Difference between revisions of "Aufgaben:Problem 12"
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| + | == 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 \( \alpha^2 = \left( \alpha^\dagger \right)^2 = 0 \). | ||
| + | |||
| + | |||
| + | === Part b) === | ||
Revision as of 13:48, 9 June 2015
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 \( \alpha^2 = \left( \alpha^\dagger \right)^2 = 0 \).
