Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Advanced Quantum Mechanics Schwabl · Problem 2.6
Bosons
Physics — Quantum Mechanics · fermi-operator-relations
Problem Derive the following relations for Fermi operators:
a)
e − α a † a e α a † = a − α 2 a † + α ( a a † − a † a ) e^{-\alpha a^\dagger}a\,e^{\alpha a^\dagger} = a - \alpha^2 a^\dagger + \alpha(aa^\dagger - a^\dagger a) e − α a † a e α a † = a − α 2 a † + α ( a a † − a † a ) e − α a a † e α a = a † − α 2 a − α ( a a † − a † a ) e^{-\alpha a}a^\dagger e^{\alpha a} = a^\dagger - \alpha^2 a - \alpha(aa^\dagger - a^\dagger a) e − α a a † e α a = a † − α 2 a − α ( a a † − a † a ) e α a † a a e − α a † a = e − α a e^{\alpha a^\dagger a}a\,e^{-\alpha a^\dagger a} = e^{-\alpha}a e α a † a a e − α a † a = e − α a e α a † a a † e − α a † a = e − α a † . e^{\alpha a^\dagger a}a^\dagger e^{-\alpha a^\dagger a} = e^{-\alpha}a^\dagger. e α a † a a † e − α a † a = e − α a † . Browse the problem library