The position of a moving particle is not the only thing that is a little fuzzy in quantum physics. It turns out that there is a very interesting interaction between the position and the momentum of a particle. The relationship between the uncertainties in these two quantities is something that does not appear at all in classical physics.
There are a few different ways to see why uncertainties in position and momentum should be related. Indeed, we just saw hints of this when we saw that a particle’s wave function can be localized somewhat (reduced uncertainty in position) by adding in a few waves with different wavelengths (or, per de Broglie’s relation, different momenta). Perhaps the most straightforward way of seeing this connection is to imagine how you would go about measuring the position of a tiny, moving particle–a falling speck of dust, say–very, very precisely.
Let’s just consider the position in one direction, the direction of the motion. Call this position x. Our objective is to determine x with the smallest possible uncertainty, which we will call Δx. This means we will determine the position to be x ± Δx. The smaller Δx is, the more precisely we’ll have determined the position of the particle.
The best way to locate an object like a speck of dust is, of course, to look at it. But in order to see it, some light has to bounce off of it. (If the speck is in a completely dark room, we have no chance of seeing where it is.) In optics, it is well known that the precision of locating an object using light is related to the wavelength (λ) of that light. It is impossible to use light to resolve features smaller than the wavelength of the light itself. Another way to say this is that the resolution (precision of the location measurement) cannot be smaller than the wavelength of the light used to make the measurement.
Let’s write this relationship in mathematical notation: λ < Δx. This simply says in symbols what we just said in words: the uncertainty in determining the position must be greater (can’t be smaller) than the wavelength of the light used to see the particle.
Arthur Compton revealed that photons carry momentum, and when they interact with matter, they transfer momentum, just like colliding billiard balls. If we are going to locate our speck of dust, at least one light photon will have to hit it. That means the dust particle will get jostled a bit, in some undetermined direction.
How big will the jostle be? Our friend de Broglie answered that one. Remember that the wavelength of a particle is equal to Planck’s constant divided by momentum. This holds for the particle as well as for the photon. We can turn this around to show that a photon’s momentum is equal to Planck’s constant divided by its wavelength. Since the scattered photon we use to measure the position of the particle could have come from any direction, the jostling results in an uncertainty in momentum Δp that is related to the wavelength. We could represent this mathematically as λ = (h ÷ Δp).
Now, if we combine this with the previous equation, we find that (h ÷ Δp) < Δx. This is the same as saying the product of the two uncertainties, in position and momentum, must be larger than Planck’s constant: h < (Δx × Δp). This is one manifestation of a very fundamental principle in quantum mechanics, the Heisenberg uncertainty principle, named after the German physicist Werner Heisenberg who published a paper on it in 1927. More generally, Heisenberg showed that in quantum physics it is not possible to measure all observable quantities to arbitrarily high precision. The accuracy of measurements is limited, not only by the quality of the experiment, but by nature herself!
DEFINITION
The Heisenberg uncertainty principle expresses the fact that the product of the uncertainties in certain pairs of observables (like position and momentum) has a minimum value of approximately Planck’s constant.
Let’s make sure we understand what’s going on here. Individually, it is possible to make a very precise measurement of the position or the momentum of a given particle. With the very best equipment, it’s possible to determine the location of that speck of dust (or atom, or electron) with an extremely small Δx. But if you do that, it will be physically impossible to measure the momentum of that particle very accurately. Conversely, if you put all of your effort into a precise measurement of the speed and direction of the particle’s motion, to make Δp extremely small, then you won’t be able to tell exactly where the particle was at the time you measured its momentum.
There is a minimum value to the product of the two uncertainties. You can always do worse, have larger uncertainties than allowed by this principle, but it is impossible to make the product smaller than the minimum (approximately Planck’s constant). That means making one of the uncertainties really small will cause the other uncertainty to be larger.
Hitting the particle with several photons to make sure you get a good position measurement is a nice idea, but it makes the momentum even more uncertain because of all the jostling. On the other hand, using only one photon with the smallest possible momentum, and you can’t locate the particle as precisely. This is because lower momentum for the photon means it must have longer wavelength (ask de Broglie). Longer wavelength means your measurement resolution is worse. No matter how clever we are, we will always be stuck with this trade-off.
QUANTUM LEAP
One interesting consequence of the uncertainty principle is the existence of so-called “zero-point energy.” A particle confined to a very small region of space can’t ever be “at rest,” because if its momentum were zero, it would violate the uncertainty principle. It must have some momentum, on the order of Planck’s constant divided by the size of the region of space in which it is located. Momentum means velocity, which means the particle must have some non-zero kinetic energy. The smaller the space to which the particle is confined, the larger its minimum possible energy.
We explored this very specific example at length in order to appreciate the meaning of Heisenberg’s uncertainty principle, but we need to stress that the principle is much broader than this one example. The important point is that the behavior described is intrinsic to the nature of the world, not merely a result of imperfections in our devices or techniques.
In fact, the most widespread interpretation of this is not just that we can’t simultaneously know the exact position and the exact momentum, but that there is no exact position or momentum to know in the first place. If a particle is in a state with a very well-defined momentum, then it doesn’t even have a definite location, only a wide range of possible locations in space. Conversely, if a particle is in a state that is well located, it doesn’t have a single momentum, only a wide range of possible momentum (and velocity) values.
WERNER HEISENBERG
Werner Heisenberg was born in Wuerzburg, Germany in 1901, which made him just one year younger than “quantum physics” and squarely among the ranks of the second generation of quantum physicists. He was therefore not driven by any attempt to visualize the inner workings of the atom—such as tiny electrons orbiting a nuclear center. Instead, he was guided by focusing on what is actually observable in the laboratory—such as atomic line spectra. This “modern” viewpoint was essential to his contributions to quantum physics.
A critical question posed by Heisenberg during the 1922 “Bohrfest” made such an impression on Bohr that he invited the 20-year-old Heisenberg to join him in Copenhagen. Heisenberg eventually agreed, and soon after moving north in the spring of 1924, he found himself strolling with his new mentor amidst the spirit of Hamlet at Kronborg Castle. A few years later, Heisenberg turned down his first offer to be a professor in Leipzig, choosing instead to continue working with Bohr in Denmark.
Heisenberg received his Nobel Prize in 1932, just as the Nazi Party was rising to power in Germany. Unlike many of his peers, he chose to remain in Germany in the run-up up to World War II. He eventually assumed a leadership role in Germany’s own effort to develop the atomic bomb. While there is speculation that its failure was due to deliberate attempts by Heisenberg to sabotage or slow down the program, it seems more likely that other factors prevented its success. In the post-war years, he continued in this role and helped plan for (West) Germany’s first nuclear reactor, which came online in 1957.
Leave a Reply