Partial differential equations. Part III: Schrödinger
This is the third a final part of the series on partial differential equation. If you are reading this, I assume you have already read the first two parts, where I talk about the wave and heat equations. If not, take a couple of minutes to familiarize yourself with these equations!
When talking about the PDE’s, it is hard to pick the most significant equations. Math professors would definitely walk you through the wave and heat equations, explain the logistic equations, and also spend several weeks discussing Laplace’s equation, since its solutions (so-called harmonic functions) are key to understanding gravity, fluid mechanics, electrostatics, and many other branches of physics.
But I would like to talk about the Schrödinger’s equation, since it is the heart and soul of quantum mechanics, the most mysterious and exciting theory in physics. And there is some magic happening when one’s trying to solve it.
Even non-physicists have heard of Schrödinger and his poor cat that is dead and alive at the same time for the sake of science: Erwin tried to illustrate the concept of superposition. But Schrödinger’s equation is telling a different story. It describes the wave function of a quantum mechanical system. Here is how it looks:
The first symbol here, i, is an imaginary unit. It is simply a square root of -1. Second symbol, the h-looking one, is a constant, specifically, the reduced Planck’s constant, that is computed in a following way:
Moving forward: d/dt here, of course, refers to the first derivative with respect to time, and Psi is a function of time, which is essentially a state vector of a quantum system — that’s what we want to solve for. Finally, H with a funny looking hat is something called the Hamiltonian operator.
Note: the formula given above is a time-dependent version of the Schrödinger’s equation, there is also a time independent equation, we’ll get to it in a moment.
Since Psi describes the state of a quantum system, we need to understand that it is a function of both space (3 by 1 vector x) and time. Linear operators acting on the function correspond to observables, and observables have well-defined values. In the case of our Hamiltonian operator (H with a funny looking hat), observable is energy, that is again, well-defined:
This is also known as time-independent version of the Schrödinger’s equation, and here E is a constant energy level of our quantum system. Another observable is the momentum operator, which is also well-defined:
In classical mechanics total energy of the system is a sum of its potential and kinetic energies, and KE and PE are also specifically defined.
If we rethink the quantum system in terms described above, we can rewrite the Schrödinger’s equation in the following way (keep in mind that the vectors are 3-dimensional, so we have 3 x’s):
To solve this equation, we need to assume that our space is a circle. Why? Because in that case we can use Fourier series! Remember, Fourier series approximate functions as series of sines and cosines, and make solving partial differential equations manageable, we’ll just need to compute some coefficients by plugging in the numbers in Fourier’s formulas. If we want solve the equation above (let’s assume this time that we are talking about a free particle, aka V=0), we can do it the following way:
Here comes magic. After rewriting this equation, we get another equation that states that the first derivative of a certain function (the one we are solving for) with respect to time is equal to some constant times the second derivative of the same function with respect to space. Let me make this even clearer:
Exactly! Our scary Schrödinger’s equation is reduced to the heat equation that is solvable using Fourier series (we talked about it last time). All we had to do is remember the definition of energy, make some assumptions, and carefully rewrite the equation. Finally, let me just give you the solution formula obtained by the same method as the heat equation:
Sources:
My personal class notes, Wikipedia, Fourier Analysis textbook.
