Operators

Quantum mechanics is a theory of probabilities and expectation values for the outcomes of experiments. We have already learned how to use the wave function to calculate expectation values of the position ${\bf x}$, the momentum ${\bf p}$, or any function of these, cf. Eq. (1.64).

We now slightly generalize our axiom 2:

Axiom 2a: In quantum mechanics, physical quantities like position, momentum, angular momentum, kinetic energy, total energy etc. correspond to linear operators $A$ that act on Hilbert space vectors. The position ${\bf x}$ corresponds to the operator `multiplication with ${\bf\hat{x}}$', the momentum ${\bf p}$ to the operator $-i\hbar \nabla$. Any other quantity depending on ${\bf x}$ and ${\bf p}$ becomes an operator $\hat{O}({\bf\hat{x}},{\bf\hat{p}})$ by this correspondence principle ${\bf x}\to {\bf\hat{x}}$ and ${\bf p} \to -i\hbar \nabla$. The commutation relation

$$[\hat{x}_k,\hat{p}_l] = i\hbar\delta_{kl}$$

$$\delta_{kl} := 1,\quad k=l,$$ (180)

holds.

Expectation values at time $t$ of any operator $A$ for a quantum mechanical system described by a vector $\vert\psi(t)\rangle$ in a Hilbert space (for example, a wave function $\Psi({\bf x},t)$), are defined by applying $A$ on $\vert\Psi(t)\rangle$ and calculating the Hilbert space scalar product

$$\langle A \rangle_t := \frac{\langle \Psi(t)\vert A\vert\Psi(t) \rangle_t}{\langle \Psi(t)\vert\Psi(t)\rangle}.$$ (181)