17.5.4 Heisenberg’s Uncertainty Principle

In 1927, Heisenberg arrived at a peculiar looking equation (while plowing through the mathematics of quantum mechanics).

Δx Δph

Δ here indicate standard deviations. And Δx and Δp represent the uncertainties in the position and momentum of a particle respectively. So the inequality seemed to suggest that the product of these two uncertainties (for a particle) cannot be smaller than a minimum value that is of the order of the planck’s constant h.

First, let’s remind ourselves that h is a very tiny number. For everyday measurements, Δx and Δp due to instrumental precision and procedural uncertainties are always much larger than h. So this inequality does not pose any meaningful constraint when you are measuring objects like a ball or a chicken. On the other hand, if you are measuring elementary particles like photons or electrons, you are likely to be dealing with Δx and Δp whose products are of the same order as h. Now the minimum limit imposed by this inequality is an imposing one. Because decreasing either one of Δx or Δp requires the other one to increase. The inequality also rules out the possibility of either Δx or Δp being completely zero.

So that’s the mathematics. But what’s the meaning of it all, physically? Heisenberg originally explained this as a consequence of the measuring process: Measuring position accurately would disturb momentum and vice versa. He offered the “gamma-ray microscope” thought experiment as an illustration: He imagined determining the position and momentum of an electron by illuminating it with gamma rays. But our new found quantum knowledge informs us that we are actually bouncing gamma photons off the electron, inadvertently changing the momentum of the electron. To reduce the uncertainty in the position measured, we could reduce the wavelength of the gamma radiation. This comes from our understanding of optics, and is related to the Rayleigh’s criterion. But de Broglie’s equation tells us that reducing the wavelength increases the momentum of those gamma photons. The electron will be hit harder and undergo a larger change due to the large momentum of the photon, thus increasing the uncertainty in the momentum measured.

Today, it is understood that Heisenberg’s original interpretation of the Heisenberg’s Uncertainty Principle (as it came to be known), is at best incomplete, and at worst totally mistaken. Today, we believe that the Heisenberg’s uncertainty does not arise because of a disturbance caused by a measurement. The HUP is inherent in the wave-particle duality nature of the particle itself. As the complete explanation will be too complicated, I will only offer an analogy here.

Let’s contrast a wave and a particle. A (continuous sinusoidal) wave has a definite wavelength and thus momentum (\displaystyle p=\frac{h}{\lambda }), but a completely undefined position. On the other hand, a particle has a definite position, but a completely undefined wavelength and thus momentum.

But wave-particle duality suggests that nothing is completely a wave nor a particle. Instead, everything is more like a wave packet. If the packet were narrower, the uncertainty in its position would be smaller, but the uncertainty in its wavelength[1] and thus momentum will be higher. Conversely, if the packet were broader, the uncertainty in its wavelength and thus momentum will be smaller, but the uncertainty in its position would be larger.

Every elementary particle exists as a probabilistic wave packet when it is not being observed. When it is not interacting with other objects, it has neither a completely defined position nor momentum. Instead, it exists as a range of possible positions and possible momentums. The Δx and Δp of its probabilistic existence is limited by the HUP

Δx Δph

That, my friend, is the essence of the HUP.

YouTube Explanations

MinutePhysics, TedEd and Veritasium

Concept Test



Heisenberg’s Homework 

Heisenberg’s Chicken

[1] Because a narrower pulse is formed by summation of higher number and higher frequency sinusoidal waves. Look up Fourier Transform if you’re interested.

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