Lesson 17: The Schrodinger Equation
The Schrodinger Equation
Before we discuss Schrodinger’s Equation, you need to understand superposition.
To do that, let’s go back two the two-slit experiment. Remember when we sent through one photon or electron at a time through the slits? You can’t ask whether the photon or electron went through slit one or slit two. They are in a state of superposition. You have to say they went through both. But how can a physical particle go through two slits at the same time? It would be like you walking through two doors at the same time.
The particle has a probability of showing up physically in one place or another (70/30, 60/40, 71.83/28.17%).
When you observe the particle, it ceases to be in a superposition state and comes into its physical state in one location. There is no more superposition. This is called collapsing the wave function.
To make more sense of this, we need to introduce Erwin Schrodinger and the Schrodinger Equation. Born on August 12, 1887, in Vienna, Austria, Erwin Schrodinger went on to become a noted theoretical physicist and scholar who came up with the groundbreaking wave equation. He was awarded the 1933 Nobel Prize in Physics.
Before we go into all the details of his equation, let’s talk about a thought experiment that Schrodinger came up with that became known as Schrodinger’s Cat. It showed just how crazy the idea of superposition could be. Again this was just a thought experiment. It was never actually done. No cats were ever harmed.
Schrodinger said to put a cat in a box with a radioactive atom. The atom has a 50/50 chance of decaying in an hour. A radiation detector is set in the box, and if it detects radiation from a decaying atom, it is to release poison in the box, and the cat dies.
But the atom is in a state of superposition until you observe it. It is in a state of being decayed and not being decayed at the same time. So does that mean until you open the box, the cat is also in a state of being both alive and dead? It is only when you open the box and collapse the wave function that the cat is actually dead or alive.
There has been a lot of debate and discussion about Schrodinger’s Cat. What do you think? We will discuss this more in later lessons.
Now let’s get to the Schrodinger Equation.
The Schrodinger Equation is a function. It gives you a wave function for finding a particle at a certain position. Remember back when we were doing calculus? You plug numbers into a function, and you get an output. Obviously, The Schrodinger Equation is a lot more complicated than the functions we did.
A wave function is a wave with varying amplitudes. The greater the amplitude, the greater the chance of finding a particle in that position. The image above is a simplistic drawing of a wave function.
The Schrodinger Equation describes quantum behavior. It lets you know how a wave function evolves over space and time. It gives you the probability of finding a particle at a certain position.
Keep in mind the weirdness of Quantum Physics. We are saying the probability of finding a particle at a certain position but don’t be misled to think that when you find the particle, that is where the particle always was. Remember superposition. The particle is considered to be in many places at one time. It is only when we observe the wave that it collapses into a particle in one physical position. So weird, isn’t it?
Look at the picture above. It is an example of a wave function. Where the wave is the highest, there is a higher probability of seeing the particle when it collapses. But that does not mean it is going to be there. It may show up where the wave is the lowest. Kind of like when the weather man says there is an 80% chance of sunny skies and only a 20% chance it will rain. It rains.
And there it is! Look at the image above that is Schrodinger’s Equation. Actually, that is one of its many forms. This one involves position in the x dimension. It does not involve time. This means the particle is not moving.
There are only a few new things you need to learn here.
The symbol ψ is the Greek letter psi. This represents the wave function of the particle.
The letter U is the potential energy of the wave. Potential energy is a type of energy an object has because of its position. A boulder on top of a hill has a lot of potential energy because it could roll down pretty fast at any moment.
The letter E is the total energy of the wave.
We are not going to work out an example of Schrodinger’s Equation. That is beyond the scope of these lessons, but we will show you that you now understand everything in the equation.
The h, as you know, is Plank’s constant. That very small number that we keep seeing over and over again. You make it negative and then square it and divide it by 2 times the mass of the particle. You then take that result and multiply it by the 2nd derivative (which is taking the derivative of a derivative) of the wave function. Remember we said when we take a derivative, we are finding the rate of change? You add that result to U(x)ψ(x), which is the potential energy of the equation, and that must all equal Eψ(x), which is the total energy of the equation.
If you didn’t get all that, read it again. We are not trying to learn to solve the equation. We are just seeing what needs to be inputted into the equation to solve it. And you know everything that needs to go in it.
Once you do all of that, you will get the wave function which will tell you the probability of finding the particle at a specific location. Again, keep in mind you are only getting the probability of where the particle might be when the wave function is collapsed. How do you collapse the wave function? You observe it. Until you observe it, the particle is in many different places at once. It is in a state of superposition.
But you are made of particles, aren’t you? Can you be in a state of superposition?
Are the particles that makeup you once waves? Were you collapsed into reality? What does that even mean? This course will answer the question, “What does all of this mean for you?”
So go explain this to someone, if you can find them. They may be in a state of superposition.