alpha

The symbol $\alpha$ is a general-purpose Greek parameter. In elementary state notation it often denotes a probability amplitude, while in atomic and relativistic contexts $\alpha$ may denote the fine-structure constant.

Default Meaning

In a two-level state, $\alpha$ is commonly used as one coefficient in a normalized superposition:

\lvert\psi\rangle = \alpha\lvert0\rangle+\beta\lvert1\rangle, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.

Here $\alpha$ is a complex number. Its phase matters relative to the phase of $\beta$ , while $\lvert\alpha\rvert^2$ gives the Born-rule probability for the corresponding basis outcome.

Other Standard Uses

In atomic physics, the same letter often denotes the fine-structure constant:

\alpha_{\mathrm{fs}} = \frac{e^2}{4\pi\epsilon_0\hbar c}

in SI conventions. In oscillator and quantum-optics notation, $\alpha$ can also label a coherent state,

a\lvert\alpha\rangle=\alpha\lvert\alpha\rangle.

This is a different use from a two-level probability amplitude, even though both are complex dimensionless numbers in common conventions.

Units

Probability amplitudes such as $\alpha$ are dimensionless after the basis state has been specified.
The fine-structure constant $\alpha_{\mathrm{fs}}$ is dimensionless.
A variational parameter, coupling, or angle named $\alpha$ has whatever units the local definition gives it.

Common Collisions

$\alpha$ as a probability amplitude versus $\alpha_{\mathrm{fs}}$ as the fine-structure constant.
$\alpha$ as a coherent-state eigenvalue versus $\alpha$ as a generic variational parameter.
$\alpha$ and $\beta$ as coefficients versus Greek angle labels in geometry or scattering.
$\alpha$ as a coupling constant whose dimensions depend on the chosen Hamiltonian.

Canonical Links

References

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.