Basics of Quantum Mechanics 2
We continue our study of basics of quantum mechanics. Part 1 of this blog series can be found here.
4. Matrix formulation of quantum mechanics
Matrix formulation of quantum mechanics was first proposed by Werner Heisenberg, Max Born, and Pascual Jordan. Later, it was unified with wave mechanics formulation introduced by Erwin Schrödinger.
Matrix formulation is particularly useful when we work with finite discrete bases. This is because by using matrix formulation, quantum mechanical quantities can be expressed mostly in matrix multiplication, which is something we understand quite well from linear algebra, and also has been highly optimized both in terms of software (e.g. leveraging structures like sparsity) and hardware (e.g. using GPUs).
Let’s start with the basics. Recall that a ket
Similarly, to express a bra
Now, we look at an operator
As an example, let’s write the expression
Then, we can express the coefficient
The last expression from the above can be expressed in matrix formulation as follows:
Therefore, we arrive at the matrix formulation for
5. Change of basis in quantum mechanics
Recall that we can represent state space with an orthonormal basis. For example, we have been working with the basis
We will look at how to express the ket
The step goes as follows: first, express the coefficient
Thus, to compute
We can also compute
To close this chapter, let’s look at how we perform a change of basis with an operator
Again, we can also go in the opposite direction. It turns out that to express
6. Eigenvalues and eigenstates in quantum mechanics
As we saw in Chapter 2 of the previous post, operators are mathematical objects that allow us to describe physical properties. Eigenvalues and eigenstates are particularly important in quantum mechanics, because when we measure the physical properties, the only possible outcome is one of the eigenvalues of the associated operator; also, the state of system, after the measurement, is in the corresponding eigenstate. We start with the third postulate of quantum mechanics that summarizes the above:
Postulate III: The result of a measurement of a physical quantity is one of the eigenvalues of the associated observable.
Mathematically, we can write down the eigenvalue equation as follows:
where