# Probability of stay

Wave function (in red), probability density (blue) and location probability (green) of the second excited state ( n = 2) of a one-dimensional harmonic oscillator .
The probability of finding a particle in the interval  A (green area: −2 < x <−1) is about 30 percent.

In quantum physics, the probability of presence indicates the probability with which a particle is to be found in a certain area of (spatial) space . It is determined by integrating the probability density over this range : ${\ displaystyle P}$ ${\ displaystyle \ rho ({\ vec {r}})}$${\ displaystyle A}$

${\ displaystyle P ({\ vec {r}} \ in A) = \ int _ {A} \ rho ({\ vec {r}}) \, {\ rm {d ^ {3}}} {\ vec {r}}}$

According to the Copenhagen interpretation of quantum mechanics , the probability density is calculated as an absolute square from the wave function : ${\ displaystyle \ Psi}$

${\ displaystyle \ rho ({\ vec {r}}) = | \ Psi ({\ vec {r}}) | ^ {2} = \ Psi ^ {*} ({\ vec {r}}) \ cdot \ Psi ({\ vec {r}})}$

with the complex conjugate wave function . ${\ displaystyle \ Psi ^ {*}}$

If one integrates the probability density in spherical coordinates over the angles and not additionally over the radius, one obtains (taking into account the Jacobi determinant ) the radial probability density function.

In contrast to the wave function itself, the probability density is accessible to observation.

The orbital model of the atomic structure is largely based on probabilities of location: the positions of the electrons (in this case to be regarded as quantum objects ) are indefinite; there are only areas in which the probability of finding an electron is greater; these are the orbitals.