Introduction: Moving States Around
Translation and rotation operators physically move quantum states through space. They're generated by momentum and angular momentum, respectively, demonstrating the deep connection between symmetries and observables.
The Translation Operator
Translation by vector \(\vec{a}\):
\[\hat{T}(\vec{a}) = e^{-i\vec{a}\cdot\hat{\vec{p}}/\hbar}\]Action on wavefunctions: \(\hat{T}(\vec{a})\psi(\vec{r}) = \psi(\vec{r} - \vec{a})\)
The Rotation Operator
Rotation by angle \(\theta\) about axis \(\hat{n}\):
\[\hat{R}(\hat{n}, \theta) = e^{-i\theta\hat{n}\cdot\hat{\vec{L}}/\hbar}\]For spin-1/2: \(\hat{R}(\hat{n}, \theta) = e^{-i\theta\hat{n}\cdot\vec{\sigma}/2}\)
Worked Examples
Example 1: Infinitesimal Translation
For small \(\epsilon\):
\[\hat{T}(\epsilon\hat{x})\psi(x) \approx \psi(x) - \epsilon\frac{\partial\psi}{\partial x} = \left(1 - \frac{i\epsilon\hat{p}_x}{\hbar}\right)\psi(x)\]Example 2: Spin Rotation
Rotate spin-up state by \(\pi\) about x-axis:
\[\hat{R}_x(\pi)|+\rangle = e^{-i\pi\sigma_x/2}|+\rangle = -i\sigma_x|+\rangle = -i|-\rangle\]Spin-up becomes spin-down (up to phase).
The Quantum Connection
The 2π rotation of spin-1/2 gives a minus sign: \(\hat{R}(2\pi)|s\rangle = -|s\rangle\). This is the famous "spinor" behavior with no classical analog. It's observable in neutron interferometry and is key to understanding fermion statistics.