x

The symbol $x$ usually denotes a one-dimensional position coordinate or coordinate value. The corresponding position operator is $\hat x$.

Default Meaning

In the coordinate representation,

$$\psi(x) = \langle x\vert\psi\rangle.$$

The position operator acts by multiplication:

$$(\hat x\psi)(x) = x\psi(x).$$

For a normalized one-dimensional wavefunction, $\lvert\psi(x)\rvert^2$ is a probability density with respect to $dx$.

Units

$x$ has units of length unless a page explicitly introduces a dimensionless coordinate.

Common Collisions

• $x$ as a coordinate value versus $\hat x$ as an operator.
• $x$ as a dimensionless variable after rescaling.
• $x$ as a generic real variable in mathematical examples.
• $x$ in Cartesian coordinates versus $r,\theta,\phi$ in spherical coordinates, where the probability measure changes.

Canonical Links

• Coordinate Representation
• Wavefunctions and Probability Density
• Position Operator
• Canonical Commutation Relations

References

• 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.