Heisenberg Uncertainty Principle

Introduction

The uncertainty principle is called the Heisenberg uncertainty principle. This principle was stated by the German physicist Werner Heisenberg in the year 1927. This principle is one of the most celebrated results of quantum mechanics and it states that we can determine different things about particles simultaneously at a time. This principle states that there is inherent uncertainty in the act of measuring a different given variable of particles. 

However, this principle mainly focuses on position and momentum of particles, the principle states that the more precisely the position is known the more uncertain the momentum is and vice versa. This principle works in contrast to classical Newtonian physics which states all variables of particles to be measurable to an arbitrary uncertainty given good enough equipment.

As per the uncertainty principle if you know the exact position of a particle it is impossible to determine the exact momentum of the particle and vice versa. The same relationship can also be applicable for energy and time, as we cannot measure the exact energy in a given finite amount of time. From this concept it is clear that uncertainties in  the conjugate pairs of momentum/position and energy/time were termed by Heisenberg as having  a minimum value corresponding to the Planck’s constant divided by 4π.

Heisenberg principle can be numerically written as: ∆X  ×  ∆p  ≥  h/4π

Here ∆ denotes the uncertainty in that variable and h is the Planck's constant and its value is  6.6 × 10−34 joule-second.

Origin of the Uncertainty Principle 

The main reason behind the origin of the uncertainty principle is the dual nature of a wave-particle. As each particle has a wave nature, the probability of finding particles is maximum where the undulations of the wave are greatest. The more the undulation of the particle, the more ill-defined becomes the wavelength, which helps in the determination of momentum of the particle. This shows that particles which have definite position have no certain or fixed velocity. Whereas the particle having a well-defined wavelength gives the precise velocity. So, an accurate measurement of one quantity leads to large uncertainty in the measurement of the other.

Derivation of the Heisenberg Uncertainty Principle

In this principle,  ∆x is considered to be an error in the position measurement whereas  ∆p is the error in momentum. Therefore, from this concept we can write:

∆X  ×  ∆p  ≥  h/4π

Since momentum, p=mv, by this Heisenberg’s uncertainty principle formula can also be written as:

∆X  ×  ∆mv  ≥  h/4π

Accurate measurement of position or momentum automatically indicates a larger error in the measurement of the other quantity.

Now, apply the Heisenberg’s Principle to an electron in an orbit of an atom, with h = 6.626 × 10⁻³⁴ Js and m= 9.11 × 10⁻³¹ Kg,

∆X ×  ∆V  ≥  6.626 × 10⁻³⁴ Js/ 4 × 3.14

∆X  ×  ∆mv  ≥  h/4π

∆X  × 9.11 × 10⁻³¹ =  6.626 × 10⁻³⁴ Js/ 4 × 3.14

If the position of the electron is measured accurately to its size (10⁻¹⁰m), the error in the measurement of its velocity will be equal or larger than 10⁶m or 1000Km.

Heisenberg’s Principle applies to only dual-natured microscopic particles and not to a macroscopic particle whose wave nature is very small.

Example of the Heisenberg Uncertainty Principle

Both electromagnetic radiation and microscopic matter exhibit a dual nature of mass/ momentum and wave nature. In the case of macroscopic particles both position and velocity/momentum can be determined simultaneously. 

Example: If the location and speed of a moving car can be determined at the same time, with minimum error. However, in microscopic particles, it will not be possible to fix the position and measure the velocity/momentum of the particle simultaneously.

We cannot see very small particles like electrons having a mass of 9.91 × 10⁻³¹kg with naked eyes. A highly intensified light collides with the electron and they illuminate and this illumination process helps in measuring the position of the electron. As intensified light source undergoes collision, this helps in increasing the momentum of the electron and makes it move away from the initial position. Thus, when fixing the position, velocity /momentum of the particle would have changed from the original value. So, if the position of the particle is exact, there will be an error in velocity or momentum or vice-versa. Whereas in the same way, the measurement of momentum accurately will change the position.

So from this, it is proved that we cannot determine both position and momentum simultaneously and measurement of both simultaneously leads to error in the measurement.