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Mathematics for Physics

133 Position Operator

An operator that multiplies a wavefunction by the coordinate that is used to represent the position observable in the position representation.

Multiplication by \(x\). In the position representation the position operator multiplies by \(x\). This principle is used to compute \(\langle x\rangle\) and to write potentials \(V(\hat{x})\).

The position operator is

\[ (X\psi)(x) = x\psi(x) \]

where

  • \(\psi\) is a wavefunction.
  • \(x\) is the position coordinate.
  • \(X\) is the position operator.

The position eigenkets. The eigenkets of the position operator satisfy \(x|x'\rangle=x'|x'\rangle\) and form a complete set. This principle is used to expand an arbitrary state in the position basis.

The position eigenvalue equation is

\[ x|x'\rangle = x'|x'\rangle \]

where

  • \(x\) is the position operator.
  • \(|x'\rangle\) is a position eigenket.
  • \(x'\) is the corresponding eigenvalue.

The wavefunction as a position-basis coefficient. The wavefunction is the overlap \(\psi(x)=\langle x|\psi\rangle\). This principle is used to pass from abstract kets to functions of \(x\).

Note: Also denoted \(X\). Also denoted \(\hat{x}\). Also denoted \(x\).

133.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \((X\psi)(x)=x\psi(x)\).
  2. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(X[f(x)]=xf(x)\).
  3. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(x|x'\rangle=x'|x'\rangle\).