Heisenberg’s Uncertainty Principle for JEE

To develop a general and appropriate atomic model, “Quantum Mechanical Model of the Atom”, two theories have contributed significantly. Those two theories of atoms are, 

1. Dual behaviour of matter

2. Heisenberg uncertainty principle.  

A German physicist, Mr. Werner Heisenberg in 1927, developed the Heisenberg uncertainty principle statement, which says that “It is not possible to estimate simultaneously, the exact position and exact momentum or velocity of an electron”. It is based on the dual behaviour of matter and radiation. 

Dual Behaviour of Matter

De Broglie, a French scientist, in 1924, developed the dual behaviour of matter, which states that “matter, like radiation, should also exhibit both particle-like and wavelike properties”. This is called the dual behaviour of matter. This means, like photons, electrons should also have momentum and wavelength. The relation between wavelength ($\lambda$) and momentum (p) of a material particle based on this theory is given by

\[\lambda =\dfrac{h}{mv}=\dfrac{h}{p}\]

Here, ‘m’ is the mass, ‘v’ is the velocity, and ‘p’ is the momentum.

Planck’s constant is $h=6.626\times10^{-34}~Js$

Origin of Heisenberg Uncertainty Principle 

According to the Heisenberg uncertainty principle, it is not possible to estimate exactly, both the position and the momentum of an electron or subatomic particle simultaneously. We can see that uncertainties in the position and momentum of large mass objects are negligible, but they hold an important role in the quantum world. 

Atoms and subatomic particles, as we know, have very small masses, and hence, any change or increase in the accuracy of their positions will cause an increase in the uncertainty associated with their momentum. The uncertainty principle was developed based on the dual nature of matter. The probability of finding particles is maximum where we can find more waveforms. 

But if the particles are under greater undulation, then there will be a vague wavelength. In this case, we can find the momentum of the particle. If the particles are in definite positions, then they will not have a fixed velocity. 

Heisenberg Uncertainty Principle Formula

The Heisenberg uncertainty principle derivation is as follows,

\[\begin{align} & \Delta x\times \Delta {{p}_{x}}~\ge \dfrac{h}{4\pi } \\ & or~\Delta x\times \Delta (m{{v}_{x}}) \ge \dfrac{h}{4\pi } \\ & or~\Delta x\times \Delta {{v}_{x}}\ge \dfrac{h}{4\pi m} \\ \end{align}\]

This is the Heisenberg uncertainty principle equation.

Here, 

$\Delta x =$ uncertainty in the position of the particle.

$\Delta p_x$ = uncertainty in momentum (or velocity) of the particle

It says that, when we know the position of the electron with a high degree of accuracy 

($\Delta x$ small), then electron velocity will be uncertain and $\Delta v_x$ will be large. If we know the velocity of the electron exactly ($\Delta v_x$ is small), then the electron position will be uncertain and $\Delta x$ will be large. Hence, we get a blurry picture of an electron’s position or velocity during physical measurement.

Significance of Heisenberg Uncertainty Principle

The main interpretation of the Heisenberg Uncertainty Principle suggests that there are no definite paths or trajectories of electrons and other subatomic particles. In the macroscopic world, we know the position of an object at a particular time and if we know its velocity and the forces acting on it, we can calculate the position of the object after some time. Hence, in this case, the position of the given object and its velocity fix its trajectory. But for an electron or any subatomic particle, as we cannot calculate the position and velocity at any given instant simultaneously to a given precision, we cannot find the trajectory of such a particle. 

Heisenberg Uncertainty Principle Example

To understand this principle, let’s say we need to measure the thickness of a sheet of paper with an unmarked metrestick. The results of this measurement will be inaccurate. To get an accurate reading, we should use an instrument like a screw gauge, which has graduations smaller than the thickness of a sheet of paper. Let us now consider an electron whose position we need to calculate. We cannot see an electron as they are small, with a mass of only $9.91\times 10^{-31}~kg$. The electron can be illuminated with a smaller wavelength of electromagnetic radiation compared to that of an electron. The photons of such radiation will have high momentum given by \[p=\dfrac{h}{\lambda }\] and on collision with such photons, the energy of electrons will be altered. The momentum of the electron will thus increase and it will move from its initial position. Hence, we would be able to determine the position of the electron, but the velocity of the electron after the collision would be difficult to calculate.

Heisenberg’s Gamma-Ray Microscope

Heisenberg’s gamma-ray microscope is based on the uncertainty principle. Let us shine an electron with the light ray of wavelength $\lambda$. The scattered light can be collected in a microscope objective, the diameter of the object subtends an angle θ with the electron as shown. 

Representation of Angle Subtended between Objective of Microscope and Electron 

$\Delta x$, the accuracy with which electron location can be determined by 

\[\Delta x=\dfrac{\lambda }{\sin \theta }\]

As the gamma ray has a small $\lambda$ and if $\sin \theta$ is made large, we can make $\Delta x$ as small as desired. Based on the uncertainty principle, we can keep $\Delta x$ as small at the cost of our knowledge of the $x$-component of electron momentum. We know that when a photon hits an electron, the electron undergoes a Compton recoil with momentum proportional to \[\dfrac{h}{\lambda }\]. Thus,

\[\Delta {{p}_{x}}=\dfrac{2h}{\lambda }\sin \theta \]

Combining with the uncertainty principle we get,

\[\Delta x\Delta {{p}_{x}} = \dfrac{\lambda }{\sin \theta }\dfrac{2h}{\lambda }\sin \theta = 4\pi h\]

Conclusion

Werner Heisenberg stated that we cannot measure simultaneously the position and the velocity of an electron or subatomic particle precisely. The dual nature of electrons helped to develop the Heisenberg uncertainty principle. In places where the undulations of the wave are more, the chances of finding the particle are higher. But here, the wavelength becomes difficult to determine and we cannot calculate the momentum. Hence, the position of the electron will be fixed. Alternatively, when a particle with definite wavelength can be found anywhere as we cannot fix its position. The Heisenberg Uncertainty Principle can be applied to the motion of microscopic particles, and its effect is negligible for macroscopic objects. The Heisenberg uncertainty principle played an important role in developing the Quantum Mechanical Model of the Atom.