Question

Question #422c4

Answer 1

Heisenberg's uncertainty principle or simply called Uncertainty Principle states that for any particle it is impossible to know simultaneously both position and momentum precisely.

Explanation:

Mathematically stated Heisenberg's uncertainty principle is

`Deltax*Delta p approxh` .......(1)

where `Deltax` is uncertainty in position, `Delta p` is uncertainty in momentum, and `h` is Planck's Constant and is `6.626xx 10^(-34) m^2 kg // s`.

Lets assume that the electron's precise location is within the atom. It is same as saying that uncertainty in position of electron will be of the size of the atom itself. The diameter of the smallest atom, hydrogen, is `2xxa_0=2xx5.29 times 10^{-11}=1.058xx10^(−10)m`.

From equation (1), uncertainty in momentum is given by
`Delta p approxh/(Deltax)`
We know that Kinetic energy `E=p^2/(2m_e)`.
Assuming `Deltap=p`, and inserting value of mass of an electron we obtain
`E=(h/(Deltax))^2/(2m_e)=((6.626xx 10^(-34))/(1.058xx10^(−10)))^2/(2xx9.11×10^(-31))=2.15xx10^-15J`

This gives us the uncertainly in the energy of an electron when it circulates around the nucleus within the size of atom. Which is more than the ground state energy of Hydrogen of about `2.2xx10^-18J`.

Due to relationship of momentum and kinetic energy as above, we see that for a precise position of electron, measurement of its momentum turns out to be highly uncertain.

Conversely we can prove from the Uncertainty Principle that for a precise value of momentum (through kinetic energy), uncertainly in locating the electron within the radius of an atom increases manifold.

This is taken care of in the modern quantum mechanics where one talks of finding the probability of an electron .