p

The symbol $p$ usually denotes a momentum value, momentum variable, or momentum eigenvalue. The corresponding operator is written $\hat p$ in one dimension or $\hat{\mathbf p}$ in vector notation.

Default Meaning

In the position representation,

$$\hat p = -i\hbar\frac{d}{dx}$$

in one dimension, with domains and boundary conditions understood. For a plane wave,

$$p=\hbar k.$$

For a nonrelativistic free particle,

$$E=\frac{p^2}{2m}.$$

Units

$p$ has units of momentum, such as mass times velocity. In SI units this is $\mathrm{kg\,m\,s^{-1}}$.

Common Collisions

• $p$ as probability, especially in statistical notation such as $p_i$.
• $p$ as pressure in thermodynamic contexts.
• $p$ as an atomic orbital label, as in $p$ orbitals.
• $\hat p$ as an operator versus $p$ as an eigenvalue or integration variable.

Canonical Links

• Momentum Eigenstates
• Momentum Operator
• Canonical Commutation Relations
• Free Particle

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.