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The electron is in the form of probability density distribution. Then in a small volume \(d\tau \) the probability will be \({ \Psi }^{ 2 } d\tau \). Where \(d\tau\) is a small volume in three dimensional coordinate space (dx, dy, dz). Therefore the total probability in the available space is unity. i.e. $$ \int _{ -\infty }^{ +\infty }{ { \Psi }^{ 2 } } d\tau = 1$$ Schrodinger equation for a wave,$$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { z }^{ 2 } } = \frac { -8{ \pi }^{ 2 }m\left( E - V \right) }{ { h }^{ 2 } } \Psi $$ When solved can give many values of \(\Psi \), Only those values which satisfy the following conditions are acceptable.
1. \(\Psi \) must be finite and single valued for all values of the coordinates. (x, y and z)
2. \(\Psi\) must be continuous function of the coordinates.
3. When the integration is carried out over the whole space $$ \int _{ -\infty }^{ +\infty }{ { \Psi }^{ 2 } } d\tau \qquad \text { must be finite }.$$
When the wave function \(\Psi\) satisfies all these conditions, then that function is called well behaved function. Such wave functions are called Eigen Functions. In case of energy E these Eigen functions are called Eigen values.
In terms of operator \(\widehat { A } \), this well behaved function can also be understood. If an operator \(\widehat { A } \) operates on a well-behaved function\( \Psi \) to give the same function by multiplying with the constant, then that function \(\Psi \) is called Eigen Function and that multiple constant is called as Eigen Value.
Let \(\Psi \) be the function and \(a\) is a constant multiplying factor. Then, \(\widehat { A } \Psi = a\Psi \) Here \(\Psi\) is an Eigen function and \(a\) is an eigen value.
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