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Inner Product Spaces

Source lecture(s): PHY621 Ch. 1

Intuition

A vector space has no geometry — no lengths, no angles, no notion of perpendicular. The inner product supplies all three at once, and it is the single piece of structure that makes linear algebra useful to physics.

\[\langle\mathbf{a},\mathbf{b}\rangle \quad\text{with}\quad |\mathbf{a}| = \sqrt{\langle\mathbf{a},\mathbf{a}\rangle},\qquad \cos\theta = \frac{\langle\mathbf{a},\mathbf{b}\rangle}{|\mathbf{a}||\mathbf{b}|}\]

Axioms

For a complex vector space:

  • Conjugate symmetry: \(\langle\mathbf{a},\mathbf{b}\rangle = \overline{\langle\mathbf{b},\mathbf{a}\rangle}\)
  • Linearity in the second argument (physics convention)
  • Positive definiteness: \(\langle\mathbf{a},\mathbf{a}\rangle \ge 0\), with equality only for \(\mathbf{a} = 0\)

The conjugate is not decoration. It is what makes \(\langle\mathbf{a},\mathbf{a}\rangle\) real and positive so that "length" means something — and it is why quantum mechanical probabilities \(|\langle\psi|\phi\rangle|^2\) come out real.

Orthonormal bases and index gymnastics

A basis is orthonormal if \(\langle\mathbf{e}_i,\mathbf{e}_j\rangle = \delta_{ij}\). This makes component extraction trivial:

\[\mathbf{v} = \sum_i v_i\mathbf{e}_i \qquad\Longrightarrow\qquad v_i = \langle\mathbf{e}_i,\mathbf{v}\rangle\]

Project to get the component. That one line is the engine behind Fourier series (where \(\mathbf{e}_i\) are sines and cosines and the inner product is an integral), the Fourier transform, and the expansion of a quantum state in energy eigenstates. It is the same operation each time.

The Schwarz inequality

\[|\langle\mathbf{a},\mathbf{b}\rangle| \le |\mathbf{a}|\,|\mathbf{b}|\]

Proof. For every \(\lambda\), \(|\mathbf{a} + \lambda\mathbf{b}|^2 \ge 0\) by positive definiteness. Expand, treat as a quadratic in \(\lambda\), and demand a non-positive discriminant. ∎

The proof is three lines and the consequences are enormous:

  • Triangle inequality \(|\mathbf{a}+\mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|\) follows immediately.
  • \(|\cos\theta| \le 1\) — the definition of angle above is consistent only because of Schwarz.
  • The uncertainty principle. \(\Delta A\,\Delta B \ge \tfrac12|\langle[A,B]\rangle|\) is Schwarz applied to two state vectors. Heisenberg's relation is a geometric inequality about inner products, not a statement about microscopes.

The inner product is a choice

The most under-appreciated point. Nothing forces the plain sum \(\sum a_i^*b_i\). Any positive definite form will do, and the physically correct one is problem-dependent:

  • Mass-weighted, \(\langle\mathbf{a},\mathbf{b}\rangle = \mathbf{a}^{\mathsf T}A\mathbf{b}\) with \(A\) the mass matrix, for normal modes — this is why the modes of unequal masses are orthogonal in that product and not the naive one.
  • Weighted integrals, \(\int w(x)f^*g\,dx\), for special functions — Legendre polynomials are orthogonal with \(w=1\), Chebyshev with \(w = 1/\sqrt{1-x^2}\), Hermite with \(w = e^{-x^2}\). Each family is orthonormal in its own inner product and not in others.
  • \(\int\psi^*\phi\,d^3x\) in quantum mechanics.

Asking "orthogonal with respect to which inner product?" is almost always the right question when an orthogonality claim looks wrong.

Common mistakes

  • Dropping the complex conjugate. Then \(\langle\mathbf{a},\mathbf{a}\rangle\) need not be real and "length" is meaningless.
  • Assuming one universal inner product. See above.
  • Forgetting the weight function. Special-function orthogonality relations are false without their weight.

Knowledge graph position

Prerequisites: vector spaces. Leads to: orthogonal transformations, Hermitian operators, all transform methods, quantum mechanics.

Quiz

Q1 (conceptual). Why must an inner product be conjugate-symmetric rather than symmetric on a complex space?

Answer

Plain symmetry with complex scalars would make \(\langle\mathbf{a},\mathbf{a}\rangle\) complex in general, so it could not serve as a squared length. Conjugate symmetry forces \(\langle\mathbf{a},\mathbf{a}\rangle\) to equal its own conjugate, hence be real, and positive definiteness then makes it a genuine norm.

Q2 (computational). Are \(P_1 = x\) and \(P_2 = \tfrac12(3x^2-1)\) orthogonal on \([-1,1]\) with weight 1?

Answer

\(\int_{-1}^{1}x\cdot\tfrac12(3x^2-1)\,dx = \tfrac12\int_{-1}^1(3x^3 - x)\,dx = 0\), since the integrand is odd. Yes — as Legendre polynomials must be.

Q3 (MCQ). The Heisenberg uncertainty principle is, mathematically:

  • (a) a consequence of the measurement disturbing the system
  • (b) the Schwarz inequality applied to two vectors in Hilbert space
  • (c) an experimental result
  • (d) a property of Fourier transforms only
Answer

(b). It is a geometric inequality about inner products. The Fourier-transform version for position and momentum is one instance of it, not the general statement.