Why is the wave function the squareroot of the probability funcition?
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Heya,
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A wave function in quantum mechanics is a description of the quantum state of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. .The state of such a particle is completely described by its wave function Ψ(x,t). So it’s the Solutions of Schrodinger Equation

and The solution is Ψ(x,t) = A e^i(kx-wt) + B e^-i(kx-wt)
where x is position and t is time. This is a complex-valued function of two real variables x and t. Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.
Since Ψ(x,t) is analogous to Amplitude of wave.
So for any kind of wave, we know I is directly proportional to A^2. The same is also for Quantum Mechanics .Here the Intensity means frequently observing the particle in a particular place in other words, Probability of finding the particle in a particular place .
Therefore chance or probability of finding the particle in place is obtained by |Ψ(x)|^2 = Ψ(x)Ψ(x)* which is Probability Amplitude Function.
Here given a graphical representation of Ψ(x) & Ψ(x)|^2 for an electron bounded in different orbits in an atom to visualize the fact.
Hope it helps.
It may be the answer:--
A wave function in quantum mechanics is a description of the quantum state of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. .The state of such a particle is completely described by its wave function Ψ(x,t). So it’s the Solutions of Schrodinger Equation

and The solution is Ψ(x,t) = A e^i(kx-wt) + B e^-i(kx-wt)
where x is position and t is time. This is a complex-valued function of two real variables x and t. Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.
Since Ψ(x,t) is analogous to Amplitude of wave.
So for any kind of wave, we know I is directly proportional to A^2. The same is also for Quantum Mechanics .Here the Intensity means frequently observing the particle in a particular place in other words, Probability of finding the particle in a particular place .
Therefore chance or probability of finding the particle in place is obtained by |Ψ(x)|^2 = Ψ(x)Ψ(x)* which is Probability Amplitude Function.
Here given a graphical representation of Ψ(x) & Ψ(x)|^2 for an electron bounded in different orbits in an atom to visualize the fact.
Hope it helps.
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