a

The symbol $a$ is context-dependent. In oscillator and quantum-optics notation, $a$ usually denotes an annihilation operator, often written $\hat a$ when hats are shown. In expansion formulas, $a$ can instead be an ordinary amplitude or coefficient.

Default Meaning

For the harmonic oscillator, $a$ and $a^\dagger$ satisfy

$$[a,a^\dagger]=1,$$

and act on number states as

$$a\lvert n\rangle = \sqrt n\,\lvert n-1\rangle, \qquad a^\dagger\lvert n\rangle = \sqrt{n+1}\,\lvert n+1\rangle.$$

The number operator is

$$N=a^\dagger a.$$

Units

In the standard dimensionless oscillator convention, $a$ is dimensionless. Its definition absorbs the oscillator length and momentum scale. Other uses of $a$, such as scattering length or lattice spacing, have units of length.

Common Collisions

• $a$ as an expansion coefficient or probability amplitude.
• $a$ as scattering length in low-energy scattering.
• $a$ as lattice spacing in condensed matter.
• $a$ as acceleration in classical-mechanics comparisons.

Canonical Links

• Ladder-Operator Solution
• Number States
• Creation and Annihilation Operators
• Scattering Length

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.