Heisenberg’s Uncertainty Principle

What is Heisenberg’s uncertainty principle? Derive expression for it. Discuss two experimental illustrations of this principle.

Give mathematical proof of Heisenberg’s Uncertainty relation between energy and time.

What are the Application of Heisenberg’s uncertainty principle?


Heisenberg’s Uncertainty Principle:

The Uncertainty Principle introduced first in 1927, by the German physicist Werner Heisenberg known as Heisenberg Uncertainty Principle, is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known simultaneously.

wave packet

“It states that you can never simultaneously know the exact position and the exact velocity of an object in microscopically region” but in microscopically it is possible to determine exactly the position of a moving particle at any instant and the momentum of the particle at that position.


Figure shows wave packet propagating in x-direction with constant amplitude with wavelength; hence from de-Broglie relation , the momentum p of the particle can be exactly determined, i.e., uncertainty in momentum  is nearly zero. But the amplitude of wave is same everywhere, hence the probability for the particle to be anywhere within the wave packet is same i.e., the uncertainty in position of the particle  is infinite.

Thus for an extended wave packet, the momentum of the particle can be exactly measured, but its position cannot be determined exactly. On the other hand, if the wave packet is small, the momentum of the particle cannot be determined with certainty, but its position can be determined with certainty. Thus at any instant it is impossible to measure simultaneously the position and momentum of a microscopic particle with certainty. According to Heisenberg’s uncertainty principle, the product of uncertainties in the measurement of position and momentum is of the order of h (or h). i.e.,

Heisenberg’s uncertainty principle

Elementary proof of Heisenberg’s uncertainty principle

Heisenberg’s uncertainty principle proof

Heisenberg’s uncertainty principle proof

Heisenberg’s uncertainty principle proof

when we consider a group consist of very large number of waves, continuously varying frequencies, the product of uncertainty comes to

Heisenberg’s uncertainty principle position momentum

This principle can also be expressed in terms of uncertainty in energy and time.

If we consider velocity of a moving particle is

Heisenberg’s uncertainty principle Energy Time

Application of Heisenberg’s uncertainty principle

Non-existence of electron in nucleus (Electron cannot resides in the nucleus)

If we assume that electron reside inside the nucleus, then since the radius is nearly 10^-14m , therefore the maximum uncertainty in the position of the electron to be inside the nucleus is

Applications of Heisenberg’s uncertainty principle

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