Class 12 > Unit # 25: Quantum Physics > Heisenberg's Uncertainty Principle

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Heisenberg's Uncertainty Principle

Uncertainty Principle

“It is impossible to determine both the position and momentum of an electron simultaneously. If one quantity is known precisely, then the determination of the other quantity will become impossible.”

Mathematical Representation

Let:

• $\Delta x$ = Uncertainty in position
• $\Delta p$ = Uncertainty in momentum

According to Heisenberg's uncertainty principle, the product of the uncertainty in position and the uncertainty in momentum is on the order of an amount involving $h$ (Planck’s constant):

$\Delta x \cdot \Delta p \approx h$     --- (i)

Uncertainty in Time and Energy

Similar to the uncertainty in position, there is another principle of uncertainty which limits the accuracy in the measurement of time. If $\Delta E$ is the energy uncertainty and $\Delta t$ is the uncertainty in time, then we have an expression similar to equation (i):

$\Delta E \cdot \Delta t \approx h$     --- (ii)

Explanation

Consider an example in which we want to determine the position of an electron. The position of an electron is measured by illuminating it with light and observing the reflected light. However, the process of scattering light disturbs its momentum.

Heisenberg considered an electron that has a definite, known momentum passing under a powerful microscope. He realized that measuring the position of an elementary particle alters its momentum in a random manner.

This technique allows the position to be specified with an accuracy comparable to the wavelength of light used in the experiment. However, when the photons are scattered from the electron, they alter its momentum because photons carry a momentum of their own. The observer cannot calculate the exact extent of this disturbance, which is random.

Increasing the wavelength decreases the disturbance, because photons of longer wavelength have less momentum and energy. However, increasing the wavelength reduces the precision of the position measurement. Conversely, decreasing the wavelength allows better position measurement, but increases the disturbance to the momentum.

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