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 |ψ=ici|ui where ci=ui|ψ, which can be thought of as the representation of the ket |ψ in the |ui basis. To write the ket |ψ in matrix form, we simply stack the representation as a column vector as follows:

[u1|ψu2|ψui|ψ]=[c1c2ci].

Similarly, to express a bra ψ|=iciui| where ci=ψ|ui=ui|ψ, we write them as a row vector:

[ψ|u1 ψ|u2ψ|ui]=[c1c2ci].

Now, we look at an operator A^=ijAij|uiuj| where Aij=ui|A^|uj, which can be expressed as a matrix:

[A11A12A1jA21A22A2jAi1Ai2Aij]

As an example, let’s write the expression |ψ=A^|ψ in terms of matrix formulation. First, recall the following:

  • |ψ=ici|ui
  • |ψ=ici|ui

Then, we can express the coefficient ci as follows:

ci=ui|ψ=ui|A^|ψ=ui|A^I|ψ=ui|A^(j|ujuj|)|ψ=jui|A^|ujuj|ψ=jAijcj.

The last expression from the above can be expressed in matrix formulation as follows:

[A11A12A1jA21A22A2jAi1Ai2Aij][c1c2ci]=[A11c1++A1jcjA21c1++A2jcjAi1c1++Aijcj]=[c1c2ci].

Therefore, we arrive at the matrix formulation for ci we began with. Similarly, we can express other quantum mechanical quantities we have seen in terms of matrix formulation.

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 |ui where ui|uj=δij. Then, we can express a ket |ψ in the |ui basis, i.e.

|ψ=ici|uiwhereci=ui|ψ.

We will look at how to express the ket |ψ in different basis, |vj, i.e.

|ψ=jdj|vjwheredj=vj|ψ.

The step goes as follows: first, express the coefficient dj as above. Then, insert an identity, and using the resolution of the identity matrix in |ui basis, we find an expression that relates the coefficient dj and ci. Mathematically,

dj=vj|ψ=vj|I|ψ=vj|(i|uiui|)|ψ=ivj|uiui|ψ=iSjici.

Thus, to compute dj given ci, we simply need to compute the quantity Sji=vj|ui, which is called an “overlap.” Using the matrix formulation we studied in the previous chapter, we can also express this relationship as follows:

[d1d2]=[S11S12S21S22][c1c2].

We can also compute ci given dj in a similar fashion. If you follow the steps similar to the mathematical derivation above, it turns out that:

ci=jui|vjdjwhereui|vj=vj|ui=Sji.

To close this chapter, let’s look at how we perform a change of basis with an operator A^. We write the elements of operator A^ in |ui basis and |vj basis as Aiku=ui|A^|uk and Ajv=vj|A^|v, respectively. Then, we can compute Ajv in terms of Aiku as follows:

Ajv=vj|A^|v=vj|IA^I|v=vj|(i|uiui|)A^(k|ukuk|)|v=i, kvj|uiui|A^|ukuk|v=i, kSjiAikuSk.

Again, we can also go in the opposite direction. It turns out that to express Aiku in terms of Ajv is:

Aiku=j, SjiAjvSk.

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:

A^|ψ=λ|ψ,

where A^ is an operator, |ψ is a “special” ket called eigenstate (i.e. eigenvector) of A^, and λ is the eigenvalue of A^. As can be seen, the operator A^ takes the ket |ψ as an input, and outputs the same ket |ψ, only scaled with λ.