Position Operator

Operator

On the line, the position operator acts in coordinate representation by multiplication:

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

In three dimensions one usually writes component operators $\hat x_i$ or the vector operator $\hat{\mathbf r}$:

$$(\hat{\mathbf r}\psi)(\mathbf r) = \mathbf r\,\psi(\mathbf r).$$

The canonical commutator with momentum is

$$[\hat x_i,\hat p_j] = i\hbar\delta_{ij}.$$

Assumptions

• The representation is coordinate space.
• The Hilbert space, measure, and domain are specified.
• The coordinate chart is appropriate for the configuration space.
• In curvilinear coordinates, multiplication by a coordinate is not enough to determine all geometric operator issues.

Common Mistakes

• Confusing the coordinate value $x$ with the operator $\hat x$.
• Ignoring the measure when using radial or angular coordinates.
• Treating generalized position kets as normalizable vectors.
• Forgetting that position can fail to be a globally meaningful coordinate on constrained spaces.

Canonical Links

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

References

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