Introduction: Why Unitary?
Quantum evolution is unitary: \(\hat{U}^\dagger\hat{U} = I\). This isn't arbitrary—it's required to preserve probability. Unitary operators are the "allowed moves" in quantum mechanics, rotating the state vector without changing its length.
Properties of Unitary Evolution
- Preserves norm: \(\|\hat{U}|\psi\rangle\| = \||\psi\rangle\|\)
- Preserves inner products: \(\langle\hat{U}\phi|\hat{U}\psi\rangle = \langle\phi|\psi\rangle\)
- Invertible: \(\hat{U}^{-1} = \hat{U}^\dagger\)
- Determinant: \(|\det(\hat{U})| = 1\)
Connection to Hermitian Generators
Every unitary operator can be written as the exponential of a Hermitian operator:
\[\hat{U} = e^{i\hat{G}}\]where \(\hat{G} = \hat{G}^\dagger\) is the generator of the transformation.
For time evolution: \(\hat{G} = -\hat{H}t/\hbar\), so \(\hat{U}(t) = e^{-i\hat{H}t/\hbar}\).
Worked Examples
Example 1: Verifying Unitarity of Time Evolution
For \(\hat{U}(t) = e^{-i\hat{H}t/\hbar}\) with Hermitian \(\hat{H}\):
\[\hat{U}^\dagger(t) = e^{+i\hat{H}^\dagger t/\hbar} = e^{+i\hat{H}t/\hbar}\] \[\hat{U}^\dagger\hat{U} = e^{+i\hat{H}t/\hbar}e^{-i\hat{H}t/\hbar} = e^0 = I \checkmark\]Example 2: Composition of Unitary Operators
The product of unitary operators is unitary:
\[(\hat{U}_1\hat{U}_2)^\dagger(\hat{U}_1\hat{U}_2) = \hat{U}_2^\dagger\hat{U}_1^\dagger\hat{U}_1\hat{U}_2 = \hat{U}_2^\dagger\hat{U}_2 = I\]Quantum gates can be composed.
Example 3: Rotation in Spin Space
Rotation about z-axis by angle \(\theta\):
\[\hat{R}_z(\theta) = e^{-i\theta\sigma_z/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\]This is unitary and represents a physical rotation.
The Quantum Connection
Unitarity is the mathematical statement of probability conservation. It's also why quantum mechanics is reversible: \(\hat{U}^{-1} = \hat{U}^\dagger\) always exists. Information is never lost during unitary evolution—only during measurement, which is non-unitary. This distinction is central to the measurement problem and quantum computing.