It is highly efficient in approximating the bound energies and tunneling rates.

It just uses the fundamental differential equation techniques with given boundary conditions of the system.

its mathematical equation :

where p(x) is the momentum of the particle or the system.

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