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Mathematics for Physics

136 Probability Density

A nonnegative function built from a wavefunction that is used to give the probability of finding a particle in a region of space.

The definition of \(\rho=|\Psi|^{2}\). The probability density is the squared modulus of the wavefunction. This principle is used to convert \(\Psi\) into a finding probability per unit volume.

The probability density is

\[ \rho(x,t) = \lvert\Psi(x,t)\rvert^{2} \]

where

  • \(\rho\) is the probability density.
  • \(\Psi\) is the wavefunction.

The integral rule for a region. The probability that the particle lies in a set \(E\) is the integral of \(\rho\) over \(E\). This principle is used to compute finite-interval probabilities.

The finding probability is

\[ P(E) = \displaystyle\int_{E}\lvert\Psi(x)\rvert^{2}\,dx \]

where

  • \(P(E)\) is the probability that the position lies in \(E\).
  • \(\Psi\) is the wavefunction.

The three-dimensional volume form. In three dimensions the same rule uses a volume element \(d^{3}x\). This principle is used to interpret \(|\Psi|^{2}\) as a probability per unit volume.

Note: Also called the position probability density.

136.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(\rho=|\psi|^{2}\) as probability density.
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(|\psi(x)|^{2}\) as position probability density.
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(P(x)dx=|\psi(x)|^{2}dx\).