Introduction: Beyond Simple Operations
What does it mean to compute \(\sin(\hat{A})\) or \(e^{\hat{A}}\) for a matrix \(\hat{A}\)? Using diagonalization and power series, we can extend ordinary functions to operators—essential for time evolution in quantum mechanics.
Definition via Power Series
For a function with Taylor expansion \(f(x) = \sum_{n=0}^{\infty} c_n x^n\):
\[f(\hat{A}) = \sum_{n=0}^{\infty} c_n \hat{A}^n\]Examples:
- \(e^{\hat{A}} = I + \hat{A} + \frac{\hat{A}^2}{2!} + \frac{\hat{A}^3}{3!} + \cdots\)
- \(\sin(\hat{A}) = \hat{A} - \frac{\hat{A}^3}{3!} + \frac{\hat{A}^5}{5!} - \cdots\)
- \(\cos(\hat{A}) = I - \frac{\hat{A}^2}{2!} + \frac{\hat{A}^4}{4!} - \cdots\)
Using Spectral Decomposition
If \(\hat{A} = \sum_n a_n |a_n\rangle\langle a_n|\), then:
\[f(\hat{A}) = \sum_n f(a_n) |a_n\rangle\langle a_n|\]The function is applied to each eigenvalue!
Worked Examples
Example 1: Exponential of σ_z
\(\sigma_z = |+\rangle\langle +| - |-\rangle\langle -|\) with eigenvalues ±1:
\[e^{i\theta\sigma_z} = e^{i\theta}|+\rangle\langle +| + e^{-i\theta}|-\rangle\langle -| = \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}\]Example 2: Exponential of Pauli Matrix
For any Pauli matrix \(\vec{\sigma} \cdot \hat{n}\) with \(|\hat{n}| = 1\):
\[e^{i\theta \vec{\sigma} \cdot \hat{n}} = \cos(\theta)I + i\sin(\theta)(\vec{\sigma} \cdot \hat{n})\]This is a rotation by angle \(2\theta\) about axis \(\hat{n}\) on the Bloch sphere.
Example 3: Time Evolution Operator
The time evolution operator is:
\[U(t) = e^{-i\hat{H}t/\hbar}\]If \(\hat{H}|n\rangle = E_n|n\rangle\), then:
\[U(t) = \sum_n e^{-iE_nt/\hbar}|n\rangle\langle n|\]The Quantum Connection
Time evolution in quantum mechanics is governed by the exponential of the Hamiltonian:
\[|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle\]This is why diagonalizing \(\hat{H}\) solves the time-dependence: in the energy basis, each component just rotates in phase at its own frequency \(\omega_n = E_n/\hbar\).