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Schrödinger suggested the wave function for the atomic particles like electrons as, $$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { z }^{ 2 } } = \frac { -8{ \pi }^{ 2 }m\left( E - V \right) }{ { h }^{ 2 } } \Psi $$ This equation gives a wave in three dimension with amplitude \(\Psi\) .
As in quantum mechanics because of uncertainty. It is not possible to determine the position and velocity of the particle accurately. Therefore, the state of the particle was expressed by function \(\Psi \). The value of \(\Psi\) (amplitude) can be positive (real) or negative (imaginary). In order to make it real function for electron it is represented as \({ \Psi }^{ 2 }\) (as \({ \Psi }^{ 2 }\) is always positive or real.) As per the classical theory \({ \Psi }^{ 2 }\) at any time is measure of energy density. But as the energy is quantised it cannot be smooth.
Therefore Einstein replaced this energy density with probability density. Thus the electron cloud is actually the probability of finding the electron in that region.
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