The probability density and the energy amplitude of a wave are not the same thing. They represent different properties of a wave.
In the context of wave functions in quantum mechanics, the probability density is a measure of the likelihood of finding a particle at a particular position in space. It is given by the squared modulus of the wave function, usually denoted as |Ψ(x)|^2, where Ψ(x) represents the wave function. The probability density tells us the probability per unit volume of finding the particle at a specific location.
On the other hand, the energy amplitude of a wave is related to the energy associated with the wave. For example, in the case of an electromagnetic wave, the energy amplitude refers to the strength or intensity of the wave, which is proportional to the square of the amplitude of the electric or magnetic field.
While the probability density and the energy amplitude may both involve squaring quantities, they have different physical interpretations and represent distinct aspects of a wave. In quantum mechanics, the probability density is directly related to the wave function, while the energy amplitude relates to the energy associated with the wave.