Basics of Quantum Mechanics Lesson 2

Last time we looked at one of the most important principles of the quantum mechanics - Heisenberg uncertainty principle. This time, we're gonna try to understand more about the micro world by analysing one of the most famous experiments - double slit experiment. Also we're gonna find out what a wave function is and how it helps to analyse the micro world.

The idea behind the double slit experiment is dead simple - a wave of light (works also for water & sound) is sent through a screen with two slits. If the size of the slits is correct (has to be comparable with the wave length of the light waves) two parts of the wave interfere with each other, creating an interference pattern on the screen. Diagram below illustrates the setup of the experiment, as well as the real view of the interference pattern in b).





So there's nothing particularly shocking about this. It's not hard to see water waves interfering in everyday life. However, there was a shocking surprise that scientists stumbled upon when using the same experimental setup with particles! But before finding out what this shocking surprise was let's think what would happen if you would try experimenting with this double slit setup while using particles. For the sake of simplicity let's imagine that our particles are like tennis balls. Let's assume you have to hit a wall with a tennis ball through a screen of two big slits. After throwing a dozen or more balls you would observe a pattern similar to this:

Once again - nothing shocking here. So after this "experiment" let's switch to the microscopic world and try the same thing with microscopic particles, let us say electrons. Of course, by using common sense, everyday experience and all your tennis knowledge you might guess that we might get the same pattern. But what we actually get is one of the most shocking discoveries of modern physics:

Yes, we get an interference pattern like in the case of waves. Now, after scratching your head for a while, you might say that this must be because a bunch of electrons interfere like waves while travelling to the screen. This is a perfectly reasonable assumption. So let's test it. Let's try sending a single electron through the slits, that way it won't interfere with anything and using common sense once again it shouldn't form an interference pattern. Hey but quantum mechanics wouldn't be quantum mechanics if that was the case would it? What we actually get is another bizarre thing. If we send electrons one by one, at first no interference pattern is visible, but eventually an interference pattern appears! Now this is really strange, using common sense (well at least in a classical physics point of view) the electron can't go through both slits at the same time and interfere with itself. So something strange has to be going on.

Naturally scientists were also shocked so they decided to find out, which slit the electron goes through. So guess what happen then. Well when they tried to observe the electron, no interference pattern appeared! Now once again this is very strange. It's actually connected to the Heisenberg's uncertainty principle - when you send a photon to collide with an electron (observing the electron) you change it's momentum. But we'll look at that  later.

So what conclusions can we draw from all of this. Well firstly, it is clear that rules of classical physics break down when it comes to the microscopic world. Secondly, particles in the microscopic world seem to have wave-like properties. And finally, due to Heisenberg's uncertainty principle, we can't be totally sure of a position and momentum of a particle. So due to these conclusions it is clear that if there was a theory that described the microscopic particles it had to describe particles with wave-like properties and would have to use Heisenberg's principle. And that theory of course is quantum mechanics. So let's look at two corner stones of quantum mechanics - Schrodinger equation and wave functions.

Now if you remember your high school days (or if you are in high school), you might recall that one equation called Newton's 2nd law was highly important. Almost all of the most important laws and equations in mechanics can be derived from or using the famous equation F=ma. For instance equations of constant acceleration motion ( s(t)=Vot +(-) (at^2)/2) can be derived using Newton's 2nd law. Furthermore simple wave equations can also be derived using the same equation. So natural question would be - is there a way to derive a equation, would tell us where to find a particle at a given time t in quantum mechanics?

The answer is yes! This equation is called the wave function. It is noted with a Greek letter psi. What does the wave function do? The answer is that it helps you to find out how likely it is to find a particle in a given place at a given time (actually it is the probability density function that does this, but the p.d.f is just the squared absolute value of the wave function).

The notation of the wave function (time independent):

An example of a graph showing the p.d.f. of a particle's position. It can be found using the wave function



A lot of quantum mechanics is about finding the wave function for different stuff, for instance the electrons orbiting a nucleus, a particle in a potential well and so on. So how do we find this wave function?? Well like in classical physics it was necessary to solve Newton's 2nd law, it is necessary to solve Schrodinger Equation to find a wave function in quantum mechanics.

The Famous Schrodinger equation (in a simpler, time independent form):

So by solving this equation you can get the wave function. But it's enough for today. Next time we're gonna look at some simple solutions to the wave function and also something called De Broglie's wavelength. Thanks for reading!

LINK to lesson 1

LINK to lesson 3

LINK to lesson 4

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