Introduction: Beyond Just Direction
A vector space tells us we can add vectors and scale them, but it says nothing about length or angle. To talk about how "big" a quantum state is or how "similar" two states are, we need an additional structure: the inner product.
The Inner Product: Formal Definition
An inner product on a vector space \(V\) over \(\mathbb{C}\) is a function \(\langle \cdot, \cdot \rangle: V \times V \to \mathbb{C}\) satisfying:
- Conjugate Symmetry: \(\langle \vec{u}, \vec{v} \rangle = \overline{\langle \vec{v}, \vec{u} \rangle}\)
- Linearity (in second argument): \(\langle \vec{u}, a\vec{v} + b\vec{w} \rangle = a\langle \vec{u}, \vec{v} \rangle + b\langle \vec{u}, \vec{w} \rangle\)
- Positive Definiteness: \(\langle \vec{v}, \vec{v} \rangle \geq 0\), with equality iff \(\vec{v} = \vec{0}\)
Note: Physicists often use linearity in the second argument (as above). Mathematicians sometimes use linearity in the first.
Norms and Distances
The inner product defines the norm (length) of a vector:
\[\|\vec{v}\| = \sqrt{\langle \vec{v}, \vec{v} \rangle}\]And the distance between two vectors:
\[d(\vec{u}, \vec{v}) = \|\vec{u} - \vec{v}\|\]Worked Examples
Example 1: The Standard Inner Product on \(\mathbb{C}^n\)
For vectors \(\vec{u} = (u_1, \ldots, u_n)\) and \(\vec{v} = (v_1, \ldots, v_n)\) in \(\mathbb{C}^n\):
\[\langle \vec{u}, \vec{v} \rangle = \sum_{i=1}^{n} \overline{u_i} v_i = \bar{u}_1 v_1 + \bar{u}_2 v_2 + \cdots + \bar{u}_n v_n\]For \(\vec{u} = (1, i)\) and \(\vec{v} = (2, 3i)\):
\[\langle \vec{u}, \vec{v} \rangle = \bar{1} \cdot 2 + \overline{i} \cdot 3i = 2 + (-i)(3i) = 2 + 3 = 5\]Example 2: Norm of a Complex Vector
For \(\vec{v} = (3, 4i)\):
\[\|\vec{v}\|^2 = \langle \vec{v}, \vec{v} \rangle = \bar{3} \cdot 3 + \overline{4i} \cdot 4i = 9 + (-4i)(4i) = 9 + 16 = 25\]So \(\|\vec{v}\| = 5\).
Example 3: Inner Product of Functions
For functions on \([0, L]\), we define:
\[\langle f, g \rangle = \int_0^L \overline{f(x)} g(x) \, dx\]This is the inner product for wavefunctions in quantum mechanics.
The Quantum Connection
The inner product \(\langle \psi | \phi \rangle\) between quantum states gives the probability amplitude for transitioning from \(|\phi\rangle\) to \(|\psi\rangle\). The squared modulus \(|\langle \psi | \phi \rangle|^2\) is the probability. The normalization condition \(\langle \psi | \psi \rangle = 1\) ensures total probability equals 1.