Introduction: Perpendicular Vectors
Two vectors are orthogonal (perpendicular) if their inner product is zero. An orthonormal basis consists of mutually orthogonal unit vectors—the ideal coordinate system for almost every calculation in physics.
Definitions
Orthogonal: Vectors \(\vec{u}\) and \(\vec{v}\) are orthogonal if \(\langle \vec{u}, \vec{v} \rangle = 0\).
Orthonormal Set: A set \(\{\vec{e}_1, \ldots, \vec{e}_n\}\) is orthonormal if:
\[\langle \vec{e}_i, \vec{e}_j \rangle = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\]Here, \(\delta_{ij}\) is the Kronecker delta.
The Power of Orthonormal Bases
In an orthonormal basis \(\{\vec{e}_1, \ldots, \vec{e}_n\}\), finding coordinates is trivially easy:
\[\vec{v} = \sum_{i=1}^{n} c_i \vec{e}_i \quad \text{where} \quad c_i = \langle \vec{e}_i, \vec{v} \rangle\]No matrix inversions needed—just compute inner products!
Worked Examples
Example 1: Verifying Orthonormality
Check if \(\vec{e}_1 = \frac{1}{\sqrt{2}}(1, 1)\) and \(\vec{e}_2 = \frac{1}{\sqrt{2}}(1, -1)\) are orthonormal:
- \(\langle \vec{e}_1, \vec{e}_1 \rangle = \frac{1}{2}(1 + 1) = 1\) ✓
- \(\langle \vec{e}_2, \vec{e}_2 \rangle = \frac{1}{2}(1 + 1) = 1\) ✓
- \(\langle \vec{e}_1, \vec{e}_2 \rangle = \frac{1}{2}(1 - 1) = 0\) ✓
Yes, they form an orthonormal basis for \(\mathbb{R}^2\).
Example 2: Finding Coordinates
Express \(\vec{v} = (3, 1)\) in the orthonormal basis from Example 1:
- \(c_1 = \langle \vec{e}_1, \vec{v} \rangle = \frac{1}{\sqrt{2}}(3 + 1) = \frac{4}{\sqrt{2}} = 2\sqrt{2}\)
- \(c_2 = \langle \vec{e}_2, \vec{v} \rangle = \frac{1}{\sqrt{2}}(3 - 1) = \frac{2}{\sqrt{2}} = \sqrt{2}\)
Check: \(2\sqrt{2} \cdot \frac{1}{\sqrt{2}}(1, 1) + \sqrt{2} \cdot \frac{1}{\sqrt{2}}(1, -1) = (2, 2) + (1, -1) = (3, 1)\) ✓
Example 3: Orthogonal Functions
The functions \(\sin(nx)\) and \(\sin(mx)\) for \(n \neq m\) are orthogonal on \([0, \pi]\):
\[\int_0^{\pi} \sin(nx)\sin(mx) \, dx = 0 \quad \text{for } n \neq m\]This is why Fourier series works!
The Quantum Connection
Eigenstates of a Hermitian operator (like energy eigenstates) are automatically orthogonal. The expansion coefficients \(c_n = \langle \phi_n | \psi \rangle\) give the probability amplitude to measure the corresponding eigenvalue. Orthonormality ensures probabilities are well-defined and sum to 1.