#Uncertainty principles

[minmin-vs-physics]

HUP, EPR, and Bell’s Theorem

Abstract

An educational document discussing the Heisenberg Uncertainty Principle, the EPR (Einstein- Podolsky-Rosen) Paradox, and Bell’s Theorem, written for an audience without a background in physics, but with their head still screwed on right.

1 Introduction

Ah, quantum mechanics. A bizarre theory which unfortunately describes our physical world exceed- ingly well. Einstein didn’t get it. Bohr didn’t get it. I don’t get it. And soon, you won’t get it either. As the saying goes, the more you know about quantum mechanics, the less you understand it.

I will be skipping around in terms of topics covered in undergraduate quantum mechanics courses to prepare you for the actual beast, Entanglement.

Entanglement, the property of quantum systems to remain correlated even when separated, is a concept which has transformed from a worrisome byproduct of a thought experiment [1] into a cornerstone of quantum mechanics itself. What is a quantum mechanics? Google is your friend, my dear reader. My time with you is limited„ and I cannot teach you the alphabet to make you read Shakespeare. I can only explain what you directly need to understand this article. Anything else shall be your homework, and if I am feeling kind at the end, I will provide a list of accessible resources on learning quantum mechanics the RIGHT way.

As we dive into the frankly confusing world of entanglement, it is vital that you remember one thing– A quantum particle is described by a wave function, Ψ. This wave function is a solution to the Schrodinger equation.

This is what they mean when they say something is both a particle and a wave; It’s behavior can be described by a special kind of wave equation, which we all know and love as the Schrodinger Wave Equation. But that’s not important right now. I’ll explain more if I need to. We need to get to HUP.

2 Heisenberg’s Uncertainty Principle

Formulated by Werner Heisenberg in 1927, the Uncertainty Principle is an indomitable tenet in the field of quantum mechanics. Its premise is simple. The more precisely a particle’s momentum is determined, the less precisely is its position. In one dimension, this can be summarized with the following mathematical statement: ∆x∆p ≤ 2

Here, ∆x is the standard deviation or “spread” of the position x, while ∆p is the standard deviation of the momentum p. As the spread of one quantity decreases, the other must increase in order to maintain the inequality. I will not bother proving the Principle in full, but I have Heisenberg’s original proof in the references.

Is that it?

Ummmmm, no. An important thing to remember about HUP is that it is not exclusive to x and p. HUP applies to any two quantum mechanical operators, A, B, which do not commute with each other i.e. [A, B] = AB − BA = 0. But that’s all mathematical nonsense, Min! What does it really mean?

Fine! I’m only doing this because it will be useful when we get to measurements in the EPR paradox and Bell theorems. In order to understand what “not commuting” means in the physical sense, let’s use our favorites, position and momentum, as an example. In quantum mechanics, xˆ andpˆ are referred to as the position and momentum operators respectively. (Why the little hats? Firstly, they’re cute, and secondly, well, you’ll see.) The whole point of calling them operators is that they act on wave functions. And in the crudest sense possible (please don’t try this at home, folks), hitting an operator on a wave function and taking the expectation value, gives a measurement of the quantum mechanical system.

There is about three semesters of quantum mechanical education I’m waving off right now, but bear with me. When we act the momentum operator on the system, in some sense we extract the momentum. Same thing for position. However, the whole deal about x and p is that they do not commute. So, the order in which you conduct the measurements absolutely does matter. First measuring x and then p would give you a different answer than first measuring p and then x. This is because the very act of measuring a quantum state changes it. That’s right! It changes. This makes all the difference when you consider the standard deviation of a bunch of measurements. If my memory of introductory quantum mechanics serves me right, after about three pages of algebra you arrive at the familiar position-momentum uncertainty principle.

The moral of the story is that the non-commutativity of these operators manifests as a sort of granularity in the accuracy of measurements you can make on a physical system. This granularity is retained between any other kinds of non-commuting measurements you can make!

On second thought, do you really need this? Probably not. But, the algebra of uncertainty principles is a pet project to me. Especially the strangest of them all, the energy-time uncertainty principle. Enough on that! Here’s the main takeaway (other than the actual HUP statement) that you need from this section:

Making a measurement on a state changes its wave function. No exceptions. None. The detached observer is not a reality in the quantum mechanical world.

3 Spin

I realized that the following sections will not make any sense if you don’t at least know what spin is. So, let’s make a short pit-stop at Spin City to learn about this nonsensical physical quantity.

We’re all aware of angular momentum– its the rotational analog of linear momentum (which we talked about the previous section). We all agree that it is a property related to the motion of an object, right? WRONG! Sometime in the 1900s (Seriously, 20th Century Physicists should chill out), it was discovered this angular momentum from motion i.e. “orbital” angular momentum, as it was called in the atomic physics context it was first described, does not account for all the angular momentum of a particle. Long story short, the remaining angular momentum, which is intrinsic to a particle, is now called Spin. Every fundamental particle has a particular value of spin, which, in quantum mechanical jargon, is the eigenvalue of the spin operator.

For understanding the following sections, we really only need to care about spin-1/2 particles, which are lovingly called fermions, and are the building blocks of all ordinary matter. The shining feature of spin-1/2 particles is that their spin can either be +1 or −1 , which is often referred to as spin-up (↑) and spin-down (↓) respectively.

Physically, the up or down comes from whether the measured spin is along the axis it is measured, or opposite to it. Yes, spin is a vector, so it does have three independent components in the three spatial directions, but it is convention to consider the z-component of the spin for calculations and experiments. Any references to up and down in the next sections are along the z-direction.

Oh, and one more thing, spin-0 particles have no intrinsic spin. This will be important when we encounter the EPR Paradox.

4 EPR Paradox

After skipping a whole bunch of most-likely important concepts in the study of quantum mechanics we arrive at the EPR paradox.

The EPR paradox is a thought experiment first described in the groundbreaking paper [1] by Einstein, Podolsky, and Rosen in 1935. Einstein was quite vocally a hater, and the EPR paradox was proposed as evidence that the description of reality provided by quantum mechanics is incomplete. Reality doesn’t care, of course, and the EPR Paradox isn’t really a paradox. In fact, it is the foundation of entanglement– a magnificent, very real feature of reality which spans black holes, quantum computers and even my field of research: Entanglement in elementary particle physics.

In fact, I’m so self-centered that the example we will use to illustrate the EPR paradox is from particle physics. Just kidding, my explanation follows Chapter 12 in Griffiths’ Introduction to Quantum Mechanics, and is a simplified version credited to David Bohm

EPRB Paradox

Suppose a pion (funky particle with spin-0) at rest, decays to an electron and positron which fly off into opposite directions. Since the pion has spin-0, conservation of angular momentum dictates that the electron and positron occupy the following spin configuration.

√(1/2) (|↑↓⟩−|↓↑⟩)

BE NOT AFRAID of the mathematical jumpscare. The fancy bracket |·⟩ is what’s called a “ket”, and is used to denote the state of a quantum system. All the expression says is that either the electron is

spin-up (+1) and the positron is spin-down (−1) or vice-versa, because the total spin of the system 22

must add up to 0. (Since the initial state is spin zero, the system must stay spin-zero even after the decay occurs. That’s what angular momentum conservation is all about.) We don’t know which combination we will get, but it must be one of the above. Measuring the spin of one of the particles will automatically tell us what the spin of the other particle is. This means that the spins of the electron and positron are correlated. In modern terms, such a state is called entangled.

Now, let’s pretend that these particles fly off in opposite directions, until say, they are several light years apart. What would happen if we found the electron and measured its spin to be +1 ? We instantly know that the positron’s spin is −1 . This is obvious. Why are we mad about this?

Naturally, we may think that the electron really was spin-up from the moment it was created and it was only that quantum mechanics did not know until we made a measurement. But by the principles of quantum mechanics, neither particle had a definite spin, until we made a measurement, causing the wave function to “collapse” and instanteously produce the spin of the positron which is lights years away!

The EPR bros were NOT having it. Einstein famously called this phenomenon “spooky action at a distance”. They stated that the quantum mechanical standpoint must be wrong! The electron and positron must have had well-defined spins from their creation, even if quantum mechanics does not know it. Quantum mechanics is not a complete description of reality and there must be some hidden variables which describe a physical system that we do not yet know.

The fundamental assumption guiding the EPR argument is that no information can propagate faster than light. This the principle of locality. In order to appease this, we can say that the wave function collapsed at some finite velocity and is not instantaneous. However, this violates conservation– If we measured the positron spin as well before the information of collapse reached it, there is a 50–50 chance that both particles are spin-up, which means the system has total spin-1. Preposterous! You can mess with anything you want in this universe, but you don’t mess with conservation laws. What do we do now?

Okay, let’s calm down. The theorists may say whatever they want, but experiment doesn’t lie. Experiment tells us that in these cases, spin is perfectly correlated. The wave function collapse is instantaneous. That’s crazy. Call your mom and tell her you want to go home. The EPR Bros are frightening you— Quantum Mechanics is NOT local so it is NOT complete.

...Except. It is. Enter, Bell’s Theorem.

5 Bell’s Theorem

Now, what’s the situation? The EPR gang is not happy. I’m not happy. You’re not happy. Is quantum mechanics wrong? No, silly! EPR said it themselves: they think it’s merely incomplete. So, in order to completely describe a quantum mechanical state, you not only need the wave function Ψ, you also need some unknown, hidden variable λ. Lots of hidden variable theories were proposed after the Einstein-Podolsky-Rosen paper, but none of them ever gained traction. It was still a respectable area of study until 1964, when J.S. Bell proved that any local (Remember locality from the last section?) hidden variable theory is incompatible with quantum mechanics.

I’ll spare you the details of Bell’s work, dear reader. One thought experiment in an essay is gruesome enough. (It is also getting quite late and I still didn’t code my calculations. I have spent far too much time on this already.)

Bell’s proof involves the wonderful use of probability, and the barest assumptions that can be made about local hidden variable theories. Basically, in any local hidden variable theory, the probabilities of various outcomes are related by what’s known as a Bell inequality. If EPR’s conjecture is right, and there really are hidden variables we don’t know about, then any physical system must obey its Bell inequality.

Except, there have been various experiments since the 1960s confirming that Bell’s inequality is indeed violated. This came as a rude shock to scientists as it is not fun to learn that reality is very much nonlocal. It was all fun and games when this was all merely a mathematical artifact, but nonlocality felt like a gateway drug to a much grimmer violation.

Causality

Bell inequality violations, no matter how surprising, are merely wonderful correlations between two sets of otherwise random data. Sure, the measurement of the spin of the electron affects the positron, but it does not cause it in any meaningful way. The person measuring the electron spin cannot use this collapse of the wave function to send a message to the person with the positron, since they don’t control the outcome of the experiment. They can decide whether to measure the electron at all, but the other person only has access to the positron’s spin and cannot tell whether the electron has been measured or not.

Phew! This sort of nonlocal influence does not transmit any energy or information, so it is exempt from the speed of light. Meanwhile, causal influences, those which do transmit information or energy, cannot travel faster than light. According to special relativity, if this was possible then, there are reference frames in which information can propagate backwards through time. And that, my dear reader, is what we call a big nono. Since the EPR paradox does not imply that causality is violated, we can lie uncomfortably on our bed of nonlocal but causal theory of quantum mechanics.

So rest easy, quantum mechanics is weird, but safe. Entanglement is not a fairytale, but also not the boogeyman. It’s probably more scared of you than you of it. Just give it some time. More answers will follow.

What Do I Do Now?

So, you want to know more? Or curl up in a ball and never think about this again? Either is fine. I won’t judge. If your answer is the former, here are some resources to guide you through the thickets of quantum mechanics.

PopSci Sources

1. IDTIMWYTIM: Heisenberg Uncertainty Principle 2. Why did Quantum Entanglement Win the Nobel Prize in Physics? 3. Bell’s Theorem: The Quantum Venn Diagram Paradox

Surely, you’ll get more out of these wonderful science Youtubers than you did from me yapping for four pages. There are a bunch more probably, but you’ll have to find them yourself.

Academic Sources

1. An Introduction to Quantum Mechanics, D.J. Griffiths.

Of course, there are other quantum mechanics textbooks that I like much more than this one. But, this is the least daunting, so I’ll leave it here.

Don’t forget to like and subscribe for more silly academic style papers.

References

[1]  A. Einstein, B. Podolsky and N. Rosen, “Can quantum mechanical description of physical reality be considered complete”? Phys. Rev. 47, 777–780 (1935) doi:10.1103/PhysRev.47.777

[2]  Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”. Z. Physik 43, 172–-198 (1927). https://doi.org/10.1007/BF01397280

[3]  D.J. Griffiths, D.F. Schroeter, “Introduction to Quantum Mechanics, Third Edition” Cambridge University Press (2018) 978–1–107–18963–8,

[blackswaneuroparedux]

Only a few know, how much one must know to know how little one knows.

- Werner Heisenberg

By 1924, the 23 year old Werner Heisenberg was already a brilliant physicist and so he went to research with Niels Bohr, the director of the Institute of Theoretical Physics at the University of Copenhagen. His paper, “Quantum Theoretical Reinterpretation of Kinematic and Mechanical Relations” was published a year later. After his return to Gottingen, he developed the matrix mechanics formulation of quantum mechanics. He was around 27 years old when in 1927, Heisenberg developed his famed Uncertainty Principal while in Copenhagen. That same year, he became a professor of theoretical physics and head of the physics department at the University of Leipzig. Werner Heisenberg received the Novel Prize in Physics in 1932 for his Uncertainty Principle, though he had been previously nominated in 1928 by Albert Einstein.

[polymorphemic-vector]

ah yes. generalized stokes theorem and generalized anxiety disorder the two genders

[zero-zoxx-international]

Elektronen am Morgen (Electrons in the Morning), 1993 by J.G.Wind

[ao3screenshotss]

[bakedbakermom]

wake up kids new principle just dropped

[toddhewitt]

4.06 / 4.10

"and we will be your partners, in the fight to free [the new world]?"

[bailey41]

I watched that whole thing for this exchange, and for the briefest of moments, it didn’t matter that we were boiling the oceans out of existence

[elvencantation]

i have a sort of question! reading dune with my mom (i’ve read it like ten times, but it’s her first) and we came across this passage

“The prescience, he realized, was an illumination that incorporated the limits of what it revealed—at once a source of accuracy and meaningful error. A kind of Heisenberg indeterminacy intervened: the expenditure of energy that revealed what he saw, changed what he saw.”

so then i got curious and looked up the Heisenberg indeterminacy, which is apparently a fundamental concept in quantum mechanics. to put it short “the more accurately one property is measured, the less accurately the other property can be known.” (from wikipedia)

which is a perfect correlation with some of the ways FH describes prescience. however, then i scrolled down a bit further and read this passage

Historically, the uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system, that is, without changing something in a system. Heisenberg used such an observer effect at the quantum level as a physical "explanation" of quantum uncertainty.

(i bolded and italicized words to hopefully make it easier to parse the blocks of text)

anyway. it sounds like frank either got the indeterminacy confused with the observer effect in this specific case, given the explanation at the end of the novel passage or im just confused. opinions?

[tmarshconnors]

"The presence of an observer in an experiment alters the outcome of the experiment itself."

Born Werner Karl Heisenberg 5 December 1901 Würzburg, Kingdom of Bavaria, German Empire

Died 1 February 1976 (aged 74) Munich, Bavaria, West Germany

He formulated the Uncertainty Principle: One of Heisenberg's most significant contributions to physics is the formulation of the Uncertainty Principle in 1927. This principle states that it is impossible to simultaneously measure certain pairs of physical properties, such as a particle's position and momentum, with arbitrary precision. This had profound implications for the understanding of quantum mechanics.

Nobel Prize in Physics: Heisenberg was awarded the Nobel Prize in Physics in 1932 for his creation of quantum mechanics, specifically the development of matrix mechanics. He shared the prize with Max Born and Pascual Jordan for this groundbreaking work.

Key Figure in Quantum Mechanics: Heisenberg was a central figure in the development of quantum mechanics during the early 20th century. Alongside other pioneers such as Niels Bohr, Erwin Schrödinger, and Max Planck, he played a crucial role in establishing the theoretical framework of this branch of physics.

German Physicist during World War II: Heisenberg remained in Germany during World War II and worked on nuclear research for the Nazi regime. However, there is much debate and controversy surrounding his involvement in the German nuclear program. Some argue that he deliberately sabotaged the project, while others believe he genuinely aimed to develop a nuclear weapon for Germany.

Post-War Influence: After the war, Heisenberg played a key role in rebuilding the scientific community in Germany. He helped establish the Max Planck Institute for Physics in Göttingen and later became the director of the Max Planck Institute for Physics in Munich. He continued to make significant contributions to theoretical physics until his death in 1976.

[sspacegodd]

[angelthefatal]

Found this beauty on YouTube

[manwalksintobar]

At North Farm // John Ashbery

Somewhere someone is traveling furiously toward you,  At incredible speed, traveling day and night, Through blizzards and desert heat, across torrents, through narrow passes. But will he know where to find you, Recognize you when he sees you, Give you the thing he has for you? Hardly anything grows here, Yet the granaries are bursting with meal, The sacks of meal piled to the rafters. The streams run with sweetness, fattening fish; Birds darken the sky. Is it enough That the dish of milk is set out at night, That we think of him sometimes, Sometimes and always, with mixed feelings?

[twistedtummies2]

Have you ever thought up KH OCs?

Actually, no. Because I really haven't seen a reason to make one so far. XD

HOWEVER, what I HAVE thought up is a KH fanfic. And I don't mean a short kink-based story, though I've written a few of those, nonono, it's a long-running project...which I'm still trying to find time to continue. 'XD So, I can't remember if the story is here on Tumblr, but I began work on a project called "Kingdom Hearts: The Uncertainty Principle." Ostensibly, it's a crossover between KH and the animated series "Rapunzel's Tangled Adventure," but it's intended to go to all sorts of other worlds and involve a bunch of other characters I've always wanted to see in KH games. The plan was to split the single story into multiple "Books," with me writing a whole book out and then uploading it chapter-by-chapter before starting the next. I finished Book I literally years ago...BUT UNFORTUNATELY, when I got started on Book II, I found it wasn't really working properly, and I had to go back to the drawing board. This, plus a TON of life stuff and constant backlogs on my commissions, has resulted in the story being...I think four years on hiatus, simply because I haven't had a chance to do anything with Book II, let alone the ones meant to follow. The sad part is that Books III to the end of the saga are already set up and I THINK still work, just looking at my plans. So Book II is really the stumbling block. If I can just find time to figure out how to make the plan work better, and then - more importantly, honestly - find time to WORK ON IT, I will continue "The Uncertainty Principle." Till then, God only knows when it will get done. I certainly hope to finish it someday. :P

[noosphe-re]

What are the elements of an act of knowing? At least two are immediately obvious: that which is known, and the consciousness which knows it. The knower's status is at least as important as that of the known, a point which modern physics has been forced to recognize in the uncertainty principle, which reveals the fact that, in order to know something about it, an observer must act upon, and thus disturb, an object. The act of knowing is thus a transaction between the object and the knower which involves physical exchanges of energy. Thus the concept of pure "parallelism”—a reflection within the knower of the known—is not tenable. We cannot think of the known as pure reflection within the knower, like the image of a flying swan on the smooth surface of a pond. When the swan lights on the water, its pure image is shattered by the broken surface.

Arthur M. Young, The Geometry of Meaning

[higgsbosom]

i pretty sure if i didnt have deadline to finish this chapter i wldve loved it