Category: Quora Answers

Geometric vs Wave Optics

A Quora question asks: How can we say that light tends to travel in straight lines, but when we squeeze it to travel through a very narrow slit, it spreads out? Doesn’t this mean the light is not traveling in a straight line anymore?

The key word is “tends”. For the most part that tendency is what dominates our experience – as can be seen for example by constructing a pin-hole camera.

But there are various situations where it does not apply. Most familiar is the change of direction when light encounters a change of refractive index, but there is also a slight tendency to bend around any obstacle. The wave theory that predicts this was demonstrated by Thomas Young’s two slit experiment but an even more impressive demonstration was perhaps the spot of light directly behind a circular barrier that was predicted by Poisson (as a supposedly ridiculous consequence of the wave theory) and then actually observed in a public demonstration by Arago (having actually perhaps been noted much earlier by Maraldi).

The reason we see don’t see these effects more often is because the wavelength of light is very small and only the part of the beam within a wavelength or so of the barrier experiences any noticeable bending, so it requires a very bright source and a special setup to actually see it. In the case of the circular barrier, it is the rotational symmetry that gives constructive interference between the bent beams from different locations on the boundary. But you can actually see a similar effect from periodic symmetry by looking at the sun through a black woven umbrella (if your twirl the umbrella the bright spots just rotate around around the line to the sun rather than the axis of the umbrella so you can see that they aren’t keeping exactly in line with the actual gaps in the weave).

Source: (1000) Alan Cooper’s answer to How can we say that light tends to travel in straight lines, but when we squeeze it to travel through a very narrow slit, it spreads out? Doesn’t this mean the light is not traveling in a straight line anymore? – Quora

YM Existence and Mass Gap (in layman’s terms)

“Yang-Mills” is just the name for a class of theories which have a certain kind of symmetry and which include as a special case parts of the “standard model” which physicists use to predict the behaviour and interactions of elementary particles .

The “existence” problem here is that the various procedures used by physicists to make calculations in these theories have never been proved to actually always work. They involve making sequences of successive modifications from some initial guess according to patterns that are known (ie proven mathematically) to work in simpler theories for producing a sequence of numbers that actually converges to a well-defined result (that is independent of the starting point). But proofs of effectiveness have never been found for the theories that are actually used to describe elementary particles. What is therefore not yet known to exist is a set of well-defined final predictions (ie an actual theory defined by the proposed procedures).

The calculations can be done in various ways, and do seem to produce useful approximations to what we actually see in experiments, but we don’t know that the results will actually converge if we keep on going. So we don’t know for sure whether or not we have a well-defined theory. (This applies even to the case of Quantum Electrodynamics, but there is some hope that the more complicated symmetries of a Yang-Mills theory may help to guarantee convergence.)

As an analogy (not to the physics but to the state of our knowledge) imagine coming across a ladder standing up in the middle of a field. It reaches up so far that you cannot see if it is stabilized in any way at the top; but you want to get a better view of what is around you, so you climb up the first few rungs and can see over the nearest hedge (and what you see from the ladder does match what you can see by walking across the ground). But now you want to look over the nearby hill. Perhaps you could climb higher, but what if the ladder is only precariously balanced? If it is infinitely long then it may have enough inertia not to be disturbed by your climbing, but on the other hand it may have enough stretch and flexibility that if you get high enough the part you are on will fall down anyhow. And even if the ladder is infinitely long and stable, on a spherical Earth there is a limit to how far you will actually be able to see (and perhaps there is important stuff happening on the far side that will eventually affect you). So the ladder may never tell you everything you need to know, and if it swings about you may never be sure that your view is ever the “correct” one, so there is no actual final prediction that it tells you.

The “mass gap” issue has to do with whether or not, if we leave out ElectroMagnetism, it is possible to clearly distinguish the vacuum as having strictly less energy than other states, and is also related to having more rapid falloff of non-EM forces such as those between nucleons. (This is actually a much weaker condition than the strict “confinement” that we actually expect for the forces between quarks within nucleons and pions, but proving it might be a first step towards that.)

One reason for combining this more specific “mass gap” issue with the more general and abstract question of “existence” is because, in some simpler cases (of just one or two space dimensions) the techniques used to prove “existence” of a well-defined quantum field theory also prove (and to some extent make use of) the existence of a mass gap.

Source: (1000) Alan Cooper’s answer to What is the Yang–Mills existence and mass gap in layman’s terms? – Quora

Chemistry

The Pauli exclusion principle allows us to approximate the wave functions of valence electrons by treating the inner electrons and nucleus together as a single source of potential; and then by treating the ionic cores as fixed we can solve the Schrodinger equation for the valence electrons and calculate its lowest energy level as a function of the relative coordinates of the cores. Minimizing that function then allows us to determine the optimal bond lengths and their relative orientations.

Source: (1001) Alan Cooper’s answer to How does quantum mechanics treat atomic bonds, and what role does the Pauli exclusion principle play in this context, considering also that electrons are everywhere in space according to their wave function rather than confined to fixed orbits? – Quora

Why do people have different definitions of quantum? Is quantum mechanics a logically consistent, self-consistent theory? 

Quantum Mechanics is not a single theory. In the past there have been other attempts to describe the fundamental aspects of physics which used the word “quantum” in various different senses, but to most physicists nowadays it is a class of theories characterized by the property of having the “pure states” of an isolated system represented by rank one projectors (or equivalently rays or unit vectors) in a complex Hilbert space – and by a rule for predicting the probability distributions of outcomes for various possible experimental observations. Each such theory is internally consistent, but that doesn’t mean either that they are necessarily correct in their predictions or consistent with either one another or with other theories about the physical world.

Source: (1001) Alan Cooper’s answer to Why do people have different definitions of quantum? Is quantum mechanics a logically consistent, self-consistent theory? – Quora

Wigner’s Friend

If we define both the observer and the “observed” as both being part of say an even bigger system, would the wave function still collapse in this system?

This conundrum is known as the Wigner’s Friend problem, though it is also often asked with reference to Schrodinger’s cat.

In my opinion its best resolution is in the understanding that the wave function or quantum state is not a property of the system itself but of its relationship to an observer, and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics [which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI)].

Source: (1001) Alan Cooper’s answer to Is the collapse of the wave function in Quantum Physics based on a system frame of reference? If we define both the observer and the ‘observed’ as both being part of say an even bigger system, would the wave function still collapse in this system? – Quora

Is observation required for collapse? 

Whether or not observation is the only way in which a wave function can collapse depends on what you mean by “collapse”, and that word is used by various people in reference to different aspects of the measurement and observation process – which can be considered as happening in two stages.

The setting involves a system in a pure quantum state which may have been prepared as an eigenstate of some observable (such as spin relative to a particular direction), and so is a nontrivial superposition of eigenstates of some other observable (such as spin relative to a different axis) which we now want to measure.

In the first stage, the system of interest interacts with a larger more complex system which is not fully known and so is in a statistical mixture of pure states (represented by a density matrix rather than a single state vector). If the larger system is suitably designed as a measuring apparatus, then the interaction leads to the state of the combined system approaching a statistical mixture of states in which the subsystem of interest is in an eigenstate of the observable and the measurement apparatus is in a related state which involves some macroscopic feature (such as a pointer, a readout panel, or a bright spot on a phosphor screen) which has a corresponding humanly visible value. Henceforth the system acts as if it is in just one eigenstate which is not yet known but is subject to classical probabilities. This process eliminates the possibility of future interference between the eigenstates that was possible while the state of the system was in a pure state (represented by a coherent wave function) and so is often called “decoherence”; and since it reduces the system to being effectively in just one eigenstate it is often identified with “collapse of the wave function”. It actually happens in almost any interaction with a complex system (even when there is no humanly visible related macroscopic property of the system). So, for those who identify decoherence as collapse, it is indeed possible for collapse to occur without observation.

But after this kind of “collapse” we still don’t know what the measured value actually is, even though we can think of it as having just one of several precise values – each with some known probability.

The second stage of the observation process is where the conscious observer notices which value is present. Some people think of this as where the “collapse” happens, but here it is not really collapse of the wave function but rather of the classical probability distribution (similar to the case of a coin toss which starts of in a stochastically mixed state and collapses to just one case when we see the result).

The difference from a coin toss is that in that case we assume that all along the system was really in whatever particular state we eventually observe, and that state could have been determined with certainty just by making more observations at the start; whereas in the quantum situation the uncertainty seems to be essential until we actually experience the result. This leads to a philosophical problem for those who think that the quantum state is a property of the system itself rather than of its relation to the observer as it seems to imply that the experience of a conscious observer has some physical effect on the universe and raises the problem of Wigner’s friend who watches an experiment before Wigner does and seems to collapse the wave function even though the friend is himself just a complex quantum system who Wigner sees with a wave function that does not collapse until the information reaches his (Wigner’s) own mind.

To my mind this is resolved by seeing the quantum state as a description not of the universe but of its relationship to the observer; and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI).

Source: (1001) Alan Cooper’s answer to Is observation the only way in which a wave function can collapse? – Quora

Does an observer modify the observed?

What many people misunderstand is that in quantum theories the “state” of a system is not a property of the system itself but rather of how it appears to an observer.

There are actually at least two stages to the observation process. One is when the system of interest interacts with the much more complex system of a measurement apparatus whose precise quantum state is too complex for the observer to keep track of and so has to be expressed as a statistical mixture. This can have the effect of causing the combined system, in which the observed subsystem was initially in a pure “coherent” superposition state (with interference still being possible between different possible observed eigenvalues), to end up very close to a statistical mixture in which each possible measured value of the observed quantity has a well defined value with no interference between them. This “decoherence” process can be caused by interaction with any sufficiently complex system (even, as Viktor Toth notes, just a brick) and it does modify the observed (as does any interaction with anything – even just another simple quantum system). But it still leaves the actual value of the observation unspecified. The “collapse” process, which identifies which particular value has occurred, only happens in the mind of the observer whose conscious experience corresponds to just one of many possible histories of the universe. But this doesn’t modify the observed – at least no more than it modifies everything in the universe that is dependent on that observed value. (For example if we are in a room together and I see a red flash then the you that I see will also see a red flash, but if you see a blue flash then the I that you see will also have seen a blue flash.)

Source: (1001) Alan Cooper’s answer to In the quantum mechanical idea that the observer modifies the observed, can the observer be an insect? – Quora

Acceleration effect on light speed

The idea that the experience of an accelerated observer might be approximated by considering its worldline as comprising many small inertial pieces is a good one. And during each inertial step the speed of light seems to be constant everywhere. But at the velocity boosts or “frame jumps” between the steps, the apparent coordinates of all events (including those on the world line of a light signal) get shifted, so the light seems to jump ahead or back. Taking the limit of these approximations leads to the conclusion that the light signal does not seem to have constant velocity from the point of view of the accelerated observer. (Since the “frame jumps” lead to coordinate changes that are proportional to the distance of the event from the observer, this does not change the fact that every light signal seems to have the same speed when it reaches the observer, so there is no local change and it is just when the signal is far away from the observer that its velocity appears to vary.)

Source: (1001) Alan Cooper’s answer to A rotating frame can be divided into an infinite number of infinitesimal inertial frames. According to SR (special relativity) the light speed in each inertial frame is constant. Is light speed therefore constant in say the Sagnac experiment with SR? – Quora

What is relative in Relativity?

What is relative in any physical theory of “relativity” are the space-time coordinates of events from the perspectives of different observers. 

One problem, I think, with the names using ‘Theory of Relativity’ is that they seem to suggest theories about what is relative, rather than (more correctly) about how the coordinates used by different observers need to be related in order to ensure that the laws of physics are invariant (ie NOT relative).

In fact the coordinates that seem most natural to any observers for the purpose of expressing their experience in quantitative terms are always to some extent relative to the observers, so just saying that they are relative without specifying how is not telling us much (though in the new theories there is “more” relativity in the sense that time as well as the spatial coordinates becomes relative).

Our intuitively expected relationship between the coordinates of relatively moving observers allows all observers to use the same time coordinate, and so to agree on which events are simultaneous (ie constitute the same moment in time). It also preserves the form of Newton’s equations of motion for observers moving at constant relative velocity – which, as Galileo noted, has the consequence that observers moving with constant relative velocities cannot, by mechanical experiments, identify any particular one as being stationary. So the question of who is moving can only be answered relative to a particular observer – but this is just one particular instance of the relativity of coordinates.

[Sometimes observers moving relative to some larger object such as the Earth might choose to agree on a fixed Origin based on that object rather than on their own positions. But Galileo noted that if they are all moving together inside a moving vessel without any view of the outside, then it makes sense for them to use the vessel itself as their frame of reference – and relative to that, anything outside would appear to be moving in the opposite direction. In the world of Galilean/Newtonian physics there is nothing aside from its greater size which makes us prefer the Earth’s frame to that of the vessel, nor anything besides Earth’s proximity which makes us prefer its frame to that of the Sun. The answer to whether or not anything is or is not actually moving was thus, even in classical mechanics, entirely relative to the observer’s arbitrary choice of a frame of reference; and so that certainly was NOT anything new in Einstein’s theory.]

The above noted preservation of form of the equations of motion is perhaps confusingly called both “Galilean invariance” and “Galilean relativity”. The confusion could be avoided by making it clear that the word “relativity” applies to coordinates and “invariance” to the laws of physics. But I think that the practice of using “relativity” for the invariance itself rather than for the coordinate transformations under which it holds was indeed a misnomer which I believe precedes Einstein (though as an aside I must add that it seems surprisingly difficult to find out who was actually the first to do this).

Einstein’s special theory describes how the spacetime coordinates must be related in order for the laws of electromagnetism to have the same form for all inertial (ie unaccelerated) observers in the absence of any gravitational field. It turns out that for this to work, observers in relative motion will not be able to use the same time coordinates, and indeed will have different notions of simultaneity; so in this theory there is indeed something more that is “relative” than in the Galilean theory (but I don’t think that is why the theory got its name).

Einstein’s theory derives the relativity of simultaneity, and the formulas relating spacetime coordinates of different observers, from the principle of invariance of Maxwell’s equations (and so in particular, invariance of the speed of light) from the points of view of all inertial observers. But in my opinion Einstein’s reference to that principle as the “principle of relativity” (as opposed to the “principle of invariance” as suggested for example by Felix Klein) was indeed a misnomer, and apparently even Einstein eventually expressed some agreement with this  (but too late to actually change it).

[The special theory of relativity also includes modifications of the laws of mechanics (excluding gravity) which are necessary for them to remain invariant under the same transformations as those which preserve Maxwell’s equations – but this has nothing to do with the name except for the fact that perhaps the thinking was that the “principle” in question was that all physical laws need to be invariant under the same relativity of coordinates.]

The general theory goes on beyond the special theory to describe how the coordinates should be related in order to preserve an invariant form for both electromagnetic and gravitational forces under more general conditions (including accelerated observers and gravitational fields). So it’s not that more things are relative in the general theory, but rather that the relativity of the same things is explored under a more general range of conditions.

P.S. It should perhaps be noted that, just as the special theory has no distinguished inertial frame, the general theory does not provide any purely local way to distinguish inertial from accelerated frames as no accelerated observer can distinguish the experience of being accelerated from that of being prevented from falling freely in some “fictitious” gravitational field – which can only be identified as truly fictitious by observing the absence of possible sources (mass-energy distributions) out to an arbitrarily great distance. So there is some sense in which acceleration vs gravitation distinction is not quite absolute in the general theory but I don’t think that this (or the absence of any distinguished inertial frame in the special theory) was ever the reason for our use of the word “relativity”.

Source: I am trying to understand the term general relativity: what is relative in GR? Gravitation (acceleration) is absolute, not relative! So what is relative in GR? Is it perhaps a misnomer? – Quora

In QM, how can all people see something and all report the same thing? Wouldn’t 1 person’s observation cause their reality to branch off?

Quantum physics, without any additional “interpretation”, is just a tool for predicting the probabilities of various possible future observations from knowledge of other observations we have made in the past. To do so, it summarizes the observer’s previous observations (up to the point of the observer’s last interaction with the system) in what is called the “state” of the system relative to that observer. Any new observation ends the period of isolation of the system from the observer and so requires that a new relative state be defined taking into account the result of the most recent observation.

(Actually the “observer” of the system here doesn’t have to be a person or any other conscious entity. Any other physical system that it could interact with will do – with observations just corresponding to changes of the state of the observing system relative to any other “external” observer.)

It turns out that all observers who are isolated from the system during an experiment, and who start with the same information about the system, can use the same mathematical object to represent its relative state and for making predictions about the outcome; and this has led to the idea that the state is somehow completely independent of the observer – with various convoluted “interpretations” being added to “explain” what is “really” going on. But none of these adds anything in the way of useful predictions, and they all lead to various kinds of seeming paradox which get seriously multiplied if you mix different “interpretations” (as pointed out in Johann Holzel’s answer ).

Actually, if some friend, or other observers, (or just other physical systems) observe (or just interact with) the system before you do, then the states of the system relative to them “collapse” in the sense that after the observation (or other interaction) the probabilities of future observations are changed (with some becoming no longer possible and others more likely). But the state of the system relative to you does not collapse until you interact with it – either directly (eg by observing it yourself), or indirectly (eg by observing or communicating with your friend).

Usually it is quite hard to keep things isolated, and so just by being in the same room and sharing contact with the same air and ambient radiation you are effectively always interacting with your friend; so even without consciously learning what the friend has observed you have access to that information and so the collapse occurs for you too at the same time as for the friend. But if we were to keep the friend completely isolated in a pure quantum state (which is not possible for a real person, or even a cat, but might be possible for another microscopic system as the “observer”), then the combination of experimental system and “friend” could be in a pure state relative to you which remains uncollapsed until you actually learn the outcome (either by observing the system directly or by checking with your friend).

But as soon as we have been in contact with one another, the you that I see will agree with me about the experiment, and the me that you see will agree with you.

Source: (250) Alan Cooper’s answer to Quantum physics question If reality or superposition is fixed at observation or measurement, how can all people see something and all report the same thing? Would 1 person obseving something different cause their reality to branch off? – Quora