Quantum mechanics | Schrödinger | Simplified
Schrödinger’s Equation: Explained
The Most Important Equation of Quantum Mechanics
The fundamentals of quantum mechanics are often inaccessible to the vast majority. Along with the counterintuition of quantum physics comes a world of prejudice. It’s conventionally taken to be particularly difficult. Naturally, it isn’t what it seems to be. This is my attempt at explaining the Schrödinger equation to the uninitiated. An equation that undeniably changed the way we perceive reality.
Quoting Wikipedia:
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system.
Though that probably made little to no sense. What does it mean to say that a function is linear? Or that it’s partially differentiated? Well answering that would mean delving deeper into the mathematics of it all. But as far as we’re concerned today:
The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time.
Conceptually, the Schrödinger equation is the quantum counterpart of Newton’s second law in classical mechanics. Given a set of known initial conditions, Newton’s second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution of a wave function over time.
Enter: The Schrödinger Equation
The equation can be written in various forms. However, in its simplest one, it looks something like this:
This daunting-looking equation is where we stand. It is by any measure, the most successful quantum mechanical postulate of all time. By the end of this article, you’ll have at least a basic understanding of what any of this really means.
Well first, let’s kick this off with psi, the 23rd Greek letter. Ψ (psi, pronounced with the /ps/ in “lapse” followed by an /eye/) here, represents the wave function of a quantum system. Well Ψ, in particular, tells us how likely we are to find a particular particle at different regions of space in our system.
Now, wavefunctions. What are they? I have dedicated a 7-minute read if you’d like to know more about it but briefly explained, a wavefunction is a mathematical description of everything we know about a particular quantum system. A quantum system is of course a “system” we happen to be studying using quantum mechanics. It could be anything. Put an electron in an electric field and that’s a system. And if we use quantum mechanics to study it, it becomes a quantum system.
Ψ² ∝ probability distribution of finding a particle at a particular point.
The Language of Quantum Physics
An introduction to kets, wavefunctions, and quantum systems
medium.com
But we don’t just write Ψ when we represent a quantum system. We actually write it like this: |Ψ⟩. A straight line to the left and an angled bracket to the right. Now, this whole “thing” is called a ket. It’s written like this for a multitude of mathematical reasons. But for the purposes of this article, what we really need to know, is that the ket allows us to understand that we’re looking at a quantum system. It’s just a notation of convenience. A ket is the quantum mechanical symbol that encodes the state of a system. So in essence the ket, which is the quantum state, tells us the state of our quantum system. Who would’ve thought?
|Ψ⟩ = state of the quantum system
Now that this is established, it’s possible to infer that the rest of the things in Schrödinger’s equation will be affecting the quantum state in one way or another. This would mean that what we’re studying is the behaviour of the quantum state.
Now let’s look at the first bit:
This part is relatively simple to understand. Firstly, it’s important to know that both i and hbar are constants. Meaning that they’re just numbers. Very important numbers but still, just numbers.
i, as it turns out, is the square root of negative 1. Now if you aren’t familiar with complex numbers, you may think that this isn’t possible. But it is. Let’s not delve too deep into how. Just take me on blind faith here.
i = √(−1)
Coming to the weird ds:
This bit is pronounced as “doh by doh t”. You would be familiar with this notation if you took calculus in high school. But if you didn’t, it simply means the change of something (in this case, our quantum state) with respect to time. Of course, there are a few mathematical intricacies that I’ve left out but the essence of it is that this differential operator here measures how fast something changes with respect to time.
So now, we have this:
Put it all together and we know that we’ve got some number, i hbar, multiplied with doh by doh t of the quantum state. The doh by doh t of the quantum state measures the change of the quantum state with respect to time. We’ve covered most of the equation now. But the good stuff’s yet to come.
The Good Stuff
The right side of the equation: Ĥ|Ψ⟩.
This (the Ĥ) is known as the Hamiltonian operator. The first letter of the “Hamiltonian” explains why it’s denoted with an H, but why the hat? Well, in quantum mechanics, there are things known as operators. They’re written with a little hat on top. They basically “operate” on quantum states. They do things to the quantum state.
The Hamiltonian operator is the big guy of the quantum operators. It’s linked with the total energy of the system. Briefly explained:
Ĥ = kinetic energy+ potential energy+ any other energy in the system
So essentially all that’s happening here is that the quantum state, |Ψ⟩, is changing with respect to time in a very specific way, depending on the total energy of the system. This equation, therefore, allows us to predict how a quantum system should behave if we can mathematically write down all the energies of our system. Child’s play.
However, there’s, like always, a caveat. For systems larger than a few particles, this becomes impossible really really quickly. In other words, it’s quite easy to find a system that cannot analytically be solved using Schrödinger’s equation. We need numerical methods beyond that. Even humans have their failures here so computers do it for us. But even now, the strongest of computers, cannot analyze quantum systems larger than a few dozen atoms.
The Helium Atom
Now to demonstrate the complexity of the Hamiltonian, let’s juxtapose it with a relatively simple atom. The Helium atom. The only atom simpler than this is the Hydrogen atom. Now already we’re going to make some assumptions. First, let’s assume that the nucleus of the atom remains fixed in space. In other words, all the particles in the nucleus are stationary. Though that’s almost never the case, the nucleus moves about a little bit. Even in this massively simplified situation, the Hamiltonian of our system looks something like this:
It contains the following terms (in order of appearance): the kinetic energy of the first electron, the kinetic energy of the second electron, the electrostatic attraction between the nucleus and the first electron, the same thing with the second electron, and the electrostatic repulsion between the two electrons. Remember that this model is so simplified that it doesn’t even begin to take into account anything happening to the quantum system. As well as this, we haven’t considered the electron spins, the proton spins, and a lot more.
So in a sense, although we have this remarkably powerful equation, the mathematics behind it can take painstakingly long. Anyway, I hope you’ve learnt something about the Schrödinger equation. Thank you for reading. If you enjoyed it, I would appreciate your support!
