When it comes to quantum mechanics, there are a few theories, laws or principles which are extremely puzzling and each time one goes through them, a new concept either takes birth or is encountered. One of the most debated laws of all time is ‘The Heisenberg’s Uncertainty Principle’.

Without involving too much mathematics, I would like to introduce this principle in a few comprehensible words. Sir Werner Heisenberg (1901-1976) derived a probabilistic relationship between any two complementary variables (such as position and momentum) which asserted a fundamental limit to the precision to which they could be known. It said that the more precisely you know the value of one variable, the more you lose the precision of the second one. Analogically speaking, the more exactly you know where the electron is situated in the energy level of an atom, the less you know its momentum. Hence, this relates the uncertainties of the two variables in the following manner,

where ħ is the reduced Planck constant.

Now, it is quite simple from the inequality that if the value of uncertainty in position gets too low, the value of uncertainty in momentum will be too high. This goes to the extent that if the position of the particle is exactly known, which means that there is no uncertainty in the value of position, then you have lost its momentum and you don’t know how fast it is moving because it’s the uncertainty in momentum which will then approach an infinite value.

So the next time you ask your friend about where he is, be mindful because he might lose his speed.

It can also be derived that the same inequality can be written for the angular position and angular momentum in a similar manner as it is written for linear position and linear momentum. It takes the form,

Using normalized wave function of the form,

we get

and

for all values of the quantum number m (the latter because the wave-function above is an Eigen-function of angular momentum).

This is where the problem begins. Computing the product of these two uncertainties will clearly give the answer zero which is certainly not greater than the half of reduced plank’s constant. This raises a very intriguing question:

**“Why do these uncertainties violate the Heisenberg’s Uncertainty Principle?”**

This question has been bothering physicists for several years and debates usually end up leading to more and more problematic questions, excluding a few who correctly win their arguments. As this is a ‘case’ of our great contributor-to-physics, Sir Werner Heisenberg, no one wants to lose his case. It appears as if Sir Werner Heisenberg is standing in the court, accused, and all physicists are providing their own pieces of evidences to vindicate him and his principle.

The answer to this question is a bit mathematical and people who are averse to mathematics should not read the article any further, because they will barely get anything after this.

The truth is that the analogy doesn’t hold up. The commutator between those two variables is actually zero, although it’s tricky to work out why.

The reason is that we have no concrete number ϕ which we are measuring. Instead, we have ϕ restricted to some domain – say, [0, 2π] [0,2π]. (Since we’re working on a manifold, technically speaking, the coordinate is not a number you can take out of context!)

Let’s call this quantity, the domain-restricted angle, Ø’, and denote the actual circle by C along which the particle is moving, with integration measure dØ and derivative **Δ**Ø around. (Both of those are well-defined even on a more general manifold!)

Let’s just commute that commutator:

Now, let’s take our example, for the sake of argument, in which ψ*ψ = 1. We have the change in Ø evaluated as we go around the circle. However, we have to include the jump back down from 2π to 0 in this integral, and so this integral is actually zero.

Equivalently,

(And the second part matters).

Therefore, the angular momentum and position have zero commutators for our state of interest, and we do not expect them to satisfy any (nontrivial) uncertainty principle for the states you consider, per the more general formulation of Heisenberg’s Uncertainty Principle.

Don’t worry because,

**“If quantum physics hasn’t profoundly shocked you, you haven’t understood it yet.”**

**Niels Bohr (1885-1962)**

A must read!

Well this was really a new look

Amazing, one should definitely read.

the quote in the end is just a perfect fit.. loved it

What if the object becomes stationary?

Requires one hell of a mind to be understood. Well-researched!