Last Christmas, my father and one of my brothers wondered how a spacecraft’s planetary slingshot works. It’s a well-known manoeuvre to get a big swing forward by passing by a planet. They realized that if the craft falls toward the planet due to its gravitational pull, it will gain momentum. However, as soon as it has flung around the planet, wouldn’t that same gravitation slow down the craft just as much? Great question, of course, so, let’s dive into this gravity assist short and simple.
First things first, you are absolutely right in thinking that the gravitational pull of the planet is a symmetrical situation. The same ‘force’ pulling the craft toward the planet making it accelerate, will also slow it down once it has reached its maximum point around the planet when it tries to escape the planet’s influence again, continuing its journey in space.
So, the whole planet’s gravity system is entirely symmetrical. What isn’t symmetrical though… is that the planet has an orbital momentum(beginfootnote)Momentum is mass times velocity in a specific direction(endfootnote) around the Sun! The planets in our solar system all orbit around the Sun. Though at different rates, their position relative to the Sun changes every single second. This means they have a great deal of momentum and we, the rather clever clumps of human cells that we sometimes are, can exploit this. So, by using the gravitational pull of the planet to get closer to it, the craft gains an additional tug by the simple fact that the planet itself is also moving through space. And so, while the momentum the spacecraft gained by the planet’s gravitational pull was lost again while escaping the planet’s gravitation, it did gain a bit of the planet’s orbital momentum!
It’s like catching a train while it’s hurling past the platform. Suppose you’re Batman and you possess a grappling hook. As soon as the train is coming through the station, you start running along with it on the platform. You pull out your grappling hook gun and grapple onto the train. Then you press the button so that the retractable cable attached to the hook pulls you toward the train (still speeding along its trajectory). At the right moment, you release the hook, spread out your batwings, and away you go, having parasitized off of the train’s momentum for a little bit.
And you’re right, this also means that the train lost a bit of its momentum due to you ‘pushing off of it’. However, due to the huge difference between your mass and the train’s mass, in combination with either velocities, the train’s momentum loss is negligible while your momentum increase is significant.
The same goes for the planet. A spacecraft doing a gravity slingshot, or, enjoying a gravity assist, as it’s also called, will decrease the orbital velocity of the planet, meaning that the planet’s orbit will get closer to the Sun. Of course, given the ratio between the spacecraft’s mass and the planet’s mass, this amount is negligible.
Voyager
Two of the most famous instances of gravity assists are the voyages of Voyager 1 and 2. In the animation below, you can see the trajectory of Voyager 2, where it gained assists from several of our solar system’s planets. Earth is the fast-orbiting blue dot. Jupiter is green, Saturn is cyan, Uranus is yellow, and orange is Neptune. And while the Voyager spacecrafts flew by the planets, they sent some of the best postcards back to Earth.
Of course, what goes around, may come around too. If you’d like the spacecraft to decelerate, you make it fly opposite the planet’s orbital motion. This way it slows down while donating a (negligible) bit to a planet’s momentum.
So, what Christmas message can we take from this? That’s right: even the tiniest entity in the Universe is capable of changing an entire planet’s momentum. No matter how small, in one way or another, its effects are significant.
It’s perhaps not as famous as Einstein’s formula but in this day and age many people may still have heard at least once of the phrase ‘Heisenberg’s uncertainty principle’. It plays an important role in quantum mechanics. You may have heard that every time you observe or measure matter, due to the crudeness or inherent inaccuracy of the measurement device, you will inevitably disturb your own observation. This would then preclude you from gaining accurate knowledge with satisfying certainty. In fact, in general, Heisenberg’s uncertainty principle states that nothing can be certain. At the risk of sounding vague and vanilla, all of these statements are completely and utterly wrong. Let’s look at what it really says, shall we?
Figure 1. Werner Heisenberg in Göttingen in 1924.
Fourier transform pairs
Trade-offs. Who doesn’t hate them? Remember when your parents told you that you could have this but then not have that or maybe just a bit of this but then less or fewer of that? Unsurprisingly, at least three famous philosophers have written a few words on this, each in their own way lamenting on the existence of trade-offs and how to deal with them. One chose to become all rebellious about it and wrote: ‘I want it all, I want it all, and I want it now!’ (May, 1988). The other two, however, chose to be more pragmatic about it as they postulated that ‘you can’t always get what you want’ (Jagger & Richards, 1968). Obviously, they knew that, sometimes, life brings you Fourier transform pairs. The more well-known example is of course Heisenberg’s uncertainty principle.
If you limit a particle’s range of possible positions in space $(\Delta x)$, you increase its range of possible momenta(beginfootnote)Momentum is the product of mass $m$ and velocity $v,$ so $p=mv.$ It’s a measure for the amount of motion of an object.(endfootnote) along the $x$-direction $(\Delta p_x),$ and vice versa.
This is formalised as follows:
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}.$$
Just to be absolutely clear: the delta-symbol $\Delta$ is a range of a certain quantity. Usually, a $\Delta$ is defined as the difference between two values. Suppose, you measure point $A$ of your garden fence to be $0.1$ metre away from your wall and point $B$ to be $0.7$ metre away from your wall, then the $\Delta$ of the distances, i.e. the length between points $A$ and $B,$ is $0.7-0.1=0.6$ metre.
In Heisenberg’s principle, it is stated that the product of the range of possible positions $\Delta x$ and the range of possible momenta $\Delta p_x$ is greater than or equal to some number. Mind you, it’s a tiny number. The symbol $\hbar$ stands for the Planck constant divided by $2 \pi,$ and the result gets cut in half yet again.
This means that whenever one is getting bigger, $\Delta p_x$ for instance, the other is getting smaller, which is then $\Delta x.$ And vice versa.
Click here if you’d like to do a bit of maths. It’s very easy.
Just to get an intuitive insight in this relation, suppose $\frac{\hbar}{2}=1,$ and so, suppose, $\Delta x \Delta p_x = 1.$ Furthermore, suppose $\Delta x = 0.5.$ What value does $\Delta p_x$ has to be to satisfy this equation? Exactly, $\Delta p_x$ has to be $2,$ because $0.5 \times 2 = 1,$ or else the equation is false.
Now, lets make $\Delta x$ smaller. In other words, we’re going to try to pinpoint the location with much more precision. So, let’s say, $\Delta x = 0.001.$ What value does $\Delta p_x$ has to become to satisfy this equation? You guessed right, $\Delta p_x$ has to become even larger: $\Delta p_x = 1000,$ because $0.001 \times 1000 = 1.$ If you were to reverse the situation – decreasing the size of $\Delta p_x$ – then, in turn, $\Delta x$ would have to become larger.
In reality, $\frac{\hbar}{2}$ is much smaller than 1. It is, in fact, about $5.273 \times 10^{-35} \text{J/s}.$ That’s thirty-four zeros behind the decimal point and then ending in 5273. It’s incredibly small. Don’t worry about this. We’ll get back to that later.
Hopefully, now you see the relation between $\Delta x$ and $\Delta p_x$ as put forward by Heisenberg’s formulation. They complement each other. Whenever a range of possible values becomes larger, in other words, the $\Delta$ or range of value-options is larger – its actual value becomes more uncertain, hence the use of the word ‘uncertainty’ in Heisenberg’s uncertainty principle(beginfootnote)In fact, it’s statistics. The $\Delta$-sign could just as well be a $\delta$-sign, so $\delta x \delta p_x \geq \frac{\hbar}{2},$ which signifies its statistical character more accurately. After all, the wave function is about probabilities.(endfootnote).
But why is this? While this principle plays a central role in quantum mechanics, it’s actually not fundamentally a quantum-mechanical law. This principle exists more generally in many instances in physics, and, even more generally, in mathematics.
In mathematics, the variables position and momentum are said to be a Fourier transform pair. Put in yet other mathematical jargon, position and momentum are said to be conjugate variables.
Sound
A well-known, non-quantum-mechanical example of the uncertainty principle is determining the pitch of a sound. How ‘high’ a note is, depends on the frequency.
The most familiar way we depict sound waves is a simple sine wave. It represents the simplest of sounds possible. Also, it’s the most boring of sounds possible.
The $x$-axis represents time. The $y$-axis represents the amplitude of the sound or the loudness, the intensity of it. As you can see, the sound wave repeats itself over time; the pattern is cyclic. One whole cycle is when the plot has completed going up, going down, going further down, and going up again. The time it takes to complete one cycle is designated by the symbol $T,$ called the period(beginfootnote)It is also possible to measure the time-distance between two peaks or two troughs.(endfootnote). So, this particular sound wave is said to be periodic.
Figure 3. A time-amplitude plot of a boring old sinusoidal sound wave. (Click to enlarge.)
The shorter the period – the quicker the cycles are – the higher the tone. Another way of saying, is that the higher the frequency, the higher the tone. The mathematical relationship between period $T$ and frequency $f$ is the following expression:
$$f = \frac{1}{T}.$$
If the period gets shorter, i.e. the value of $T$ becomes smaller, then the value of $f$ becomes larger, which means higher, which means a higher tone.
Seeing as the time period $T = 2 \pi$ seconds, the frequency diagram looks like a spike at $\frac{1}{2 \pi}$ Hz. In this frequency-diagram, the $x$-axis is the frequency and the $y$-axis is still the amplitude.
Figure 4. A frequency-amplitude plot of the sound wave of Figure 3. It shows the exact frequency at which that sound wave exists.
So, there are now two ways in which we can describe the sound wave: either by frequency (Figure 4) or by change over time (Figure 3).
Notice that the sound wave plotted as a function of time (Figure 3) has no beginning nor end. For all we know, that plot could just go on forever, to an infinite amount of time, in both directions. Suppose, we would ask the question at what time exactly does the sound exist? The answer is: always. There is no particular, specific time at which it exists.
In other words, we could write that $\Delta t = \infty.$
Notice, however, that the frequency plot looks very finite: just one stroke. One well-defined, finite stroke. If we were to ask the question what frequency exactly does the sound have? The answer is: there is a particular, specific, exact frequency at which it exists and it is $\frac{1}{2 \pi}$ Hz $( \approx 0.16).$
Fourier analysis
In reality, no sound is going to be infinitely long. Pluck a guitar string and it will fade out as the energy dissipates slowly. Also, at some point it started – meaning, before that, it didn’t exist. In other words, in reality, a sound wave usually exists in a finite range of time.
Let’s limit our sound wave to a range in time, so it looks more like the sound of a ‘blip’ and less like an infinite tone of boredom. Again, the $x$-axis represents time and the $y$-axis represents the amplitude.
Figure 5. A time-amplitude plot of a so-called wavelet, a short sound burst. Contrary to the sound wave in Figure 3, it’s not infinitely long. It’s now also more difficult to assess its frequency.
As you can see, the sound now exists in a more defined range of time – roughly 1.5 seconds. In other words, $\Delta t \approx 1.5$ seconds. That’s a whole lot smaller than the old $\Delta t = \infty.$
Now, we ask ourselves, what is its frequency? The difficulty now is that it’s hard to pinpoint an exact period $T$. The evolution of the plot is quite different from our infinitely long sine wave. Yes, we can identify kind of those cycles we’re looking for, however, no cycle has the same shape, so, technically, we’re dealing with multiple cycles at once. And guess what, its frequency-amplitude plot looks like this.
Figure 6. The frequency-amplitude plot of the wavelet in Figure 5. It’s far from being a specific, exact frequency. At varying degrees, it’s actually a few frequencies at the same time.
As you can see, it has become difficult to pinpoint the exact frequency of our wavelet. It exists at a variety of frequencies and amplitudes.
So, while the ‘time window’ of the sound wave has become more exact, the frequency has now become ‘less certain’.
The brilliant mathematician Joseph Fourier discovered that a wavelet such as in Figure 5 can actually be constructed by adding many infinite waves at many frequencies. Put differently, Fourier analysis shows that our wavelet is the culmination of a superposition of many waves at many frequencies.
Figure 7. The wavelet at the bottom is constructed by many infinite waves at many different frequencies superposed onto each other. This automatically means that the wavelet’s exact frequency is fundamentally harder to determine than the frequency of the sound wave in Figure 3.
Now you see why the frequency-amplitude plot has changed from a very specific value in Figure 4 to the wider set of frequencies in Figure 6. In the latter case, the wavelet ‘contains’ multiple waves at multiple frequencies, so when you Fourier transform its time-amplitude plot to its frequency-amplitude plot, the frequency has become ‘uncertain’.
The relation between time $\Delta t$ and frequency $\Delta f$ in ordinary classical physics is fundamentally complementary. No quantum mechanics needed.
The term ‘Uncertainty principle’ pertains to the general phenomenon that Fourier transforms (such as between time and frequency) entail a fundamental, mathematical trade-off between types of information carried by the two transformed variables. Heisenberg then showed that this principle also holds in quantum mechanics. And so, the uncertainty principle in quantum mechanics is called Heisenberg’s uncertainty principle.
The De Broglie relation
Time to go back to quantum mechanics. Remember that a particle’s best description is a wave function? A wave function is the mathematical expression of a particle containing all possible states it can assume once we measure it.
Instead of a time-amplitude plot, let’s represent a particle by a space-amplitude plot. To make it a little bit easier, let’s take the wave function of a particle of which the amplitude only varies along one dimension of space, $x.$
Here is a representation of a particle’s wave function along one dimension of space (along a ‘straight line’). The $x$-axis represents a position in space. The $y$-axis represents the amplitude of the wave function (which is proportional to the probability of finding the particle in that particular position $x$).
Figure 8. A representation of a wave function of a free particle. Note that this is not what it actually looks like. For one, an actual wave function exists in complex space, which we didn’t plot here. The goal is to illustrate, not to map accurately. Also note that the free particle has no specific position yet as it’s a free particle!
It was the eminent French physicist Louis de Broglie(beginfootnote)Many physicists have tried and mispronounced his last name. It should sound like ‘broy’ where the r is produced at the back of the throat, like the French r – a ‘dry’ kind of r. In this interview with him, you can hear the French presenter pronouncing his name (just after 0:16 seconds). It’s not ‘brog-ly’ nor ‘bro-ly’. Thank you.(endfootnote) who formulated the relationship between a particle’s wave function’s wavelength $\lambda$ and its momentum $p.$
$$\lambda = \frac{h}{p},$$
where $h$ is the Planck constant. Incidentally, this is the equation better known as De Broglie’s matter wave hypothesis, stating that matter, such as electrons, possess a wave-like characteristic(beginfootnote)Do note that this same equation shows that this wave-like behaviour of large bodies such as our bodies, brains, bowling balls, tennis balls, and animals is completely and utterly negligible as we will demonstrate at the end of this post.(endfootnote). This won him the Nobel Prize, no less.
If we rewrite this to solve for $p,$ we get
$$p = \frac{h}{\lambda}.$$
So, clearly, a wave’s momentum is determined by its wavelength. The smaller the wavelength, the greater the momentum. What is the wavelength? It’s the length between two peaks (or two troughs). The higher the frequency, the smaller the wavelength. Now have a look at Figure 8 again. As you can see, the infinite wave of a free particle has a well-defined wavelength. The logical conclusion is that the momentum is also well-defined. Nevertheless, Figure 8 also shows that the particle’s position is not defined at all!
Let’s turn this on its head and limit the range of possible positions of our particle. No longer is it a free particle. It is now confined within a finite range of locations.
Figure 9. Our former free particle’s position is now restrained between $x = 0$ and $x= \pi.$ In other words, $\Delta x$ is now limited to $\pi$ wide. There is no well-defined wavelength as the wave function has different values in different places. It’s there, but not as well-defined as in the wave function in Figure 6.
What we’ve done in Figure 9 is making $\Delta x$ smaller than it was in Figure 8 (where it was infinitely large). In fact, $\Delta x = \pi$ wide. By the same Fourier transform mechanism as with the time-frequency pair, the complimentary sister of position space $\Delta x$, namely momentum space $\Delta p_x$, will now become less certain.
To construct a limited wave function such as the one in Figure 9, Fourier analysis shows that you need – again – a bunch of waves at different frequencies in superposition (added on top of each other).
Figure 10. A Fourier deconstruction of the wave function in Figure 9. Many waves, many frequencies. Hence, the momentum is less well-defined.
So, when it comes to quanta, Heisenberg’s uncertainty principle states that there’s a fundamental trade-off between information on position and momentum(beginfootnote)Another pair is energy and time. This is interesting in the context of Hawking radiation. We’ll get to that, don’t worry.(endfootnote). This is due to the fact that they are a Fourier transform pair or conjugate variables.
This also means that if you constrain a particle to a minuscule $\Delta x,$ its wave function will start to contain momenta $\Delta p_x$ all over the place. It will occupy many more velocity possibilities, including the much faster velocities. If you were to subsequently perform a measurement, the probability of finding it moving at higher speeds is now much larger!
Scale and effect
At the scale of the big bad world, we never see this effect. If you would confine a bowling ball in a limited space, you will not see its momentum increase dramatically. It won’t suddenly start bouncing up and down. Conversely, if you swoop the bowling ball with considerable momentum, it won’t suddenly start appearing everywhere and nowhere at the same time: its position is still quite clear. You won’t suddenly quantum tunnel through the pins or be rolling on all bowling lanes of the neighbouring players at the same time. If it doesn’t hit a single pin, then that’s not because it’s suddenly in a state of superposition with regard to its possible locations of existence. You’re just not that good.
You won’t notice any of these quantum effects in your everyday-scaled objects. Only when you’re dealing with particles. Or atoms. However, as soon as the mass increases, it all changes. Why? Partly because Planck’s constant is so darn small(beginfootnote)And because the number of interactions between atoms increase exponentially, causing any quantum effect to disappear due to decoherence.(endfootnote). It’s just $5.273 \times 10^{-35} \text{ J/s},$ remember? That’s small.
All this knowledge does allow for some fun calculations. For instance, if you were to confine a bowling ball with a mass of $7.2$ kg (16 lb) inside a box where $\Delta x = 22$ cm (8.66 inches), by Heisenberg’s uncertainty principle, the ball’s speed will be $3.283 \times 10^{-35} \text{ m/s}.$ That means that after $965.9$ billion years it might have moved a distance equal to the diameter of a proton. That amount of time is seventy times the age of our current universe. Granted, quantum-mechanical effects aren’t zero, but as you can see (or rather, as one can calculate), on our everyday scale, these effects are quite meaningless.
Sometimes, weird films such as What the #$*! Do We (K)now!? and What the Bleep!?: Down the Rabbit Hole will want to make you believe such quantum things can happen anyway. They will mention Heisenberg’s uncertainty principle like it is a magical law allowing us to do whatever. I hope that this post has shown that Heisenberg’s uncertainty principle is not about that. Nor does the uncertainty principle itself have its roots in quantum mechanics. It’s basically wave mechanics, the classical stuff, which all first-year undergraduates in physics have to learn in their first or second semester.
A few months ago, I stumbled across a video showing an Australian senator’s question to the head of the Commonwealth Scientific and Industrial Research Organisation, an Australian federal government agency responsible for scientific research. Clearly, the senator had – shall we say ‘read something about Heisenberg’s uncertainty principle’. During a senate hearing for a legislative committee, the senator questioned if research done in climate change should be taken with precaution as Heisenberg’s uncertainty principle stands in the way of accurate measurements(beginfootnote)He basically sought a ‘scientific’ way to put climate science in doubt – which, apparently, he is not a proponent of. I do not claim to know anything about Australian politics, or even at great depth about climate science, however, when a legislator starts talking quantum physics – well, I do know stuff about that.(endfootnote).
I suspect this discussion pertained to a study where a satellite uses infrared radiation to perform surface and/or atmospheric remote sensing. He continued to state that as infrared light has lower frequencies than visible light, it’s ‘very difficult’ to understand the properties of infrared radiation based on Heisenberg’s uncertainty principle.
Many things were going on (wrong) in this one short bit of speaking time of the senator, as is usual when someone hasn’t caught up on quantum physics as much. Which is understandable, but no less gnawing to watch (the link opens a new tab and leads to a short video on Twitter).
In any case, I genuinely hope that this article contributed at least a sliver of knowledge to educate the electorate of the world, so we can all vote as informed and responsible as possible for the right persons for the right jobs, besides one’s preferred socioeconomic idealism.
If you should take one thing from this post, it’s that Heisenberg’s uncertainty principle is not about anything spiritual nor does it have anything to do with scientific measurement mistakes: it’s good, old wave mechanics and Fourier analysis taught to undergrads in their first year at university. It works and it works well. It does not lead to science not being able to know things about the universe. In fact, it increased our knowledge of it. In fact, no modern information device would have worked without it. After all, you’re reading this with an electronic device which exists thanks to Fourier, Heisenberg, and De Broglie, among others. All that with a bit of more maths and more physics at the same time.
Photo Werner Heisenberg by Friedrich Hund, a German physicist who took this photo in Heisenberg’s place of residence, Göttingen, in 1924. It was uploaded to Wikimedia Commons under CC BY 3.0 by Friedrich Hund’s son, Gerhard Hund, a German mathematician, computer scientist, journalist, and chess player. We have used a colour-corrected version by Martin Geisler.
We will focus on a few simple problems where we will manipulate the equations for relativistic energy and momentum.
This could be seen as a second-year university-level post.
Einstein had shown that the Lorentz transformations were the correct way to switch between the coordinate systems of different frames of reference [1]. He also taught us that Newton’s laws weren’t at all proper relativistic laws. For instance, Newtonian momentum $ \mathbf{p} = m \mathbf{v} $, and energy $ E = mv^2 / 2 $ were not at all accurate at speeds approaching that of light.
Instead, we have all come to learn that the relativistic momentum is written as
Prove, for a particle travelling at $ c $, that the magnitude of the relativistic energy is given by $ E = pc $.
Show that the energy-momentum relation for a particle with any mass $ m $ travelling at any speed $ v $ is correct and do mind it is not the famous $ E = mc^2 $ we are referring to. Use the correct one, if you please.
Given that the mass of a proton is $ m_p $, calculate its exact speed when its relativistic translational kinetic energy (which is the relativistic total energy minus its relativistic mass energy) is four times its relativistic mass energy.
Problem I
Since $ E $ is expressed in terms of $ p $, we need to rewrite Eq. $ \eqref{eq:relativistic momentum} $ by solving for $ m $:
\[ m = \frac{p \sqrt{1 – \dfrac{v^2}{c^2}}}{v}. \]
Note, we do not use the vector quantities, just the magnitudes. We can now proceed to substitute this into Eq. $ \eqref{eq:relativistic energy} $:
which is about $ 0.98c $ rounded to two decimals, which means that the proton zips at about 98% of the speed of light through the fabric of the cosmos.
Image Hydrogen bubble chamber Fermilab: Proton with 300 GeV energy producing 26 charged particles in the 30 inch hydrogen bubble chamber at Fermilab. Source: Wikimedia Commons
[1] Einstein, A. (1905) ‘Zur Elektrodynamik bewegter Körper’, Annalen der Physik, 322(10), pp. 891–921. doi: 10.1002/andp.19053221004.
This is a repost. Slight errors in the parsing of LaTeX in the original article of 24 December 2018 have been corrected.
.jpg){kind=link}