1. Intensity ~ (amplitude of the wave)2
What are matter waves?
Recall that for a wave:
Intensity ~ (amplitude of the wave)2
EM waves is a good example, since we know that they can be considered also as beams of particles – photons, and the intensity in the photon model is
If N is the [average] density of photons, in the beam (density of photons is the number of photons per unit volume)

2. What are matter waves? Here Therefore,
Speed of light, c, is a constant
For a given EM wave, photon energy, hν, is a constant
The only variable “responsible” for change in the light intensity is the average density of photons, N
Therefore,

3. What are matter waves?
N may also be considered from a statistical point of view, by asking the question:
What is the probability of finding a given number of photons in a given volume of the beam?
The answer is of course:
The probability of finding X number of photons in the volume V is given by ratio X/V
The average photon density in this case is calculated in the same way: N = X/V !

4. Here |Ψ(r,t)|2 = Ψ*(r,t)Ψ(r,t)
Definition of Ψ(r,t)
The probability P(r,t)dV to find a particle associated with the wavefunction Ψ(r,t) within a small volume dV around a point in space with coordinate r at some instant t is
P(r,t) is the probability density
For one-dimensional case
Here |Ψ(r,t)|2 = Ψ*(r,t)Ψ(r,t)

5. Since, N is also proportional to the square of the amplitude of the corresponding wave, we conclude that
The probability of finding photon within a given volume of the beam is proportional to the square of the amplitude of the wave associated with this beam

6. What are the matter waves?
Thus, a wave (function describing the wave) can be considered as a mathematical function that measures the photon probability density in the beam of light
Since not only EM radiation has a dual nature, but also matter, Max Born extended this interpretation to the matter waves proposed by De Broglie, by assigning a mathematical function, Ψ(r,t), called the wavefunction to every “material” particle
Ψ(r,t) is what is “waving”

7. Definition of Ψ(r,t)
The probability of funding a particle somewhere in a volume V of space is
Since the probability to find particle anywhere in space is 1, we have condition of normalization
For one-dimensional case, the probability of funding the particle in the arbitrary interval a ≤ x ≤ b is