Perturbation Theory: Physically Explained & Applied

[Ed Quanta]

I have been studying Perturbation theory in my Quantum class but my professor has not really explained why physically it comes into play. The book says that perturbation theory is used to help come up with approximate solutions to the Schrödinger Equation. Is this analagous to how we use Fourier series to express solutions to partial differential equations as infinite series? Only instead of using an infinite number of terms, we take the first few for an approximation ? Is there any physical significance or is perturbation theory just pure mathematics?

 

[selfAdjoint]

The reason physicists use perurbation theory is that they CAN'T DO the full non-perturbative theory - can't derive and solve the equations. Non-perturbative means with the full strength of the interactions included. So they start with a free theory - no interactions at all, and add just a tiny interaction, called a perturbation, and expand solutions for this in terms of a power series in the coupling constant. If the constant is small, like alpha ~ 1/137, the series will converge, or at least be asymptotic (good results out to some hopefully high power, where it blows up). Perturbation is an approximation technique and should not be compared to Fourier analysis which is a theoretically exact technique (though of course it has its own approximation schemes!).

 

[R0man]

Perturbation theory is a powerful mathematical tool that is commonly used in physics, particularly in quantum mechanics, to approximate solutions to complex systems. It allows us to study the behavior of a system that is slightly different from a known, simpler system. This is why it is often used to solve the Schrödinger Equation, which describes the behavior of quantum systems.

To understand the physical significance of perturbation theory, let's first consider the example of a simple harmonic oscillator. This system can be described by a single energy level and has a known, exact solution. However, in many real-world systems, the behavior of the oscillator may be influenced by external factors, such as an additional force or interaction. In these cases, the exact solution is no longer applicable and perturbation theory is used to approximate the behavior of the system.

In essence, perturbation theory allows us to break down a complex system into simpler, known components and then add in the effects of the perturbation to obtain an approximate solution. This is similar to how we use Fourier series to express solutions to partial differential equations as infinite series. Instead of using an infinite number of terms, we can use a few terms to get a good approximation of the solution.

However, unlike Fourier series, perturbation theory has a physical significance as it allows us to understand the effects of external factors on a system. This is especially important in quantum mechanics, where small perturbations can have significant impacts on the behavior of a system. For example, perturbation theory has been used to study the effects of an external magnetic field on the energy levels of an atom.

In conclusion, perturbation theory is not just pure mathematics, but it has a strong physical significance in understanding the behavior of complex systems. It is a powerful tool that allows us to approximate solutions and gain insights into the effects of external factors on a system. So, while it may seem like just another mathematical technique, perturbation theory plays a crucial role in our understanding of the physical world.