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Heisenberg uncertainty principle equation and formula

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What is Heisenberg  uncertainty principle?

Uncertainty principle is defined as:”It is impossible to measure simultaneously both the position and momentum of a microscopic particle with accuracy or certainty.”

When we are studying a large moving object say a planet,then we can follow its definition path on which it travels.If we know its initial position and momentum,then we can predict its position and momentum at any other time.But this is not possible for electron,proton and neutron which are microscopic particles. Heisenberg has given a principle in this connection.

 Heisenberg  uncertainty principle proof

In order to know the position of an object,we throw the photons of light upon them.If we want to have the idea for the position of electron,then the photons of X -rays region have to be used because their wavelengths are very small and the possibility for the hitting of electron is there.During this hitting,the photon transfers some of its energy to the electron.Therefore,the velocity and hence the momentum of electron changes.

If we use the photons of longer wavelengths say of visible region,the velocity and the momentum will not change appreciably because longer wavelengths rarely find the chance to hit the electron.But its position can not be determined because object will not be visible.

Keep it mind that,the uncertainity is not due to lack of better techniques for the measurement of position and momentum.It is due to the reason that we cannot observe the microscopic objects without disturbing them. Uncertainity  principal is not applicable to stationary electron because in stationary state the velocity of an electron is zero.As a result,position of electron can be accurately determined.But both positions and velocities of electron cannot be determined accurately.

Heisenberg uncertainty principle formula

Let the motion of the particle is along x axis,then according to the de Broglie Hypothesis:

mathematical form of uncertainity principle

If the motion of the particle depends upon the three coordinates x,y,z the generalizing above relation ,we have:

uncertainity

These are known as Heisenberg’s uncertainity relationships.According to these relationships:It is impossible to determine both the position and the momentum of a particle with ultimate precision.

The width Δx of the wave packet indicates the problem location of the particle,and Δpis the range in momentum.So,the uncertainity principle may also be stated as:

“A particle cannot be described by a wave packet in which the position and momentum have arbitrary small ranges.”

It means that due to the wave nature,the exact position x of a particle cannot be determine,but it will be in the range Δx.Similarly,the true or exact momentum of the particle px cannot be determined,but it will be in the range of Δpx.

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