Math: Linear Operators

Def.: A linear Operator $A$ acting on vectors $\vert\psi\rangle$, $\vert\phi\rangle$ of a Hilbert space ${\cal H}$ has the property

$$A[\vert\psi\rangle + c \vert\phi\rangle] = A\vert\psi\rangle + c A\vert\phi\rangle, \quad c \in C.$$ (190)

Examples for linear operators: 1. $(n \times n)$-matrices $A$, acting on vectors ${\bf x } \to A({\bf x})$ (linear mappings)

2. The momentum operator $\hat{{\bf p}}$, acting on a Hilbert space of differentiable functions as

$$\hat{{\bf p}}: f\to \hat{{\bf p}}f, \quad \hat{{\bf p}}f({\bf x})=\frac{\hbar}{i}\nabla f({\bf x})$$

Example for a nonlinear operator: the operator that squares a function $f$, $A: f \to f^2$.

Expectation values of observables $A$ in particular should be real numbers because they represent the outcome of an average over many measurements. We have to introduce one additional definition to clarify this concept:

Def.: The adjoint operator $A^{\dagger}$ of a linear operator $A$ acting on a Hilbert space ${\cal H}$ is defined by

$$\langle \psi\vert A\phi\rangle = \langle A^{\dagger}\psi\vert\phi\rangle, \quad \forall \phi,\psi\in {\cal H}.$$ (191)

Def.: A linear operator $A$ on the Hilbert space ${\cal H}$ is called hermitian, if the following relation holds:

$$\langle A\psi\vert\phi\rangle = \langle \psi\vert A\phi\rangle, \quad \forall \phi,\psi\in {\cal H}.$$ (192)

Examples:

1. For complex two-by- two matrices

$$A = \left( \begin{array}{cc} a & b\\ c & d \end{array}\right) \... ...dagger} = \left( \begin{array}{cc} a^* & c^*\\ b^* & d^* \end{array}\right).$$

This means that the adjoint matrix $A^{\dagger}$ of a given matrix $A$ is given by the transposed conjugate complex of $A$, i.e. $A^{\dagger}=(A^*)^T$.

2. For complex two-by-two matrices

$$A = \left( \begin{array}{cc} a & b\\ c & d \end{array}\right) = A^{\dagger} \leadsto a=a^*, d=d^*, b^*=c.$$

3. The momentum operator $\hat{p}$ in one dimension, acting on wave functions $\psi(x),\phi(x), x\in R$ that vanish at $x\to \pm \infty$, is hermitian:

$$\langle \psi \vert \hat{p}\phi \rangle = -i\hbar \int dx \psi^*(x... ...= \int dx [-i\hbar \psi'(x)]^* \phi(x) = \langle \hat{p}\psi \vert\phi\rangle.$$

The expectation values of hermitian operators $A$ in any Hilbert space state vector $\psi$ are real indeed because

$$\langle A\rangle = \langle \psi\vert A\psi\rangle = \langle A\psi\vert\psi\rangle = \langle \psi\vert A\psi\rangle^*=\langle A\rangle^*.$$ (193)

This gives rise to the second part of axiom 2:

Axiom 2b: Physical observable quantities correspond to hermitian linear operators $A$ acting on Hilbert space vectors.

Furthermore, the following theorem holds:

Theorem: The eigenvalues of hermitian operators $A$ are real. This is because

$$A\vert\psi\rangle =\lambda \vert\psi\rangle \leadsto \lambda = \frac{\langle \psi \vert A\vert\psi\rangle }{\langle \psi\vert\psi\rangle} \in R.$$ (194)

This leads us to the most central part of ours axioms: