What is the difference between probability and probability density of finding a particle in space?

I am at basics of the wave function and this thing confused me.

Probability = $[\psi(x, y,z)]^2(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$

Probability density at point $(x,y,z)$ in space = $[\psi(x, y,z)]^2$.

The latter one is the probability of finding particle in an infinitesimally small volume element $\mathrm{d}V=\mathrm{d}x, \mathrm{d}y, \mathrm{d}z$ situated at $(x,y,z)$. Then what's the former one?

  • 1
    $\begingroup$ The probability of finding particle at any particular coordinate is zero. $\endgroup$ – Ivan Neretin Mar 30 '18 at 14:21

The latter is the limit of probability per volume, as volume approaches zero, at a particular point in space.

  • $\begingroup$ What is exactly probability? I mean probability of finding particle where? $\endgroup$ – Iceberry Mar 30 '18 at 16:39
  • $\begingroup$ @DivyankaS.Chaudhari if an experiment is repeated a large number of times, the fraction of the outcomes the met condition X is the probability of X. So for a hydrogen atom, the probability that the electron is within all space is 1. And for any finite volume the probability of the electron being in that volume is a fraction that is less than 1. The probability that the electron is at any specific point, is zero, as Ivan said. But the probability per volume, in the limit that the volume approaches zero at that point, is non-zero. $\endgroup$ – DavePhD Mar 30 '18 at 16:49
  • $\begingroup$ So, the former is the probabilty of finding electron in a given volume and the latter is the same but for very small volume? $\endgroup$ – Iceberry Mar 30 '18 at 16:57
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    $\begingroup$ @DivyankaS.Chaudhari The former is a probability of being in an infinitesimal volume element, and is a pure number (no units). The latter has units of per volume ( such as per cubic meter). The former needs to be integrated over a volume to get a non-zero number. The latter can be non-zero at a specific point. $\endgroup$ – DavePhD Mar 30 '18 at 17:01

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