Basics of Quantum Mechanics Lesson 4

So last time we calculated the wave function of a free moving particle and got the following answer:

SE for a free moving particle:

Solution for this equation:  

So what do these equations tell us about the free moving particle? To find the answer to this question we need to find the probability density function. If you recall, the probability density function is basically just the square of an absolute value of the wave function. So by calculating the probability density function we would get:


|Psi |^2 = Psi x complex conjugate of Psi =
= A(cos(kx − wt) + i sin(kx − wt))A(cos(kx − wt) − i sin(kx − wt))
= A^2(cos2(kx − wt) + sin2(kx − wt))
= A^2


Now if you look at the result more carefully, you should realise that there's some strange stuff going on. The probability density function is constant, that means that the probability of finding a the particle in any point of space is equal. In more simple words, you have the same probability of finding the free moving particle in your room and somewhere in the other side of the universe. This is due to the fact that we have not taken into account the uncertainty of the momentum of a particle.



So you might be asking  - why are we studying free moving particles, while in reality most of the matter is situated in confined atoms and molecules. And you're definitely right, so let's look at another system, which might help us understand the behaviour of electrons, which are "trapped" in an orbit of an atom.


The easiest way of understanding simple atoms or similar systems is using a thing called infinite potential well. That is a system in which a particle is trapped between infinite potential "walls", which can be for instance be various electric of magnetic fields or anything similar, which confines the particle in a given space. After all, in a sense electrons orbiting a nucleus are also in a similar potential well - if they get too close to the nucleus, they are pushed back, if the get too far they are attracted back. 


Infinite potential well:

So how does particle behave in such a well? Well let's firstly imagine what would happen in a classical situation, let's say for a tennis ball trapped between two walls. It's clear that (neglecting gravity, which would eventually bring the ball down) any position between the walls has the same probability of finding the ball there.

However, in quantum mechanics, things are different. We already know that particles tend to behave as probability waves, so it's clear that the probability of finding the particle in any point of space will not be equal( a good analogy of a wave in a sort of potential well is a rope fixed at two places and oscillating).

So how can we find out at which points of space we are most likely to find the particle? By solving the Schrodinger equation of course.

And since this is a basic course we're not gonna bother ourselves with the process of solving the SE. We're just gonna skip right to the solution which is:

Wave function for a particle confined in a infinite potential well: Psi(x) = Asin kx

Now to calculate the probability density function, we need to apply the initial conditions of the system - as we know the particle can't be in points of space, which are beyond the boundaries.

In an infinite potential well the boundary conditions imply:
 Psi (0) = Psi  (L) = 0.
Resulting in the limitation that k must have one of the discrete set of values
kn = (pi . n)/L , where
n = 1, 2, 3 . . .


Now when we know k, we can find allowed energies.
Using  E= (p^2)/m and De Broglies wavelength p= h/ lambda = hk/ 2pi gives the following:

En = (h^2 . n^2)/(8mL^2)
 n = 1, 2, 3 . . .


The allowed energies for the trapped particle are quantised according to the value of
n which is known as a quantum number.


Note that we can calculate the quantum number using simple calculations, which we'll look at later. It's important to notice that for macroscopic objects n is very larger, which gives rise to the so called correspondence principle - the idea that if n is very large (which is definately the case for macroscopic objects), quantum mechanics prediction become similar to classical physics prediction.

And we can see this by finding the probability function of the infinite potential well, and graphing it with different values of n:

 

As we can see as n grows, the wavelenght of the wave in the graph becomes shorter. In n was very very big, it would look as every point in the confined space has the same probability (like in the classical physics case).


So that's all for now, thanks for reading!


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Basics of Quantum Mechanics Lesson 3

So last time we looked at the corner stone of Quantum Mechanics - Schrodinger equation:

* Note it's a simplified version for 1 dimension.

This time we're gonna look at some simple solutions of this equation, which will show us how microscopic particles behave.

So let's start by simplifying Schrodinger equation for a case of a free moving particle. Since the zero of potential is an arbitrary choice we choose to make U0 = 0. This gives:
(1)


Also it is useful for our solution to express the energy E in terms of momentum p (where Ek= (mv^2)/2 & p=mv), which gives:

 (2)

Substituting (2) into (1) gives:

This can be simplified further using De Broglie's wavelenght (something we covered last time: p=h/lambda) and defining a quantity called wave number k: k= 2pi/lambda:

( Schrodinger equation for a free moving particle)


So how can we solve this equation? We'll it's not a trivial process, and it doesn't really matter for learning the basics, so let's skip to the answer:

(where A is some constant, e is the exponential function, i is the square root of -1 from complex numbers, k is the wave number)

Note that this is a time-independent solution. The time dependence of the wave function comes from solving the time dependent Schrodinger equation, which gives:

So ok, we solved the SE for a free moving particle and found the wave function, but what does it tell us about the particle? Don't worry we're gonna look at it next time. Thanks for reading!


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Basics of Quantum Mechanics Lesson 1

So I've been learning quantum mechanics for some time now and I can say that it's a charming subject. It's hard to find another part of physics which makes you panic and adore it at the same time. So that's why I decided to create these simple lessons of the basics of quantum mechanics.

I tried to study quantum mechanics by myself a couple of times in the past and what I found out was that it is really hard to find good lessons, lectures or books about the basics online. I mean you can easily find good e-books about the whole philosophical part of quantum mechanics, which explain the basic principles, but hardly give you the real taste of the subject with whole the maths part of it. On the other hand, you can find a lot of hardcore stuff, like lectures and videos which require you to know the subject at a very high level. So as you can see it's really hard to come up with some real basic quantum mechanics with all the appropriate maths for a beginner. So I decided to make these basic lessons that give you all the basic stuff you need.

So firstly let's look at some basic information about quantum mechanics: Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales, the so-called quantum realm. In advanced topics of quantum mechanics, some of these behaviors are macroscopic and only emerge at very low or very high energies or temperatures. The name, coined by Max Planck, derives from the observation that some physical quantities can be changed only by discrete amounts, or quanta, as multiples of the Planck constant, rather than being capable of varying continuously or by any arbitrary amount. For example, the angular momentum, or more generally the action, of an electron bound into an atom or molecule is quantized. While an unbound electron does not exhibit quantized energy levels, an electron bound in an atomic orbital has quantized values of angular momentum. In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons and other atomic-scale objects.


So let's start easily. At the heart of quantum mechanics there is the famous Heisenberg uncertainty principle. This principle appears in all parts of quantum mechanics so it's important to know it by heart and get it out of the way early.

Heisenberg uncertainty principle:      

  


 So what does this mean? It means that the product of the uncertainty in position of a particle and momentum must always be higher than some constant quantity, which is h bar/2, where h bar= h/2pi (Planck constant). This means, that if you try to detect a particle you can only do it with a certain precision. For instance, if you detect it's position with a very small uncertainty then you will have a very big uncertainty in momentum of the particle. 

Now this is very strange, because we are so used to the "fact" that we know the position of anything very accurately. And now suddenly we realise that when it comes to microscopic world, we can't even find the exact position of the particle. But this is just the beginning of the whole strangeness of the quantum world.

But why can't we detect particles position or moment with a big precision? Well it's mostly because of the way we detect particles. Long story short, we detect a particle by shooting a photon at it (a particle of light). However, a photon also has some momentum and as strange as it sounds you can simply "knock" a particle away with a photon. Also, if you want to find out the position of a particle with a high precision, you must use a high frequency light wave (a high energy photon, where the energy of a photon E is given by E=hf, where h- Planck constant and f- frequency) which naturally has a bigger momentum, which causes a particle to be knocked away at a high velocity. This is exactly what causes the uncertainty in the momentum.


Now if this looks strange to you then don't worry. I mean the whole world was in shock when quantum mechanics was established. One of the most shocking things was the collapse of the Newtonian determinism. This is an idea that if you know enough information about a particle, you can calculate it's position and velocity in the future, that is if you have enough information you can tell the future. You can imagine how shocked the world of science was when after a couple of hundread years suddenly they realised that you can't actually tell the particle's position and momentum with an exact precision (thus you can't tell the future of a particle with an exact precision). Long story short, the world went upside down in the beginning of the XX century - particles became wave - particle probability waves, nothing was certain anymore, electrons decided to show up at two different places at one time, other particles started to travel back in time and so on. Whole hell broke through in the world of science back in those days.  All of these crazy things brought to the world of science by the quantum mechanics can really cause headaches, but hey don't worry, it can be explained in a very elegant way. But we'll look at it next time.Thanks for reading!

PS. I'm not an expert of quantum mechanics, so if you notice any mistakes feel free to tell me. Thanks!

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