The mathematics behind quantum mechanics

By Ryan Ng

This article will cover the mathematical foundations of quantum mechanics. You may know quantum physics by superposition, uncertainty, and so on—we’ll look at how the scientific community has turned these into a rigorous mathematical framework. If you’re new to quantum mechanics, it may be helpful to read the uncertainty principle article first.

We know quantum mechanics is about measuring the properties of particles. There are three components to measurement: a particle and its states, the observable (and measuring device), measuring device, and the value of its states. If this sounds confusing, keep reading.

What is a state? For example, 3 m/s is a velocity state. A position state could be the x-coordinate 3.5. Examples of observables are momentum, position, spin, acceleration, and energy. This is important terminology.

Onto the mathematics. First, we need to define the concept of an operator. Each observable has an operator: for example, there is a momentum operator, p. All operators can be expressed as a matrix. 

Second, we’ll define something called an eigenket, or eigenstate. States are expressed as eigenkets. For example, there could be a momentum eigenket for the value 2 kg m/s. It would be written as |2〉and it can be expressed as a vector.

When an operator acts on an eigenket, or in other words, when the matrix is multiplied by the vector, we yield an equation like the following:

p|2〉= 2|2〉

The momentum operator applied to (multiplied with) the eigenket yields the state value itself, multiplied by the eigenket. This is known as an eigenvalue equation. Physically, all this means is that if a particle is in the momentum state 2 kg m/s, its momentum is, 2 kg m/s. I know it sounds circular, but this is how we define it.

Let’s look at something else. |〈a|b〉|^2 is the probability of a particle being in the a state, given that it is in the b state.〈a|, which is called an eigenbra, is the transposed version of |a〉, meaning that its rows and columns are switched, and all the entries of the vector are replaced by their complex conjugate. Here, we’ve multiplied〈a|  by  |b〉.

Going back to our above momentum eigenkets, we have |〈2|2〉|^2 = 1. All this means is, if a particle has momentum 2 kg m/s, the probability of its momentum being 2 kg m/s is 1. Again, very circular. The reason we include an absolute value, is because〈2|2〉by itself is a complex number, which we call the amplitude; its absolute value squared yields the probability. 

As we know, particles actually have uncertainty. We don’t know exactly where a quantum particle is. So we can represent a particle as a linear combination of eigenkets multiplied by amplitudes. Suppose our particle is called α. Then we could have, in terms of momentum:

|α〉= √(0.4) |1〉+ √(0.6) |2〉

This means the particle α has a 40% chance of having a momentum 1 kg m/s, and a 60% chance of having a momentum 2 kg m/s. Let’s try applying a〈1| eigenbra to its left. 

〈1|α〉= 〈1|√(0.4) |1〉+ 〈1|√(0.6) |2〉

〈1|α〉= √(0.4)〈1|1〉+ √(0.6)〈1|2〉

I a particle has momentum 2 kg m/s, the probability of its momentum being 1 kg m/s is 0. So 〈1|2〉= 0 (〈1|2〉is called a bra-ket) and as we knew before, 〈1|1〉= 1.

Finally, we have

〈1|α〉= √(0.4)

Taking the absolute value squared, we find that the probability of the particle α having momentum 1 kg m/s is 0.4. 

These are the mathematical foundations of quantum mechanics. Although I still don’t know how physicists decided on using matrices and vectors, they do work; as it turns out when you follow the rest of quantum mechanics, we’re able to represent all physical properties and processes with this notation.