In the previous parts we talked about row and column vectors, without referring to them as a bunch of numbers. Now we will do the same for matrices, which we will call operators.
An operator $O$ is defined by two properties:
- Given an operator $O$ and a ket $|\psi\rangle$, we can create the ket $O|\psi\rangle$. This operation is linear in the sense that $O(k|\psi\rangle)=k(O|\psi\rangle)$ and $O(|\psi\rangle+|\phi\rangle)=O|\psi\rangle+O|\phi\rangle$.
- Given an operator $O$ and a bra $\langle\psi|$, we can create a bra $\langle\psi|O$. This operation is linear.
However, this makes it sound like the action of an operator on a bra and the action of an operator on a ket are completely independent. However, we know that one action is supposed to determine the other. In fact, the two actions are related by $(\langle\psi|O)|\phi\rangle=\langle\psi|(O|\phi\rangle)$. This nicely implies the expression $\langle\psi|O|\phi\rangle$ is unambiguous.
We can do all the familiar operations from matrices on operators. We can add operators by saying $(O+P)|\psi\rangle=O|\psi\rangle+P|\psi\rangle$ and similarly for bras, we can multiply operators by numbers via $(kO)|\psi\rangle=k(O|\psi\rangle)$, and we can also multiply operators by $(OP)|\psi\rangle=O(P|\psi\rangle)$.
Operators have all the properties of matrices: multiplication is associative, some operators do not commute, …
The dagger operation is also defined for operators. The dagger $O^\dagger$ is defined by $O^\dagger|\psi\rangle=(\langle\psi|O)^\dagger$ and $\langle\psi|O^\dagger=(O|\psi\rangle)^\dagger$. With these definitions, the dagger has the following property:
Theorem. Suppose we have a bunch of bras, kets, and operators. Then the dagger of the product is the product of the daggers, but in reverse order.
An important class of operators consists of the outer product of bras and kets. Given a bra $\langle\psi|$ and a ket $|\phi\rangle$, they have the outer product $|\phi\rangle\langle\psi|$. This operator acts on bras and kets in the obvious way, e.g. $(|\phi\rangle\langle\psi|)|\chi\rangle=|\phi\rangle\langle\psi|\chi\rangle$.
Types of operators
Two classes of operators are particularly important in quantum mecahnics.
An operator $U$ is unitary if the length of $U|\psi\rangle$ is the same as the length of $|\psi\rangle$. Equivalently, an operator is unitary if $U^\dagger U=I$ where $I$ is the identity matrix. Proof: the length of $U|\psi\rangle$ is $(U|\psi\rangle)^\dagger U|\psi\rangle=\langle\psi|U^\dagger U|\psi\rangle$.
An operator $H$ is hermitian or self-adjoint if $\langle\psi|H|\psi\rangle$ is always real. Equivalently, $H$ is hermitian if $H=H^\dagger$.
In the following sections, we will see that hermitian operators correspond to physical quantities and unitary operators describe time evolution.