What is Heisenberg’s uncertainty principle? Derive an expression for it. Discuss two experimental illustrations of this principle.
Give mathematical proof of Heisenberg’s Uncertainty relation between energy and time.
What is the application of Heisenberg’s uncertainty principle?
Answer:
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.
“It states that you can never simultaneously know the exact position and the exact velocity of an object in microscopical 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.
The figure shows wave packet propagating in the xdirection with constant amplitude with wavelength; hence from deBroglie relation, the momentum p of the particle can be exactly determined, i.e., uncertainty in momentum is nearly zero. But the amplitude of a wave is the same everywhere, hence the probability for the particle to be anywhere within the wave packet is the same i.e., the uncertainty in the particle’s position is infinite.
Thus for an extended wave packet, the momentum of the particle can be exactly measured, but its position cannot be determined exactly. 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.,
An elementary proof of Heisenberg’s uncertainty principle
when we consider a group consist of very large number of waves, continuously varying frequencies, the product of uncertainty comes to

Energy – Time Uncertainty principle : Heisenberg’s uncertainty principle
This principle can also be expressed in terms of uncertainty, in energy, and time.
If we consider the velocity of a moving particle is
Application of Heisenberg’s uncertainty principle
Nonexistence of an electron in the nucleus (Electron cannot reside in the nucleus)
If we assume electron reside inside the nucleus, then since the radius is nearly 10^14m, therefore the maximum uncertainty in the electron’s position to be inside the nucleus is
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