Use only the replacement rule and the ladder action , .
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs
Problem
Use only the replacement rule aa†→a†a+1 and the ladder action a†∣n⟩=n+1∣n+1⟩, a∣0⟩=0.
((a)) Bring (a+a†)3 to normal order, showing each replacement.
((b)) Use your result to write (a+a†)3∣0⟩ as a combination of number states, and hence find ⟨0∣(a+a†)6∣0⟩ without expanding the sixth power.
((c)) With x=ℏ/2mω(a+a†), give ⟨0∣x6∣0⟩ and check it against the sixth moment 15σ6 of a Gaussian of variance σ2=⟨0∣x2∣0⟩.