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Sometimes, certain problems in quantum mechanics become unsolvable due to their mathematical complexity. But we still have techniques for approximating their solutions! One such technique is perturbation theory - let's see how we can use it. #perturbation #quantum #approximation
To begin this video, we will look at how we study quantum physics problems in the first place. We recall that every system has an associated wave function. For example if our system is an electron in space, then the wave function of that electron will give us the likelihood of finding the electron at different points in space. This is discussed in more detail in my wave functions video!
But how do we actually find the wave function of a system? Well, we have to solve the Schrodinger equation of course! This is the governing equation of the theory of quantum mechanics, and we plug in information about our system (such as kinetic energy and potential energy or potential well of the system), in order to solve for the allowed wave functions. Specifically, we plug the information about the system into the Hamiltonian of the Schrodinger Equation.
If we know how to solve the Schrodinger equation once we plug in the system's properties, then we can calculate the allowed wave functions (and energy levels) of the system. The energy levels are of course discrete rather than continuous, which is what is referred to as quantization.
But what happens when we cannot solve the Schrodinger equation for a given system? What if we don't have enough mathematical skills or techniques to solve a particular differential equation? One way to solve such problems is numerically, using a computer. And what about if we don't have a computer?
In such scenarios, physicists have developed some clever techniques to find approximate solutions to our equation. One such technique is perturbation theory. It works best for systems that are very close to other systems that we DO know the solutions for. In this scenario, the phrase "very close" means the new system can be described as the original system plus some small change. The example used in this video is the addition of a small dirac delta function (spike) in the middle of a square potential well.
Then, the new system's Hamiltonian can be written as the old system's Hamiltonian plus some small change. Usually we also multiply the new / added small change by a factor lambda, that helps us in our upcoming mathematical steps. Lambda takes values between 0 and 1 as we go from the unperturbed, original system (lambda = 0) to the perturbed, new system (lambda = 1).
We can then say that the new system's allowed wave functions are equal to the old system's wave functions plus a small term proportional to lambda, plus a smaller term proportional to lambda squared, and so on. This forms an infinite series of "corrections" to the original wave function. We don't have time to calculate infinitely many terms, but luckily for most situations just the first new term is enough. And exactly the same logic applies for energy levels.
Luckily, the first order correction just depends on the change between the old and new systems, and the wave functions of the old system. And nothing else. The first order energy level correction is something we know how to calculate, meaning we don't have to deal with an "impossible" differential equation whilst still getting a very good approximation.
And this is why perturbation theory is a very valuable technique for solving (or at least approximating) "impossible" to solve quantum mechanical systems.
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Timestamps:
0:00 - How Problems are Solved in Quantum Mechanics (Wave Functions, Schrodinger Eqn)
3:12 - Energy Levels and Wave Functions for Quantum Systems
4:53 - Perturbation Theory (for a Perturbed System)
6:30 - Sponsor Message (and magic trick!) - big thanks to Wondrium
8:55 - Approximating the new Wave Functions and Energy Levels
10:00 - First Order Approximation - EASY!
#ad - This video was sponsored by Wondrium
