Heisenberg’s Principle of Uncertainty

On Copenhagen Interpretation

TThe Quantum Theory is a theory which manifests itself in the scale of the very small — we call this scale “The Plank Scale”. The Plank’s scale is defined by the Plank’s Constant, having a value of 6.62607015×10^(−34) Js (Jule-Seconds) and in equations, it’s symbolized by h. In the theory of Parallel Universes, there’s often arguments of the constants of nature — Gravitational Constant (G), Speed of light (c), and the Plank’s Constant (h). Have they been constant throughout the evolution of the cosmos? Are they the same throughout the parallel universes? These are deep questions. If, for instance, the scale of Plank’s Constant becomes, say, equal to 1, that particular universe would be very different from our own because of one quite stirring phenomenon: Uncertainty. This article is about this uncertainty on the Quantum Realm.

The Uncertainty Principle is a beautiful theory devised by Werner Heisenberg while facing intense pressure by “The Great Dane”, Neils Bohr. The theory flows on regarding aspects of experimentation and observation in the atomic realm. Because of this very fact, we have to bring up thought experiments, that is, experiments we perform by imagination, but in accordance to the proven theories. The theory tends to go beyond macroscopic detail of the world and dive into the very small and yield a world which is entirely different. However, we’ll try to reconcile the concepts of the quantum realm with the macroscopic world and try to understand why they are not manifested in the world we see.

Tracing an Electron’s Path

Let’s say we want to trace a path of a ball. A “path”, is a continuous set of positions of an object — in this case, the ball — in space and time. So, if we want to trace the path of a ball upon observation, given the position x in space at time t, and the velocity vector v, we must determine where the ball would be in t + δt, where δt is a very small increase of time. In order to determine the position of the ball in space at this minuscule increment of time, we must invoke the principle of causality — “when we know the present precisely, we can predict the future, as Heisenberg famously stated in his final uncertainty paper. That is, we must assume that we know about the present, which we do, and predicts its future position. In this case, we know the position of the ball in space, x, and velocity vector v, it is possible to predict where the ball would be in the future. Sum all the δt increments, and we can yield a path that traces the positions of the ball.

How we observe anything, is by using our fields of vision — photons reflecting off of the surface of that particular object. However, to observe an electron, due to its minuscule scale, we cannot use visible light, since its wavelength is larger (i.e. the frequency is smaller) with respect to the size of the electron. Notice that an electron is a microscopic object. It’s very small, and we require special apparatus to observe it. In the instance we discussed on the ball, we can observe the ball without any instrument. We need a different kind of light in the spectrum to observe the scale of the very small — enter Gamma rays.

Electromagnetic Spectrum: Source — http://www.astronomersgroup.org/EMspectrum.html

When a Gamma ray photon is bombarded at an electron, it collides with the electrons and ricochet — a game of billiards at the very small scale. And how we see the electron, therefore, is through the ricocheted photon. This collision, however is not a smooth one — it’s a sudden jolt, which results in the electron changing its momentum vector in an unpredictable manner. In order to minimize this change, we need to use low-frequency (high-wavelength) light. Now, the first assumption that we arrived was that in order to observe the position of an electron, we must use low-wavelength light. However, in order to keep the change of momentum of the electron smooth to predict the momentum vector of the electron at time t + δt, we arrive at a contradiction to use high-wavelength light.

Werner Heisenberg, Source — https://en.wikipedia.org/wiki/Werner_Heisenberg

This was the base for Heisenberg’s initial formulation. He posited that more accurately we observe the position of an electron (through low-wavelength light), less accurately we can measure the momentum change of the electron. He put this phenomenon into an equation, where Δp represents the “uncertainty” of the measurement of momentum of the electron, and Δx represents the same of the position. h is the Plank’s Constant.

Heisenberg’s Uncertainty Equation

Remark: Now, imagine a universe where h is larger, leading to a larger value at the right hand side of the equation. This leads to the multiplication of the uncertainty of position and momentum’s lower bound to become higher. That is, if in our universe (U), the minimum value for ΔxΔp would be 1, in the universe which h is larger (U*), the minimum value for the same quantity would be, say, 2. This means, that for a given quantity, the other quantity’s uncertainty would be higher in U* than in U. U* therefore, is a very “uncertain” universe indeed. All we had to do was to tweak h to become larger than its value at U.

Wolfgang Pauli explained this phenomenon like this: “One can see the world with a p-eye, and view it with a q-eye, but when he opens both eyes together, then one goes away”.

Heisenberg unified his theorem into the already-known fundamental equation of quantum physics, where p and q has the same meaning we’ve discussed above.

Heisenberg argued that the non-commutativity behind the equation (i.e. pq not being equal to qp) is explained by his thesis. That is, if we measure pq we cannot measure qp using the same uncertainties. Measuring the first quantity caused a disturbance, as he called it, which affects measuring the second quantity, just like the photon did for the electron. Therefore, the left hand side of the equation is never equal to 0 in quantum mechanics. But in Newtonian Classical Mechanics, this equation does not serve a purpose.

Enter, Neils Bohr

If you would have been keen on the passages above, you would have noticed one little subtlety that Bohr noticed. If we measure the angle and velocity of the ricocheted photon, given we know the mass and velocity of the electron at time t, we can invoke the principle of the conversation of momentum and calculate its momentum at t + δt! But the fundamental thesis for this calculation cannot be observed — that is, we cannot measure the angle of the photon entering the electron. Bohr argued that this was the essential phenomenon that invokes this uncertainty. Bohr’s argument was that the initial momentum of the electron when the photon hits it is uncertain and it is precisely what causes the uncertainty principle in the first place, which was something that Heisenberg failed to take into account.

Neils Bohr, Source — http://www.missingtheforest.com/niels-bohr/

Edwin Schrodinger had already posited his wave interpretation. In this interpretation, he presented a wave function ψ which is an imaginary number which cannot be observed (like x + iy where i is the square root of -1). A part of Schrodinger’s interpretation was that the quantity |ψ(x, t)|² was a manifestation of the charge density of the electron cloud at point x and time t. However, this interpretation was flawed and Max Born came up with an alternative, much more interpret-able and valuable thesis, to interpret |ψ(x,t)|² as the “probability of finding an electron at point (x,t) in space and time”.

Bohr thought that this wave interpretation of an electron — or any particle, de Broglie has already posited his wave-particle duality — must not be excluded from the picture of quantum uncertainty. Schrodinger’s initial thesis posited that a “particle electron” was nothing more than a set of matter-waves superimposed into one “wave packet”. Therefore, an electron in motion is nothing but a traveling wave packet. It was this very same phenomenon of resolving wave-characteristics into Heisenberg’s particle uncertainty theorem that caused tremendous pressure between the mentor and student.

A “Wave Packet” — Source: https://www.youtube.com/watch?v=ngGpSyWgDcY
Superimposition of multiple waves, Source: https://www.quora.com/What-is-the-difference-between-wave-function-and-being-a-wave-Doesnt-wave-function-determine-probability-density-If-so-how-can-we-say-that-particles-that-have-wave-function-are-waves-etc-photon

Bohr’s interpretation stated that it was not the “measurement” of either p or q that made this uncertainty — or disturbance — but what the experiment observes. The experimenter, Bohr believed, chose one side of wave-particle duality to be observed — or measured — forsaking the other.

De Broglie Equation, Source: https://www.qsstudy.com/chemistry/significance-of-de-broglie-waves

When superimposition of groups of waves produces a wave packet, it makes the wavelength indeterminate. That is, this superimposition makes a localized wave that has a wavelength which cannot be determined. According to de Broglie’s equation where p defines the momentum and λ defines the wavelength, we can posit that when λ becomes indeterminate, and for a given position, we can say that the velocity v is therefore indefinite. A wave with a well-defined wavelength is a spread-out wave, which has a velocity v but it is so spread out we cannot determine its position. We, therefore, come back again to Heisenberg’s initial hypothesis: we cannot measure exactly position and velocity of a particle (or in this case, the wave-manifestation of a particle) simultaneously. This is how uncertainty is applied to wave packets.

There’s a fascinating experiment called The Double-Slit Experiment. In this experiment, first, a wave-source produces waves to be traversed through two slits, and at the other end produces an “inference pattern” due to the superposition of waves, which is expected. When we use a gun with bullets as the source, the probability distribution of a “screen” at the other end is a summation of probabilities of bullets coming from either slit, which is expected. However, when we use electrons instead of bullets, the probability distribution that we see at the screen is the same as we saw with waves! But if we use a bright light source to detect the path of electrons (i.e. “observe” electrons), we’d get the probability distribution same as bullets. Bohr modified this experiment to accommodate Heisenberg’s uncertainty principle by introducing a vertical slider for the first plate, which you can read about in here.

Remarks

Now that we have come across the uncertainty principle, we’ll do some hands-on calculations to observe how it is at play in the real and quantum world. I’ll be taking the exercise in this article, and going through it to emphasize the insights. I’ll be using symbols for brevity.

Consider a ball of mass m, and an electron traveling at speed v. Now, assume that we allow for δp uncertainty of momentum in both instances. We can therefore calculate Δp as p×δp. If we plug this to Heisenberg’s uncertainty inequality, we obtain an inequality as follows.

For the ball, the inequality becomes:

Comparing these two inequalities, we can see what determines the lower bound for Δp are m, ρ and x.

Now, we can apply the same principles to an electron. Assuming the mass of an electron is m*, we come to the same formulation:

Since m >> m* the right hand side (RHS) of the second inequality is higher than the RHS of the first inequality. Since the ratio m : m* is a very large quantity (i.e. the mass of the ball is many orders of magnitude the mass of the electron), we can deduce that the lower bound for Δx* is very high for the electron, in comparison with the ball. This means, essentially, that the degree of uncertainty of measuring the position is very high when compared with a large-scale object like a ball. If we assume m = 400 grams and m* = 9×10^(−31) kg, the ratio between the uncertainties Δx* and Δx becomes in the order of 10³⁰! This means the uncertainty of measuring the position of an electron is 10³⁰ times that of a ball of 400grams! This is we aren’t aware of any uncertainty going on at the macroscopic realm.

The Copenhagen Interpretation on Quantum Mechanics heightened a fundamental importance for the observer of an experiment. In other words, it gave a privileged position for the observer in an observer-dependent reality. This interpretation was posited by Bohr and the contemporaries in Copenhagen. Einstein believed in an observer-independent reality, and he continuously argued Bohr on the topic. One of his thought experiments was called The Light Box experiment, which Bohr cleverly refuted using Einstein’s own arguments on time dilation (and relativistic physics on general relativity). After the second world war, Einstein came up with another experiment, which was called the EPR paradox that categorizes the Copenhagen interpretation of quantum mechanics as an incomplete theory, by introducing definitions for the elements of reality. Discussions which followed continued to the deathbeds of both the great physicists — Bohr and Einstein. When his wife ran into Bohr’s room after him calling out to her, she saw one last drawing he had drawn on his blackboard: the drawing was a diagram of Einstein’s Light Box.

Bohr’s last diagram, Source: http://lvdang2003.blogspot.com/2013/07/einstein-portrait_9139.html?view=snapshot

Software Engineer @ CodeGen International, CSE Graduate @ University of Moratuwa

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