Orthogonality of Eigenvectors in Quantum Mechanics

Why are eigenvectors orthogonal in Quantum Mechanics?

In quantum mechanics, the eigenvectors (or eigenstates) corresponding to different eigenvalues of an observable are orthogonal. This is a fundamental property that arises directly from the mathematical nature of the operators representing physical observables. These operators are Hermitian.

The Hermitian operator

A linear operator \(\hat{A}\) is defined as Hermitian if it is equal to its Hermitian conjugate (or adjoint) \(\hat{A}^\dagger\). The defining property of a Hermitian operator in bra-ket notation is:

\[ \langle\phi|\hat{A}\psi\rangle = \langle\hat{A}\phi|\psi\rangle \]

for any two state vectors \(|\phi\rangle\) and \(|\psi\rangle\) in the Hilbert space.

Proof of orthogonality for distinct eigenvalues

Let’s consider a Hermitian operator \(\hat{A}\) and two of its eigenstates, \(|\psi_1\rangle\) and \(|\psi_2\rangle\), with corresponding eigenvalues \(\lambda_1\) and \(\lambda_2\).

\[ \hat{A}|\psi_1\rangle = \lambda_1|\psi_1\rangle \] \[ \hat{A}|\psi_2\rangle = \lambda_2|\psi_2\rangle \]

We can use these relations to prove the orthogonality.

1. Start with the inner product involving the operator and the first eigenstate.

   \[ \langle\psi_2|\hat{A}|\psi_1\rangle = \langle\psi_2|\lambda_1\psi_1\rangle = \lambda_1\langle\psi_2|\psi_1\rangle \]

2. Use the Hermitian property to rewrite the same inner product.

   \[ \langle\psi_2|\hat{A}|\psi_1\rangle = \langle\hat{A}\psi_2|\psi_1\rangle \]

3. Substitute the second eigenstate equation into the rewritten inner product. Since eigenvalues of a Hermitian operator are real (\(\lambda_2^* = \lambda_2\)), we have:

   \[ \langle\hat{A}\psi_2|\psi_1\rangle = \langle\lambda_2\psi_2|\psi_1\rangle = \lambda_2^*\langle\psi_2|\psi_1\rangle = \lambda_2\langle\psi_2|\psi_1\rangle \]

4. Equate the results from steps 1 and 3.

   \[ \lambda_1\langle\psi_2|\psi_1\rangle = \lambda_2\langle\psi_2|\psi_1\rangle \]

5. Rearrange the equation.

   \[ (\lambda_1 - \lambda_2)\langle\psi_2|\psi_1\rangle = 0 \]

6. Conclude the orthogonality. If the eigenvalues are distinct, meaning \(\lambda_1 \neq \lambda_2\), then \((\lambda_1 - \lambda_2)\) is a non-zero number. For the equation to hold, the inner product must be zero.

   \[ \langle\psi_2|\psi_1\rangle = 0 \]

   This is the definition of orthogonality.

The physical implication

The orthogonality of eigenstates means that if a quantum system is prepared in a state corresponding to a definite value of an observable (an eigenstate), then a measurement of that observable will always yield that value. The probability of measuring any other value (corresponding to a different eigenstate) is zero. This guarantees that measurement outcomes for distinct eigenvalues are mutually exclusive.