Quantum Mechanics 101

Quantum Mechanics 101

Overview

Quantum mechanics is hard! Developing a quantum simulator is even harder! It requires knowledge of advanced math and physics. Fortunately, you don't need a major in quantum physics if you want to use the engine. You need the basics. In this page we will guide you through the world of quantum mechanics. Remember: only the basics.

What is a quantum ?

Classical physics is based on the idea of continuity. This means that you can always subdivide an object. For example the derivative. Here is the definition of the derivative:

$$ \frac{dy(x)}{dx} = \lim_{\Delta x \to 0} \frac{y(x + \Delta x) - y(x)}{\Delta x} $$

Lets say that we choose $\Delta x = 0.1$. Our derivative won't be precise. But for smaller $\Delta x$ we will "eventually" approach the correct value. But what does "eventually" mean? Lets say that every minute our $\Delta x$ gets 10 times smaller. When we will reach this "eventually"? 10 minutes? 1 hour? The end of time? This question makes no sense! Why? Because the number line is continuous. We can always subdivide it. There is no "smallest" number.

Back in the days this paradox wasn't a big deal. Science managed to make pretty accurate predictions. Until the late 19th century when the science community faced a catastrophe: the ultraviolet catastrophe to be precise. Science failed to predict the spectral radiance of black bodies. This seemed weird at the time.

The catastrophe was resolved by Max Plank. With his solution he predicted the color of the sun. How did he do that? He quantized light. To quantized something means to subdivide it to finite subparts called quanta. A quantum is an object that you cannot subdivide. For example the pixels on your screen are quanta. This little trick saved centuries of science work. We have to thank him.

What is a quantum state ?

In the beginning of the 20th century Albert Einstein found out that the quantum of light is the photon: the particle of light. This shocked the world. It was well known that light is an electromagnetic wave. This is a contradiction. Particles are not waves and waves are not particles. How could this be possible?

Firstly, how do we know that light is a wave? Because of Thomas Young's double slit experiment. Here it is: we have a wall with two slits. Behind the wall we have a screen that records everything that reaches it. If we shoot a beam of light trough the slits we will get an interference pattern on the screen. This pattern could only be generated by a wave source. So light is a wave? But Max Planck and Albert Einstein proved that light is made of particles. So what is light? But here is something strange. If we shoot electrons through the slits we will still generate the interference pattern. So electrons are also waves? That on earth is going on?

Physicists developed quantum mechanics to explain the wave-particle duality. And here is their answer: probability. Quantum mechanics is based on probability. To see what's going on under the hood let's study the simplest quantum system: a single bit. A single bit has two possible states: on and off, or 1 or 0. How could we express this mathematically? Let's try this:

$$ \ket\psi = \alpha\ket0 + \beta\ket1 $$

Don't fear the Greek letters and the odd brackets. Here is the explanation. We will use these triangular brackets (called "kets") to denote "state". For example $\ket0$ means "the 0 state of the bit" and $\ket1$ is "the 1 state". The $\ket\psi$ is our bit state. And $\alpha$ and $\beta$ are either 0 or 1. If $\alpha = 1$ and $\beta= 0$ the bit state is $\ket\psi = \ket0$. If $\alpha = 0$ and $\beta= 1$ the bit state is $\ket\psi = \ket1$.

Now let's get quantum. It was mentioned that "Quantum mechanics is based on probability.". So let's study a bit that has unknown value. This is also called a qubit (quantum bit). We don't know whether it is 0 or 1. Mathematically we will do this by allowing $\alpha$ and $\beta$ to have any value. Their values must be related to the probability of "being in that state". For example $\alpha = 0.6$ has higher probability than $\alpha = 0.2$. How can we relate the value and its probability?

Here is a fact: the sum of the probability of "being 0" and the probability of "being 1" must be 1. This is true because the qubit has only two states: 0 or 1. Only two. So if we sum up the probabilities we will get 1.

Let's look again at the qubit equation: $\ket\psi = \alpha\ket0 + \beta\ket1$. Probability is a scalar but we have these weird kets. How can we get rid of them? If you are have mastered vectors you may recognize this equation. It looks like the relation between a vector, its components and the coordinate basis. If you don't understand this we recommend you to learn more about vectors. Anyway, back to the question: relationship between a ket and a scalar? Vectors are related to scalars by the dot product. Could we invent a "ket dot product"? Actually, yes. Let's say that our "ket dot product" is written like this:

$$ \braket{\psi|\psi} $$

The left psi is called a "bra", so the whole expression is a "bra-ket". This explains the weird naming. The whole "bra-ket" notation is also called "Dirac notation". It is very useful and we will use it from now on. Let's now define the "bra-ket product" as the "inner product". It will be similar to the dot product:

$$ \braket{\psi|\psi} = \alpha^2 + \beta^2 $$

Now we are setting the product to the total probability of 1:

$$ \alpha^2 + \beta^2 = 1 $$

This looks like the sum of probabilities. So we can say that the state's coefficient is the square root of the state's probability.

Before we end this section, we will explore what does the bra mean. Let's define that the bra is:

$$ \bra\psi = \bra0\alpha + \bra1\beta $$

If we multiply it by its ket we get:

$$ \braket{\psi|\psi} = \alpha^2\braket{0|0} + \alpha\beta\braket{0|1} + \alpha\beta\braket{1|0} + \beta^2\braket{1|1} $$

To get the dot product equation we must set: $\braket{n|n} = 1$ and $\braket{n|m} = 0$. We can use the Kronecker delta to state that:

$$ \braket{\mu|\nu} = \delta_{\mu\nu} $$

One last thing. If $\alpha$ and $\beta$ are complex numbers the math is a bit different. We have to define that:

$$ \bra\psi = \bra0\alpha^* + \bra1\beta^* $$

The star is the complex conjugate of the coefficient. When the coefficients are complex they are called "probability amplitudes". In the complex realm the relation between probability: $p$ and probability amplitude: $\alpha$ is:

$$ p = \alpha^*\alpha $$

This concludes our quantum state journey. But how does these probabilities explain the double slit experiment?

What is a wave function ?

In the last chapter we saw that the quantum state encapsulates all possible events and their probability amplitudes. Let's consider the general case. We have a quantum state with $N$ events. The probability amplitude of state $n$ is $\alpha_n$. So the equation for the quantum state is:

$$ \ket\psi = \sum_{n=0}^N \alpha_n \ket n $$

For shorthand we will denote the sum from $0$ to $N$ as just the sum over the dummy variable $n$. Ok, let's now find the state's inner product:

$$ \braket{\psi|\psi} = \sum_n \alpha_n^* \alpha_n $$

Don't forget the Kronecker delta equation. We can use the vector interpretation of the ket to claim that the probability amplitude is related to the inner product of the basis and quantum states:

$$ \alpha_n = \braket{n|\psi} $$

This is enough. Let's study a new system based on this general case. We will think about a quantum particle in a box. One dimensional box to be precise. Our "box" will start from $x = -1$ and end at $x = 1$. Our particle could be between $x = -1$ and $x = 1$. The number of points between -1 and 1 is the number of possible states. But now we arrive at the paradox: how many points are in this domain? Fortunately, we can use a very useful tool: the integral. The integral is an infinite sum. So here is how our state equation looks like:

$$ \ket\psi = \int_{-\infty}^\infty \alpha_x \ket x dx $$

Here $\alpha_x$ is actually a function of the position $x$. So let's replace it with the function $\psi(x)$:

$$ \ket\psi = \int_{-\infty}^\infty \psi(x)\ket x dx $$

This function is called the wave function. It encodes the probability amplitude of "being in the $x$ position". For example let $\psi(0) = 0.5$. This means that the probability amplitude of the particle being in the 0th position is 0.5. If we square the value we will get the probability which is 0.25 or 25%. But remember that $\psi(x)$ must be normalized.

Let's see how could we normalize the wave function. Look up the generalized inner product formula. In our wave function state we can do something similar:

$$ \braket{\psi|\psi} = \int_{-\infty}^\infty \psi^*(x)\psi(x) dx $$

This formula may seem weird at first but remember that integrals are sums where $N = \infty$. Ok, let's say that our quantum state is normalized: $\braket{\psi|\psi} = 1$. This means that the integral must also be 1. Now let's think about probability density. The probability density functions tells us how probability is distributed. Let's think of the density function $\rho(x)$. If we integrate it we should get the total probability: 1.

$$ \int_{-\infty}^\infty \rho(x) dx = 1 $$

This is the same as $\int_{-\infty}^\infty \psi^*(x)\psi(x) dx$ term. We can equate probability density and probability amplitude:

$$ \rho(x) = \psi^*(x)\psi(x) $$

This is also known as Born's rule. So, what is a wave function? It is the square root of the probability density.

In 1925 Erwin Schrödinger formulated an equation that predicts how the wave function evolves over time. Here it is:

$$ i\hbar\frac{\partial}{\partial t}\psi = \frac{-\hbar^2}{2m}\nabla^2\psi + U\psi $$

This is the famous Schrödinger equation. We won't derive it. Actually it is impossible to fully derive. But you can justify it. Anyway, if you know the wave function at one moment the equation will predict its future. This equation is responsible for the double slit experiment. If you plug in the double slit system in the equation you will get an interference pattern. This is why the answer to "is it a particle or a wave?" is "probability". This weird phenomena is a compromise of wave/particle duality. It is not very satisfying but this is how nature operates on the micro level.

What is a spinor ?