Overview

In quantum mechanics, observables (like position, momentum, energy) are represented by operators. Requiring those operators to be Hermitian (more precisely, self-adjoint) is not arbitrary—it follows from a few fundamental physical requirements.

1. Measurement outcomes must be real numbers

A physical measurement always gives a real value.

If an operator A^\hat{A}A^ represents an observable, its possible measurement outcomes are its eigenvalues.

A Hermitian operator guarantees:All eigenvalues of A^ are real\text{All eigenvalues of } \hat{A} \text{ are real}All eigenvalues of A^ are real

A^=A^†\hat{A} = \hat{A}^\daggerA^=A^†

If the operator were not Hermitian, you could get complex eigenvalues like 3+2i3 + 2i3+2i, which have no physical meaning as measurement results.

2. Expectation values must be real

Even before measurement, we often compute the expectation value:⟨A⟩=⟨ψ∣A^∣ψ⟩\langle A \rangle = \langle \psi | \hat{A} | \psi \rangle⟨A⟩=⟨ψ∣A^∣ψ⟩

For a Hermitian operator:⟨ψ∣A^∣ψ⟩∈R\langle \psi | \hat{A} | \psi \rangle \in \mathbb{R}⟨ψ∣A^∣ψ⟩∈R

If A^\hat{A}A^ were not Hermitian, the expectation value could be complex—which would make no physical sense as an “average measurement.”

3. Orthogonality of eigenstates (clean measurement structure)

Hermitian operators have a powerful property:

  • Eigenstates corresponding to different eigenvalues are orthogonal

This gives us a clean decomposition:∣ψ⟩=∑ici∣ai⟩|\psi\rangle = \sum_i c_i |a_i\rangle∣ψ⟩=i∑​ci​∣ai​⟩

Where:

  • ∣ai⟩|a_i\rangle∣ai​⟩ are eigenstates of the observable
  • ∣ci∣2|c_i|^2∣ci​∣2 are probabilities

Without Hermiticity, this orthogonal structure breaks down → probabilities become ambiguous.

4. Probability interpretation requires it

Quantum mechanics relies on:P(ai)=∣⟨ai∣ψ⟩∣2P(a_i) = |\langle a_i | \psi \rangle|^2P(ai​)=∣⟨ai​∣ψ⟩∣2

This only works cleanly if:

  • Eigenstates form an orthonormal basis
  • The operator is Hermitian

Otherwise, you lose a consistent probability framework.

5. Connection to unitary time evolution

Hermitian operators also generate unitary transformations.

Example: the Hamiltonian H^\hat{H}H^U(t)=e−iH^t/ℏU(t) = e^{-i \hat{H} t / \hbar}U(t)=e−iH^t/ℏ

If H^\hat{H}H^ is Hermitian:

  • U(t)U(t)U(t) is unitary
  • Total probability is conserved

If not:

  • Probability could grow or decay → physically unacceptable

6. Deeper insight (physics intuition)

You can think of Hermitian operators as enforcing:

  • Reality → measurements are real
  • Stability → probabilities don’t explode
  • Consistency → repeatable measurements give structured outcomes

In a deeper sense:

Hermiticity ensures that the mathematical structure of quantum mechanics aligns with the physical requirement that observations are real, probabilistic, and consistent.