Free particle Wave equation | Energy and momentum operator

What is the Free particle Wave equation? Derive Energy and momentum operator with the help of Free particle Wave equation


Free particle Wave equation:

If we assume that matter waves are produced due to variation in the wave function in a similar manner as sound waves are produced due to variation in pressure and electromagnetic waves are produced due to variation in electric and magnetic fields. The wave function is a function of both the position and time. Now, let us assume that wave function for the wave moving along positive x direction can be represented as

Free partical wave function

Free partical wave function 1

This is the free particle wave equation.

Energy and momentum operator:

On partially differentiating with respect to t to equation (3), we get

Energy operator

Momentum operator

This is known as momentum operator.

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