Category: Quora Answers

“Basis Vector” Confusion

A Quora question asks: “The wave function is contained in a Hilbert space while its basis vectors aren’t because plane waves are not square-integrable functions. Is this true for all Hilbert spaces or only for the square-integrable sub-space?”

My response: There are a number of ways you are misusing the language here, and I thought at first that your main misunderstanding may be to think that an element of a Hilbert space has its own set of basis vectors – while in quantum theories the choice of a relevant basis is more often related to an observable than to a state. But perhaps you are not associating the “basis” with a particular wave function, and rather thinking of it as associated with the position space representation as a whole. That makes more sense so let’s go back and try to describe the situation more clearly.

A wave function is a representation of an element of the Hilbert space of quantum states by a square integrable function (but the space of such functions is actually isomorphic to the entire state space as a whole and not just a proper sub-space). There are many such representations, sometimes with associated ways of identifying bases and sometimes not. In particular for any observable with purely discrete spectrum (such as the Hamiltonian of a harmonic oscillator) there is a basis of eigenvectors, and every state is represented by a square summable sequence corresponding to its spectral decomposition. Unfortunately the position and momentum observables have pure continuous spectrum and no eigenvectors, so the corresponding representations involve elements from some larger space. The usual “position space” wave function corresponds to the spectral decomposition of the position operator, and the analogue of basis vectors are actually not functions at all but rather distributions (in particular delta functions). The plane waves on the other hand are (position-space representations of) eigenfunctions of the momentum operator (but again not eigenvectors since not in the Hilbert space).

So in the end I might answer your question by saying that there is only one Hilbert space of states but that any complete set of observables can be used to represent it in terms of square integrable functions (or sequences); and that it is only in the case of pure point spectrum that the resulting spectral decomposition can be described in terms of an actual basis, while observables with continuous spectrum generally require some kind of generalized basis involving functions or distributions that do not actually correspond to normalizable states.

Source: (1000) Alan Cooper’s answer to The wave function is contained in a Hilbert space while its basis vectors aren’t because plane waves are not square-integrable functions. Is this true for all Hilbert spaces or only for the square-integrable sub-space? – Quora

Virtual Particles and Conservation of Energy

Virtual particles are never observed directly, so (subject to the limits of experimental error) we don’t actually ever see any violation of conservation of energy.

What virtual particles are is just a part of one particular method for calculating probabilities of events that we do see; but this is not even the same as saying we observe them indirectly, as the various possibilities with different numbers of such particles all contribute to the overall calculation – with no specific numbers ever being required to actually exist.

The use of virtual particles is analogous to Feynman’s path integral approach to quantum mechanics where, as an alternative to solving the Schrodinger equation by traditional methods, Feynman noted that the probability amplitudes predicted from it for going from one event to another could also be calculated by adding up contributions from all conceivable paths between the two events (including unphysical ones). But neither the unphysical paths nor the unphysical particle number histories need to be considered as anything that actually happens.

Another point that is often made in answers to this question is that the contributions from paths or particle histories that violate conservation of energy are inversely proportional to the time durations of those violations in a way that is consistent with Heisenberg’s uncertainty principle \Delta E \Delta t < \frac{h}{4\pi}. But I am not sure how much this helps – other than to explain how (as pointed out in yet a third set of answers) “laws” of physics are not absolute but just expressions of the limits of what, according to current theories, we expect to see – and indeed conservation of energy can appear to be violated if we try to measure things too quickly (though the “violation” can be interpreted as just due to our inability to measure both energy and time with sufficient simultaneous accuracy).

Source: (1000) Alan Cooper’s answer to I’m a 17-year-old boy from Turkey. I recently read your article about virtual particles and conversation of energy. Could you explain why these particles don’t break the this law scientifically? – Quora

Waves vs Particles

A Quora question asks: Can we measure wave properties of particles or is my contention that ‘waves travel but particles are detected’ correct?

You are right that the actual values of the quantum “wave function” are not generally observable and that the things we can actually measure are usually more like properties of particles.

But there are some ways of getting partial information about the wave function itself. For example chemistry and molecular structure gives us a way of learning about the squared magnitude of the wave function when it is a “standing wave” and a scanning tunneling microscope even gives us a more direct picture of that. And the relationships between phases at different points sometimes lead to observable effects in solid state theory and in the Aharonov-Bohm effect.

Source: (1000) Alan Cooper’s answer to Can we measure wave properties of particles or is my contention that ‘waves travel but particles are detected’ correct? – Quora

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