Generally speaking, the wave function is a complex function. But the probability density is a real function, and the integral sum in space is 1, which is called the probability density function. Therefore, the probability of finding particles in this area is 1.
Since particles exist in space, the probability of finding particles in space is 1. So integrating in the whole space will get 1. If the probability of wave function obtained by analyzing Schrodinger equation is limited, but not equal, we can multiply the wave function by a constant to make the probability equal to 1.
Or, if there is already an arbitrary constant in the wave function, you can set the value of this arbitrary constant to make the probability equal to 1.
Extended data:
In quantum mechanics, in order to quantitatively describe the state of microscopic particles, wave function is introduced into quantum mechanics, which is expressed by ψ. Generally speaking, the wave function is a function of space and time, and it is a complex variable function, that is, ψ =ψ (x, y, z, t).
The relationship between Einstein's "ghost field" and photon existence probability is generalized. Born assumes that ψ * ψ is the probability density of particles, that is, the probability of finding particles in a unit volume near time t (x, y, z), and the square of the absolute value of wave function ψ is therefore called probability amplitude.
The probability density of electrons appearing in various positions on the screen is not constant: in some places, the probability is high, that is, there are "bright stripes" in the interference pattern; However, the probability of occurrence in some places can be zero, and no electrons arrive, showing "dark stripes".
It can be seen that the probability distribution of a large number of events is observed in the electron double-slit interference experiment, which is born's explanation of the physical meaning of wave function, that is, the square of wave function mode corresponds to the probability density of microscopic particles appearing somewhere:
In other words, the probability density of microscopic particles appearing everywhere has obvious physical significance.
Accordingly, it can be considered that the wave function represents a probability fluctuation. Although this is an understanding that people can make of matter wave, the formation of the concept of wave function is a sign that quantum mechanics is completely out of the classical concept and mature. Wave function and probability density are the most basic concepts that constitute the theory of quantum mechanics.
References:
Baidu encyclopedia-wave function