Quantum observables

Quantum observables as Hermitian matrices

Quantum observables are represented by Hermitian matrices, which act on states represented by vectors. However, all experiments result in observations of real numbers, which is why we use Hermitian matrices to describe them.

Let $M$ be a quantum observable. Then, the measurable values of an observable are exactly its eigenvalues, $\lambda$, which are real (as we saw in the previous section), and are associated with eigenstates, $\ket{a}$, such that $$M \ket{a} = \lambda \ket{a}$$

We can interpret eigenstates and eigenvalues in the following way: if a system is prepared in the eigenstate, $\ket{a}$, and a measurement is made of the observable, $M$, then the value of the measurement will be the eigenvalue, $\lambda$, with 100% certainty.

This is a special case of a more general postulate of quantum mechanics, called the probability postulate.

The probability postulate

Suppose we have some observable, $M$, with eigenvalue, $\lambda$, associated with eigenstate, $\ket{a}$, and suppose that a system has been prepared in some arbitrary state, $|b\rangle$.

The postulate states that the probability that we will measure the observable to have the value $\lambda$, written $$|\braket{a}{b}|^2$$ to be the square of the inner product of $\ket{a}$ and $\ket{b}$.

That is, $$|\braket{a}{b}|^2 = \braket{a}{b}^* \braket{a}{b}$$

Recall, quantum states, $\ket{b}$, are unit states, meaning that $$\braket{b}{b} = 1 = \braket{b}{b}^*$$ which makes sense of the interpretation of the eigenstate above, since $$|\braket{b}{b}|^2 = \braket{b}{b}^* \braket{b}{b} = 1 \cdot 1 = 1$$