07 - Complex Inner Product Space

Complex Inner Product

Let $\psi ,\varphi$ be normalisable wave functions

$$⟨\varphi \mid \psi ⟩\equiv {\int }_{-\infty }^{\infty }\overline{\varphi (x)}\psi (x)dx\in \mathbb{C}$$

Schwarz Inequality for Integrals

$${\left|{\int }_a^b\overline{\varphi (x)}\psi (x)dx\right|}^2\le {\int }_a^b|\varphi (x){|}^{2}dx{\int }_a^b|\psi (x){|}^{2}dx$$

Square Norm

Let $\varphi$ be a normalisable wave function then

$$||\psi |{|}^2\equiv ⟨\psi \mid \psi ⟩$$

with norm $||\psi ||=\sqrt{||\psi |{|}^2}$

Note that $\varphi$ is normalised if $||\psi ||=1$

Properties of Complex Inner Product Space

1. $\overline{⟨\varphi \mid \psi ⟩}=⟨\psi \mid \varphi ⟩$

2. $⟨\varphi \mid {\alpha }_1{\psi }_1+{\alpha }_2{\psi }_2⟩={\alpha }_1⟨\varphi \mid {\psi }_1⟩+{\alpha }_2⟨\varphi \mid {\psi }_2⟩$ for all ${\alpha }_1,{\alpha }_2\in \mathbb{C}$

3. $⟨{\beta }_1{\varphi }_1+{\beta }_2{\varphi }_2\mid \psi ⟩=\overline{{\beta }_1}⟨{\varphi }_1\mid \psi ⟩+\overline{{\beta }_2}⟨{\varphi }_2\mid \psi ⟩$ for all ${\beta }_1,{\beta }_2\in \mathbb{C}$

4. $||\psi |{|}^2\ge 0$

5. $||\psi |{|}^2=0$ if and only if $\psi =0$

Complex Inner Product Space

Let $\mathcal{H}$ be a complex vector space with

1. Complex Inner Product $⟨\varphi ,\psi ⟩\in \mathbb{C}$ between vectors $\varphi ,\psi \in \mathcal{H}$

2. Satisfying the Properties of a Complex Inner Product Space

Then $\mathcal{H}$ is a complex inner product space

States as Rays of $\mathcal{H}$

States of a Quantum System are elements of a Complex Inner Product Space $\mathcal{H}$

with proportional vectors representing the same state

Hilbert Space

Hilbert Space is a Complete Complex Inner Product Space

where the states of a Quantum System are elements of the Hilbert Space

Linear Operator on $\mathcal{H}$

Linear Operator is a pap $A:\mathcal{H}\to \mathcal{H}$ on complex vector space $\mathcal{H}$ satisfying

$$A({\alpha }_1{\psi }_1+{\alpha }_2{\psi }_2)={\alpha }_{1}A{\psi }_1+{\alpha }_{2}A{\psi }_2$$

for all ${\alpha }_i\in \mathbb{C}$ and ${\psi }_i\in \mathcal{H}$ for $i=1,2$

Adjoint of a Linear Operator

Let $\mathcal{H}$ be a complex inner product space
Let $A$ be a Linear Operator on $\mathcal{H}$ then

Adjoint $A^{\ast }$ of $A$ satisfies (by definition)

$$⟨A^{\ast }\varphi \mid \psi ⟩=⟨\varphi \mid A\psi ⟩$$

for all $\varphi ,\psi \in \mathcal{H}$

Self-Adjoint Linear Operator

Let $\mathcal{H}$ be a complex inner product space
Let $A$ be a Linear Operator on $\mathcal{H}$ then

If $A^{\ast }=A$ then

$$A\text{ is Self-Adjoint }$$

Complex Inner Product in Three Dimensions $\psi ,\varphi$ be normalisable wave functions

Let

$$⟨\varphi \mid \psi ⟩\equiv {\iiint }_{{\mathbb{R}}^3}\overline{\varphi (\mathbf{x})}\psi (\mathbf{x})d^{3}x$$

Self-Adjoint Operators from Real-Value Functions

Consider real-valued function $f(x)$
Consider operator $f(X)$ by

$$(f(X)\psi )(x)=f(x)\varphi (x)$$

Then
$f(X)$ is self-adjoint as

$$⟨\varphi \mid f(X)\psi ⟩={\int }_{-\infty }^{\infty }\overline{\varphi (x)}f(x)\psi (x)dx={\int }_{-\infty }^{\infty }\overline{f(x)\varphi (x)}\psi (x)dx=⟨f(X)\varphi \mid \psi ⟩$$

Properties of Self-Adjoint Linear Operators

Let $A,B$ be self-adjoint linear operators then

1. $(A+B)$ is self-adjoint where

$$(A+B)\psi (\mathbf{x})=A\psi (\mathbf{x})+B\psi (\mathbf{x})$$

2. $\alpha A$ is self-adjoint where $\alpha \in \mathbb{R}$ is a real constant

3. Composite Operator $AB$ is self-adjoint if $A$ and $B$ commute ($AB=BA$)

Proof

$(i)$ and $(ii)$ follow from definition almost immediately

For $(iii)$ then

$$⟨\varphi \mid AB\psi ⟩=⟨A\varphi \mid B\psi ⟩=⟨BA\varphi \mid \psi ⟩$$

Hence adjoint $(AB{)}^{\ast }=BA$ so by self-adjointness then $AB=BA$