Aufgaben:Problem 12

Problem 12

Hamiltonian of a 1D fermionic oscillator $$ H_F = -i\omega\psi_1\psi_2, $$ where the fermionic wave functions anticommute $$ \{\psi_i,\psi_j\} = \hbar\delta_{ij}. $$ We introduce the lowering and rising operators $$ \alpha = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 - i\psi_2 \right), \quad \alpha^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left( \psi_1 + i\psi_2 \right). $$

Part a)

Show that \( \{\alpha,\alpha^\dagger\} = 1 \), \( \{\alpha,\alpha\} = \{\alpha^\dagger,\alpha^\dagger\} = 0 \) and \( \alpha^2 = \left( \alpha^\dagger \right)^2 = 0 \).