By Sunil Bhardwaj

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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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