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Quantum Mechanics

Creation and Annihilation Operators

Operator

Section titled “Operator”

For one harmonic oscillator,

a=mω2ℏ x^+i2mℏω p^,a = \sqrt{\frac{m\omega}{2\hbar}}\,\hat x + \frac{i}{\sqrt{2m\hbar\omega}}\,\hat p,

and

[a,a†]=1.[a,a^\dagger]=1.

The number operator is

N=a†a,N=a^\dagger a,

with actions

a∣n⟩=n ∣n−1⟩,a†∣n⟩=n+1 ∣n+1⟩.a\lvert n\rangle = \sqrt n\,\lvert n-1\rangle, \qquad a^\dagger\lvert n\rangle = \sqrt{n+1}\,\lvert n+1\rangle.

In many-body mode notation, ai†a_i^\dagger creates an excitation in mode ii and aia_i annihilates one, with bosonic commutators or fermionic anticommutators depending on statistics.

Assumptions

Section titled “Assumptions”
  • The oscillator normalization or mode normalization is specified.
  • Bosonic and fermionic algebras are not interchangeable.
  • The vacuum and one-particle mode basis are specified in many-body contexts.
  • Unbounded operator domains are suppressed in compact physics notation.

Common Mistakes

Section titled “Common Mistakes”
  • Using bosonic commutators for fermionic modes.
  • Forgetting square-root factors in number-state actions.
  • Treating creation operators as creating particles without specifying the mode basis.
  • Confusing oscillator ladder operators with angular-momentum ladder operators.
Section titled “Canonical Links”

References

Section titled “References”
  • R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
  • A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.