1.1.3 Hermitian Operators#

Prompts

Why must observables be represented by Hermitian operators rather than arbitrary operators? What special property of Hermitian operators makes them suitable for describing measurements?
Find the eigenvalues and eigenvectors of \(\hat{X}\) and \(\hat{Z}\) by solving the characteristic equation. Why are all Pauli eigenvalues \(\pm 1\)?
For a general qubit state \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\), write down the expectation values \(\langle\hat{X}\rangle\), \(\langle\hat{Y}\rangle\), \(\langle\hat{Z}\rangle\) in terms of \(\alpha\) and \(\beta\). What do these three numbers represent geometrically?
Prove that eigenvectors of a Hermitian operator with different eigenvalues are orthogonal. Does this result depend on the eigenvectors being normalized?
Interpret the Bloch vector: for a qubit state parameterized by angles \((\theta, \varphi)\), explain how the expectation values of the Pauli operators encode position on the Bloch sphere.

Lecture Notes#

Overview#

In quantum mechanics, observables (measurable quantities) are represented by Hermitian operators. This is a fundamental postulate: for measurement outcomes to be real numbers, the operators must satisfy special properties. This section establishes eigenvalues, spectral decomposition, and the Pauli matrices—the mathematical foundation that makes quantum predictions testable.

Why Hermitian?#

A quantum state is described by complex numbers (superposition coefficients). Yet every measurement outcome is a real number. How do we extract reality from complexity?

For a complex number \(z = x + \mathrm{i}y\), the real part is \(x = \frac{z + z^*}{2}\). The key: \(x = x^*\) (real part equals its complex conjugate). The same principle applies to operators: for an observable to yield real outcomes, we require:

(9)#\[ \hat{O}^\dagger = \hat{O} \]

This is the definition of a Hermitian (self-adjoint) operator.

Definition: Hermitian Conjugate

For an operator \(\hat{O}\), the Hermitian conjugate \(\hat{O}^\dagger\) is defined by:

\[ \langle \psi \vert \hat{O} \phi \rangle = \langle \hat{O}^\dagger \psi \vert \phi \rangle \]

for all states \(\vert \psi\rangle, \vert \phi\rangle\).

Operators as Matrices#

Quantum operators act on the Hilbert space (spanned by basis states \(\vert 0\rangle, \vert 1\rangle\)). Represent them as matrices using:

(10)#\[ O_{ij} = \langle i \vert \hat{O} \vert j \rangle \]

A matrix is Hermitian if all diagonal elements are real and off-diagonal elements are conjugate pairs: \(O_{ij} = O_{ji}^*\).

Example: matrix elements of \(\hat{X}\)

For \(\hat{X}\) on a qubit (basis \(\vert 0\rangle, \vert 1\rangle\)):

\[\begin{split} \hat{X}\vert 0\rangle = \vert 1\rangle \quad \Rightarrow \text{column 0} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad \hat{X}\vert 1\rangle = \vert 0\rangle \quad \Rightarrow \text{column 1} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{split}\]

Thus \(\hat{X} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\).

Eigenvalues and Eigenvectors#

Some states \(\vert \psi\rangle\) are only rescaled by an operator:

(11)#\[ \hat{O}\vert \psi\rangle = \lambda \vert \psi\rangle \]

Here \(\vert \psi\rangle\) is an eigenvector and \(\lambda\) is its eigenvalue.

Theorem: Hermitian Operators Have Real Eigenvalues

If \(\hat{O}\) is Hermitian and \(\vert \psi\rangle\) is an eigenvector with eigenvalue \(\lambda\), then \(\lambda \in \mathbb{R}\).

Proof: real eigenvalues of Hermitian operators

Take \(\langle \psi\vert \) times the eigenvalue equation:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle = \lambda \langle\psi\vert \psi\rangle = \lambda \]

Complex conjugate both sides:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle^* = \lambda^* \]

But by Hermiticity:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle^* = \langle \psi\vert \hat{O}^\dagger\vert \psi\rangle = \langle \psi\vert \hat{O}\vert \psi\rangle = \lambda \]

Thus \(\lambda = \lambda^*\), so \(\lambda\) is real.

Theorem: Orthogonality of Eigenvectors

If \(\hat{O}\) is Hermitian and \(\vert \psi_1\rangle, \vert \psi_2\rangle\) are eigenvectors with different eigenvalues \(\lambda_1 \neq \lambda_2\), then \(\langle\psi_1\vert \psi_2\rangle = 0\).

Proof: orthogonality of eigenvectors

From the eigenvalue equation:

\[ \langle \psi_1\vert \hat{O}\vert \psi_2\rangle = \lambda_2 \langle\psi_1\vert \psi_2\rangle \]

Also:

\[ \langle \psi_1\vert \hat{O}\vert \psi_2\rangle = \lambda_1 \langle\psi_1\vert \psi_2\rangle \]

(using Hermiticity). Equating gives:

\[ (\lambda_2 - \lambda_1)\langle\psi_1\vert \psi_2\rangle = 0 \quad \Rightarrow \quad \langle\psi_1\vert \psi_2\rangle = 0 \]

since \(\lambda_2 \neq \lambda_1\).

Spectral Decomposition and Pauli Operators#

Any Hermitian operator can be decomposed as:

(12)#\[ \hat{O} = \sum_i \lambda_i \vert \psi_i\rangle\langle \psi_i\vert \]

where \(\lambda_i\) are eigenvalues and \(\vert \psi_i\rangle\langle \psi_i\vert \) is the projector onto eigenstate \(\vert \psi_i\rangle\).

A general \(2\times 2\) Hermitian matrix has 4 real degrees of freedom—thus 4 linearly independent Hermitian \(2\times 2\) matrices: the identity \(\hat{I}\) and the three Pauli operators \(\hat{X}, \hat{Y}, \hat{Z}\). Every Hermitian \(2\times 2\) operator is uniquely \(a_0\hat{I} + a_x\hat{X} + a_y\hat{Y} + a_z\hat{Z}\) with \(a_0, a_x, a_y, a_z\in\mathbb{R}\).

Table: Pauli Operators and Spectral Decompositions

Operator	Matrix	Eigenstate (\(+1\))	Projector \(\hat{P}_{+1}\)
\(\hat{X}\)	\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)	\(\vert +\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{X})\)
\(\hat{Y}\)	\(\begin{pmatrix} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{pmatrix}\)	\(\vert \mathrm{i}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \mathrm{i} \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{Y})\)
\(\hat{Z}\)	\(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)	\(\vert 0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{Z})\)

Pauli Algebra#

The Pauli matrices satisfy elegant algebraic relations:

(13)#\[ \hat{\sigma}^i \hat{\sigma}^j = \delta_{ij}\hat{I} + \mathrm{i}\epsilon_{ijk}\hat{\sigma}^k \]

Commutation relations:

(14)#\[ [\hat{\sigma}^i, \hat{\sigma}^j] = 2\mathrm{i}\epsilon_{ijk}\hat{\sigma}^k \]

For example: \([\hat{X}, \hat{Y}] = 2\mathrm{i}\hat{Z}\) and cyclic permutations.

Pauli operators do not commute

The Pauli operators do not commute. Measuring along one axis disturbs the result of measuring along another—a manifestation of the uncertainty principle.

Anti-commutation relations:

\[ \{\hat{\sigma}^i, \hat{\sigma}^j\} = 2\delta_{ij}\hat{I} \]

where \(\{\hat{A}, \hat{B}\} = \hat{A}\hat{B} + \hat{B}\hat{A}\) is the anti-commutator.

Discussion: must observables be Hermitian?

Are all physical observables necessarily Hermitian?

We require Hermiticity for real measurement outcomes. Yet creation operators \(\hat{a}^\dagger\) are non-Hermitian but physically meaningful. Recent work on non-Hermitian PT-symmetric Hamiltonians shows certain non-Hermitian operators can have entirely real spectra. What is the minimal mathematical requirement for an observable?

Expectation Values and the Bloch Sphere#

The expectation value of a Hermitian observable is:

(15)#\[ \langle\hat{O}\rangle = \langle\psi\vert\hat{O}\vert\psi\rangle \]

This is the average outcome over many measurements on identically prepared copies of \(\vert\psi\rangle\).

For a qubit \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) (with \(|\alpha|^2 + |\beta|^2 = 1\)):

\[ \langle\hat{X}\rangle = 2\operatorname{Re}(\alpha^*\beta), \quad \langle\hat{Y}\rangle = 2\operatorname{Im}(\alpha^*\beta), \quad \langle\hat{Z}\rangle = \vert\alpha|^2 - |\beta|^2 \]

In the Bloch parametrization \(\vert\psi\rangle = \cos(\theta/2)\vert 0\rangle + \mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2)\vert 1\rangle\):

(16)#\[ \langle\hat{X}\rangle = \sin\theta\cos\varphi, \quad \langle\hat{Y}\rangle = \sin\theta\sin\varphi, \quad \langle\hat{Z}\rangle = \cos\theta \]

The Bloch vector \(\boldsymbol{n} = (\sin\theta\cos\varphi,\,\sin\theta\sin\varphi,\,\cos\theta)\) lies on the unit sphere and completely specifies the qubit state.

Derivation: Pauli Expectation Values

From \((\alpha,\beta)\) to \((\theta,\varphi)\):

Substitute \(\alpha = \cos(\theta/2)\), \(\beta = \mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2)\):

\[ \alpha^*\beta = \cos(\theta/2)\,\mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2) = \tfrac{1}{2}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta \]

Thus:

\[ \langle\hat{X}\rangle = 2\operatorname{Re}\!\left(\tfrac{1}{2}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta\right) = \sin\theta\cos\varphi \quad \checkmark \]

\[ \langle\hat{Z}\rangle = \cos^2(\theta/2) - \sin^2(\theta/2) = \cos\theta \quad \checkmark \]

Poll: Measurement outcomes are eigenvalues

For a qubit in state \(\vert\psi\rangle = \frac{3}{5}\vert 0\rangle + \frac{4}{5}\vert 1\rangle\), you measure \(\hat{Z} = \vert 0\rangle\langle 0\vert - \vert 1\rangle\langle 1\vert\). What outcome(s) are possible?

(A) Always \(-7/25\) (the expectation value).

(B) Either \(+1\) or \(-1\) (the eigenvalues of \(\hat{Z}\)).

(D) Either \(3/5\) or \(4/5\) (the amplitudes of the state).

Summary#

Hermitian operators have real eigenvalues and orthogonal eigenvectors—essential for physical observables.
The Pauli matrices (\(\hat{X}, \hat{Y}, \hat{Z}\)) represent spin measurements along three orthogonal directions.
Spectral decomposition expresses any observable as a sum of projectors onto eigenstates.
Non-commuting observables cannot be simultaneously diagonal—measuring one disturbs the other.
Pauli expectation values (\(\langle\hat{X}\rangle, \langle\hat{Y}\rangle, \langle\hat{Z}\rangle\)) encode the qubit state as a point on the Bloch sphere.

Homework#

1. Hermiticity of Pauli matrices. Show that the Pauli matrix \(\hat{Y} = \begin{pmatrix} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{pmatrix}\) is Hermitian by explicitly computing \((\hat{Y})^\dagger\) and verifying it equals \(\hat{Y}\). Then find its eigenvalues and normalized eigenvectors.

2. Hermitian operator properties. Consider the operator \(\hat{A} = \begin{pmatrix} 1 & \mathrm{i} \\ \mathrm{i} & 1 \end{pmatrix}\). Is \(\hat{A}\) Hermitian? If not, find \(\hat{A}^\dagger\) and construct a Hermitian operator from \(\hat{A}\) using the combination \(\frac{1}{2}(\hat{A} + \hat{A}^\dagger)\).

3. Eigenvalue calculation. A qubit is in the state \(\vert \psi\rangle = \frac{1}{\sqrt{3}}\vert 0\rangle + \sqrt{\frac{2}{3}}\vert 1\rangle\). Compute the expectation values \(\langle\hat{X}\rangle\), \(\langle\hat{Y}\rangle\), and \(\langle\hat{Z}\rangle\). Verify that \(\langle\hat{X}\rangle^2 + \langle\hat{Y}\rangle^2 + \langle\hat{Z}\rangle^2 = 1\).

4. Composition property. The spectral decomposition of \(\hat{X}\) is \(\hat{X} = (+1)\vert +\rangle\langle +\vert + (-1)\vert -\rangle\langle -\vert \), where \(\vert +\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\) and \(\vert -\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle - \vert 1\rangle)\). Verify this by explicitly computing the right-hand side as a \(2\times 2\) matrix.

5. Projector eigenvalues. Show that the eigenvalues of any projector \(\hat{P} = \vert \psi\rangle\langle \psi\vert \) are 0 and 1. What are the corresponding eigenstates? Is a projector Hermitian?

6. Observable measurement prediction. A spin-1/2 particle is in the state \(\vert \psi\rangle = \cos\alpha\,\vert 0\rangle + \sin\alpha\,\vert 1\rangle\) (with \(\alpha\) real). You measure \(\hat{Z}\). What are the possible outcomes and their probabilities? Then compute \(\langle \hat{Z}\rangle\) directly from the definition \(\langle\psi\vert \hat{Z}\vert \psi\rangle\) and verify it equals \(\sum_i \lambda_i P_i\).

7. Pauli operator powers. The operator \(\hat{n}\cdot\hat{\boldsymbol{\sigma}} = n_x \hat{X} + n_y \hat{Y} + n_z \hat{Z}\) represents spin along an arbitrary unit vector \(\hat{n} = (n_x, n_y, n_z)\) with \(\vert\hat{n}\vert = 1\). Show that \((\hat{n}\cdot\hat{\boldsymbol{\sigma}})^2 = \hat{I}\) (the identity matrix). What does this imply about the eigenvalues of \(\hat{n}\cdot\hat{\boldsymbol{\sigma}}\)?

8. Pauli matrix eigenvalues. Using the result of Problem 7, show that the eigenvalues of \(\hat{n}\cdot\hat{\boldsymbol{\sigma}}\) are \(\pm 1\) for any unit vector \(\hat{n}\). Find the eigenvector corresponding to eigenvalue \(+1\) for the case \(\hat{n} = (\sin\theta, 0, \cos\theta)\) (a vector in the \(xz\)-plane). Express your answer in terms of \(\theta\).

9. Reality of expectation values. Prove that the expectation value of any Hermitian operator \(\hat{O}\) is real-valued for any state \(\vert \psi\rangle\). That is, show \(\langle \hat{O}\rangle^* = \langle\hat{O}\rangle\) using the definition \(\langle\hat{O}\rangle = \langle\psi\vert \hat{O}\vert \psi\rangle\) and the Hermiticity condition.