Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.
So — what do you want to know? What are the really interesting questions about QM that we should be talking about?
One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.
Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?
(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)
Comments
165 responses to “Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask”
The inner product in 3-d is simply the dot product we know so well from freshman calculus with its simple geometric meaning. QM is in 3-d. The Born interpretation is simply that the complex conjugate of psi is a 3-d probability density function, no different than the pdf’s used in probability. What could be simpler? We have to pick solutions to the Schrödinger equation that are normalizable, drop off as e^-x. But then atoms do not have surfaces, they extend to infinity in all directions. Sean, my question about QM that I was never afraid to ask, is how can atoms not have surfaces when all macroscopic systems do have surfaces?
The arena: the discrete, the dynamic, + and -, space and time.
James,
To make a brief observation about time, the question is whether it is the basis of motion, or a consequence of it. If physical reality moves along a dimension of time, then it proceeds from past to future, but if time is being produced by motion, than one event is replaced by the next and the linear process of time is going from future potential to past circumstance.
My opinion is that distance and volume are features of the vacuum, while time and temperature are effects of fluctuation.
Celestial mechanician,
TimG didn’t mean that some differential equations don’t have solutions, he meant that they don’t have solutions that can be written down as a nice simple formula – that’s what’s meant by the term “analytic solution”. Although I’m not sure how “analytic solution” is defined in a precise way.
Hi,
Mainly to Lawrence B. Crowell,
Apologies – I should not have mentioned WD as I am not an expert “a little knowlege is a dangerous thing” but it occurred to me when typing and it was too hard to resist – but still I think it is odd.
I did not think of the QM parallel at the time i.e. E/t (despite its somewhat delicate nature – you have probably seen the John Baez discussion (too lazy to find link)) this is never taught at uni (well, not in the lectures I went to) and seems to just be left be taken for granted by people (A similar complaint might be levelled at E=mc^2 which many people seem to think is a consequence of Lorentzian-relativity without any conservation laws)
Still, I think QM has to be re-interpreted – not with math(s – UK!) necessarily but with something that allows intuition (I’m no philosopher but I’m tempted to call that “understanding”) to explain how you can solve the “unanswered question” while taking a walk in “Central Park in the dark”.
-James
Diocletian on Jul 9th, 2008 at 8:00 pm
Please discuss the paradox that arises when one tries to attribute expansion of the universe to the energy of quantum vacuum fluctuations. The fact that a discrepancy of 10^120 is found … physicists have expended a great effort in trying to reconcile this argument. What am I missing here? Why should I insist that vacuum energy causes the universe to accelerate?
————–
The equation between the vacuum energy density and the cosmological constant (CC), or negative pressure assigned as responsible for the expansion of the universe, is a model. It does not necessarily hold as a fundamental theory in my opinion, but as some effective theory or approximation. There are a number of things which bring questions to this whole enterprise. I have sat through talks where people on slides have pictures of virtual loops and the rest. The one problem is that these things are an artifact of quantization. These objects which couple to real particles (on shell etc) are really pertubative terms that are summed over. Just as in elementary calculus a summation variable is a dummy variable, in many ways these terms are much the same. Yet there is a tendency to see these things as “real.”
These quantum loops in space are the result of quantization. Yet as an exercise one can with the classical variables add the negative of the commutator which happens in quantization to get the a^*a + 1/2. There is nothing wrong with this since classically all you are adding is zero. If you then quantize the ZPE term is removed. This is one reason so called normal ordering works. The whole ZPE term can be removed.
The Casimir effect might be seen as an interaction of plates with the vacuum. It could also be seen as a quantum response or nonlocal effect due to the presence of the plates in some proximity with respect to each other.
Assigning the cosmological constant to the vacuum energy is a convenient thing to do. It is tempting to say that there exists a symmetry between positive and negative energy terms that is slightly broken to give the CC we observe that is small. The problem is that all of these ideas are increasingly to my mind fundamentally wrong. My reason for saying this is that the cosmological constant is the result of pure spacetime curvature. It is a curvature term R_{ab} = /g_{ab} and defines an Einstein space. We can of course cast about and say that / = 8piG(e + 3p) and assume the CC is determined by a “fluid” that has certain properties. Yet this muddies up the picture of general relativity.
There is a more basic reason why these ideas may diverge from reality. A cosmology is a Petrov type-O solution that has no Killing vector system. In effect cosmologies in their lack of such isometry permits the creation of momentum and energy — aka the big bang. We have been through this some on this list, with how cosmologies fail to admit a global conservation of energy. Assume that the spacetime of the universe has a source by the Einstein field equation
R_{ab} – (1/2)Rg_{ab} = 8pi G,
here I have written this in a semi-classical form, then the symmetries of the spacetime (the left hand stuff) is given by the symmetries of the source by its momentum energy tensor. If we assume that the source is given by a gauge field then
T_{ab} = F_{ac}F^c_b – (1/4)g_{ab}F^{cd}F_{cd},
has a set of symmetries which determine the symmetry of the spacetime. This is a very difficult sort of analysis to enter into, but the punchline is that the spacetime symmetries are type-N. Due to the nature gauge symmetries if they define a large enough of a source (say a vacuum state etc) then they couple to spacetime in the form of gravity waves or related structures. You can’t build cosmologies from this source, not without adulterating general relativity in some subtle ways.
It is for this reason and others that I have had growing doubts about these types of models. In spite of having upheld these ideas in years past I have called these models into question — in spite of their population growth on the arXiv like bunnies. In order to do these things right we have to be honest about what we are talking about, and on a fundamental level these models appear to have some inconsistencies.
Lawrence B. Crowell
LC said…and this paragraph is so conceptually central to cosmology it is worth repeating and studying carefully…
“Assigning the cosmological constant to the vacuum energy is a convenient thing to do. It is tempting to say that there exists a symmetry between positive and negative energy terms that is slightly broken to give the CC we observe that is small. The problem is that all of these ideas are increasingly to my mind fundamentally wrong. My reason for saying this is that the cosmological constant is the result of pure spacetime curvature. It is a curvature term R_{ab} = /g_{ab} and defines an Einstein space. We can of course cast about and say that / = 8piG(e + 3p) and assume the CC is determined by a “fluid” that has certain properties. Yet this muddies up the picture of general relativity.”
To repeat the critical point one final time…LC points out…
“My reason for saying this is that the cosmological constant is the result of pure spacetime curvature”….
Hello all,
About the Schrödinger equation, I also had much difficulty to understand it. I didn’t find any textbook that could explain it intuitively. It took me tens of times re-reading Feynmans presentation on how states change with time or due to external forces before coming to the conclusion that it is a common sense law for any arrow-like object. Feynman taught us that all we do in QM is drawing little arrows on a piece of paper… that’s all. The differential of a 3-D arrow is always perpendicular to the arrow itself and the proportionality factor is the differential of the angle. If the arrow represents |psi>, then it’s common sense to have:
d|psi>=i.d(angle).|psi>
which is a generalized form of the Schrodinger equation. I present this at my blog and youtube channel.
I think that if you’re looking to explain quantum mechanics to the masses, an explanation of just what a wave function means would be a useful precursor to talking about “the wave function of the universe”. Having recently TAd introductory quantum mechanics (i.e. end of a freshman sequence), it seems that interpreting the wave function is a huge stumbling block for a lot of people. If you have a good way of explaining it to the laiety in ten minutes, it might be the most worthwhile thing you could do in that context.
Eric said “I think that if you’re looking to explain quantum mechanics to the masses, an explanation of just what a wave function means would be a useful precursor to talking about “the wave function of the universe”.”
Yes, that’s a good precursor, I’ll going to explain the wave function in one of my next sequences. But I needed first to explain the ket (arrow) and operations on kets (arrows). The wavefunction may than be seen as the projection of the ket on the base kets. Feynman explained that very well. That projection is an ondulatory function when you vary time or position (or any other observable).
Lawrence, (Sam,)
Since the rate of expansion, with the added dark energy assumption, seems to mirror a cosmological constant, if it is pure curvature, what does this do for the Big Bang assumption that redshift is due to recessional velocity?
The gravitational curvature causes lensed sources of light to shift apparent location, but it is an optical effect. So is redshift an optical effect?
Just one final gripe…
I have always disliked the (Heisenberg) uncertainty principles (to some extant this refers back to earlier comments from Lawrence B. Crowell where the energy/time relation one was alluded to).
These “principles” often seem to be taken as somehow basic in QM. However, as far as I can see (and please correct me since this is the reason for sending this in the first place) I thought that all the uncertainty relations say are:
1. The Universe is quantum mechanical
2. It is NOT classical mechanical
3. If you will insist on trying to force a classical picture on the universe due to deficiencies in your cognition resulting from evolution in a non-QM (or non-GR for that matter) environment, then, well, OK, – this will be the best that you can do… Here are the (uncertainty limits) that are possible.
4. This doesn’t strike me as fundamental to QM – perhaps a bit too human
5. Of course, this relates to lack of any understanding of the measurement problem.
-James
@Celestial mechanician
In celestial mechanics there is no simple analytical solution to the many body problem for the differential equations of three or more massive objects. Approximate solutions have to be obtained.
In quantum mechanics the same many body problem applies to systems of particles such as multielectron atoms and molecules. Only two body systems such as the hydrogen atom and the He+ ion have analytic solutions. Considerable effort has been expended in attempting to get ever more accurate solutions for the electonic structure of atoms and molecules over the past four decades. Of particular importance here is the application of the variational principle to the QM of such systems.
If you want to get a good overview of the approaches taken here I would recommend Szabo and Ostlund’s book Modern Quantum Chemistry: Introduction to Advanced Electronic Structure.
Hi John,
Lets go back to Einsteins original work. He found that in his 4D model of GR, cosmologically there was a left-over tendency toward expansion, which with GR is ridiculous, because by definition the GR universe is conceptually everywhere (quasi-static) and there is nothing outside it to expand into.
The CC was introduced to keep the concept logical…to insure the math made sense, NOT just to “make the universe static”.
This fact, that the CC is not really a vestigial artifact, but an essential element of the concept has been kind of “glossed over” or just plain ignored by many.
LC’s point is that gravity and the CC are related conceptually. The CC is not some material ingredient of the universe, it is a characteristic of the structure of a GR universe.
The “force” which causes the universe to (seem to) accelerate outward is is a result of the way we observe the structure of the universe.
The universe overall is not going anywhere, certainly not by the GR definition…we just observe it to seem to expand and accelerate outward from our frame of reference in 4D, which also by the GR defintion is “real”…to us as observers.
We are familiar with the basics, the concept of free fall, space-time curvature, invariant frames and so forth, and understand the possibility of an extra three-space in the structure. The universe (according to Einstein, at least) just “is”.
Best Wishes, Sam
@Sean
May the landscape problem be related to the manywords interpretation of QM?
Sam,
One of the big problems with understanding physics in the first place is that there are any number of versions. When I first heard of the CC, say early ’80’s, it was described something Einstein added because according to his calculation, gravity would cause the universe to collapse to a point, so he added it to maintain a static universe. Since then, I’ve heard a fair number of other descriptions. That he did reject it upon learning of the redshift didn’t correspond to my original understanding, as the redshift would seem to be reasonable evidence of something balancing the contraction of gravity, so it would seem to be evidence of a cosmological constant! Then again that the most recent measurements of the rate of expansion attributed to dark energy have been said to match a CC would seem to imply this original understanding was accurate.
So, yes, I do see gravity and a CC as intimately related. So far as I can understand, while gravity is the contraction of space, the CC does seems to be a corresponding expansion of space in those areas not dominated by gravity. That’s pretty much been my point all along; How, if space is expanding between gravitational wells, at approximately the same rate it is collapsing into these wells, how is it that the universe as a whole is expanding? These two effects are happening simultaneously, not sequentially, so the idea that the universe is expanding and then collapse into a Big Crunch doesn’t make any sense. To use your(I think it was yours.) analogy of the merry-go-round, it’s like saying that because the horses close to us are moving to the left, the entire merry-go-round moves to the left, but since those further away are moving to the right, eventually it might move back to the right, if it doesn’t move too far to the left(fadeout scenario).
That’s why I describe it as a convective cycle of expanding energy and collapsing mass.
I agree it does seem to expand outward from our frame of reference, but our only measure is of light that has managed to traverse the most distances and thus the paths least blocked by gravitational obstacles, therefore the space that is expanding the most. The effect of this expansion is eventually absorbed by those gravity wells, including the telescopes collecting the light we measure.
The cosmological constant came about from the continuity equation on the momentum-energy tensor
$latex
nabla_a T^{ab}~=~0,
$
where the differential is covariant. An integration of this recovers the momentum-energy tensor plus an ingration constant
$latex
frac{8pi G}{c^4}T^{ab}~+~Lambda g^{ab}~=~R^{ab}~-~frac{1}{2}Rg^{ab},
$
where the second term on the left hand side is the integration constant. For the observable universe the density of matter and radiation is very small and so the momentum-energy tensor can approximately be set to zero. The result is an equation that says, “Ricci = constant x metric.” This suggests that the structure of the universe is entirely due to spacetime (classical) vacuum physics with no sources.
Historically Einstein included this integration constant in order to model a static universe. This was a sort of bias that existed then, and to keep the universe from collapsing in this constant was inserted. Hubble’s results on redshift called this into question and the term was abandoned as Einstein’s greatest blunder. Then in 1997 the SNI results by Perlmutter caused the term to be reconsidered.
Many models regard the cosmological constant / as due to the quantum vacuum, or the vev. This certainly sounds reasonable. The problem is two fold in my opinion. The GR problem is that the vacuum fields will result in a spacetime of type N, due to some grand summation over all these virtual waves etc. It is hard to know how one gets a solution with no Killing vector structure, cosmologies of type O, built up from type N solutions with a 4-fold null direction or Killing direction. The other problem frankly are the idea of all these virtual waves as a source of anything. The argument is similar to the Casimir result. Yet this result is usually taken as due to a difference in vacua — that between the plates and that outside. With a cosmology there is only one space or spacetime.
Now, it might be argued that since cosmologies have no Killing structure that globally we can’t define a constant vev. As such the vacuum here might be different from the vacuum “there.” If so there might be a difference in vacua structure globally across the universe. These different vacua are likely to be unitarily inequivalent, similar to what happens with Hawking radiation and the Unruh effect. Remember, it is a difference in vacuum energies in separated regions that can give rise to dynamics!
However, this leads to some rude questions. This physics is constructed by a difference between Killing fields. Wald spells this out in his little book on spacetime thermodynamics and BH QM, where the Unruh effect is due to a deviation between the coordinate time and a Killing time. But with cosmologies we have no Killing symmetries to work with! But then again maybe we do have one thread to grab onto, the anisotropy of the universe. If we were to consider some spacetime with Killing vector fields K_a we might consider some series
$latex
K_a~=~xi_a~+~sum_pC^p_{abcd}[xi_p^b,~xi_p^c]xi_p^d~+~O(xi^3).
$
For a type O solution we will have
$latex
xi_a~=~0.
$
I write the first order term with a “C,” for I suspect this would be related in some ways to the Weyl tensor. The index “p” means a summation over the other Petrov-Pirani types. This means a cosmology is being perturbed by a series term built up from other solution types, type D for black holes, N for gravity waves and I, II, III types etc. This of course could be “jazzed up” to include in the first order some harmonic sum of C’s and so forth which can be made to fit with the WMAP data. The anisotropy could then be modelled this way. This then might provide the Killing structure required to understand how the universe might have different vacua, and how these perturbation terms are related to the quantum vacuum.
Lawrence B. Crowell
erratum,
I just realised I make a typo on the order of indices in the perturbation series with the “C.” It is easy to make the correction.
L C.
LC, Awesome and very interesting…a very nice, brief and understandable review and technical evaluation. I appreciated your look at the anisotrophy of the universe. Your approach makes possible a mathematical treatment of a subject which may cosmologically be quite a bit more complex overall, with smooth cosmic variations rather than sets of contant ones. Computer analysis and modeling might be possible?
John, You touch on the “tightrope” aspect… that the universe “rides the line” between a tendency to either collape or expand. (Astronomically we observe the universe to expand and accelerate outward). The merry-go-round analogy in which the different parts of the mery-go-round are observed to be heading in different directions at the same time, when in fact the overall structure stays in the same place (in GR, everywhere, all the time) is appropriate.
Sean was discussing the single Wave Function of one Electron, two Electrons- and the universe. I am reminded of possible matter/antimatter occillations at the quark level of scale, in the Planck Realm, everywhere, all the time….that all order, information, complexity and observation in the universe is everywhere and all the time (eternally) entangled.
I think the not so subtle reality that at the farthest extent of our (theoretical)astronomical observations, 13.7BLY distant, the universe is everywhere geometrically singular (white hole…big bang) at 360 degrees is important to keep in mind. Any observer, anywhere in the universe has this common horizon of existence and experience.
Very interesting!
Sam Cox:
Computer analysis and modeling might be possible?
—————
This approach, if it really works, might allow this problem to be converted into a perturbation problem with Lie algebras. This is similar to what is done with astro-mechanics, where pertubation theory is worked according to symmetries in a problem. Jacobi found that the gravitational three body problem has some remarkable symmetries which permit it to be reduced to four integrals of motion. This removes some of the mystery of the problem. Such techniques have been used in other perturbative problems. If this can extend to cosmology then the problem can be considerably simplified.
Before I continue, the O(xi^3) should be O(xi^5).
If this works then the perturbative Killing fields are on comoving frames, and the difference between them from region to region might then lead to inequivalent vacua from which some of these types of analyses might be performed. This might be particularly important in the early universe. Rather than the cosmological constant being of the form
$latex
Lambda~=~Lambda(rho),
$
it would be of the form
$latex
Lambda~=~Lambda(deltarho)
$
where this variation is across two overlapping regions. This variation would be computed according to differences between Killing vectors across these regions. There are clearly some scaling or conformal issues here, but … .
Yet if the CC depends upon small variations in the vev this might at least ameliorate some of these 10^{123} orders of magnitude problems.
There is one problem with this — it is wrong on a most fundamental basis. For various reason I think this can only be at best a better approximation or effective theory.
Lawrence B. Crowell
Lawrence,
Wouldn’t this raise the question of whether the universe actually is static?
Sam,
I take it to mean that at 13.7 billion light years, everything appears to be receding at the speed of light? So if there were anything radiating from beyond this distance, we would only record black body radiation and not be able to pinpoint the source. For me, a horizon line effect remains a reasonable possibility, especially in light of such proposals as Inflation that are required to make the BB work.
John Merryman: Wouldn’t this raise the question of whether the universe actually is static?
—————–
Remember, the CC came about as an integration constant. As such one can set this to what ever you want, ignoring data and … . With the so called eternal inflation the / is set so that the equation of state w = -1 holds and the CC pushes the universe out. If cosmology has demonstrated anything it is that the universe is not static.
Lawrence B. Crowell
John Merryman: I take it to mean that at 13.7 billion light years, everything appears to be receding at the speed of light? So if there were anything radiating from beyond this distance, we would only record black body radiation and not be able to pinpoint the source.
————–
This ain’t necessarily so. The spatial manifold is itself stretching out, so the light cones at distant points are pointed in different directions. As a result the universe up to the CMB opaque boundary is about 80 billion light years out. This sounds funny, since this is further out than the time for the universe.
Google Ned Wright, he has a good cosmology site which gives some really good pictures and explanations of these things.
Lawrence B. Crowell
“There is one problem with this — it is wrong on a most fundamental basis. For various reason I think this can only be at best a better approximation or effective theory…”
Lawrence B. Crowell
Yes, but you may be on the right track. After all, the universe IS anisotropic. We don’t have to look through a telescope or use WMAP to see that! The very existence of order, energy density information and complexity in the cosmos belies anisotrophy.
Does the Planck Realm exist at the same level of local scale in the center of the Sun as it does on the surface of the Earth? Is any place in the universe so remote as to have no Planck Realm at the local scale? Black holes ARE the Planck Realm, and the space nearby them has a measured Planck Realm much higher in local scale than, say here on the surface of the Earth.
The Planck realm, however, regardless of its observed location in local scale, is the same thing cosmologically and is universally quantum entangled, just as the photonic realm is quantum entangled. Whether we observe photons or the singular is but a matter of the direction we observe within the manifold, and the “size” and “mass” of the Planck Realm, but a matter of our relative observing location in the manifold…our coordinates.
John,
Of course you are correct. The fact that galaxies close to the big bang are observed and measured to be accelerating and receding at almost the speed of light, however relates to the fact that at those coordinates, near the singular realm (with respect to us), light experiences the effects of gravitational time dilation and red-shifts. The closer we observe things to be, relative to the big bang, the more rapidly they accelerate. In spite of the time gravitational time dilation involved, the acceleration outward can (obviously) still be measured.
As we have discussed, all the implications of these phenomena and the geometry for the universe which they imply, are profound, and relate directly to the quantum nature of reality as espressed within a two-sphere GR manifold, with energy densities as observed from our frame, behaving in an SR manner- according to the grand proportion.
I’m sorry for the very much and quite over-simplified perspective of a retired high school math and physics and college research paper writing teacher! I really appreciate Lawrence as a teacher. It is a privilege to have these opportunities to share ideas with him and you as well, John. I’m looking forward to the upcoming discussion of Quantum Mechanics.
Sam Cox
Lawrence,
This would be implicit to the concept of a horizon line, as it is a function of perspective.
If the CC is pure spacetime curvature, then the further light travels, the more this effect is compounded as it crosses more space, so that distant sources appear to recede at a greater rate than closer sources, but if we were closer to them, they wouldn’t appear to be traveling faster, as it is only because those distant sources are at the edge of what is our horizon line that they do appear to be traveling away from us at the speed of light. This makes much more sense as an lensing effect than actual recessional velocity. Since it would be a very gradual effect that only applies outside the gravitational field of the galaxy and across enormous distances, it would seem at least worth considering.
“This ain’t necessarily so. The spatial manifold is itself stretching out, so the light cones at distant points are pointed in different directions. As a result the universe up to the CMB opaque boundary is about 80 billion light years out. This sounds funny, since this is further out than the time for the universe.”
I have a profound respect for Ned Wright, and have carefully studied his tutorial. Of course he has carefully thought out conceptual reasons for making the above statement. However 13.7 billion Earth years ago at the “Big Bang” everything in the universe was singular. To state that anything in the universe can, at any time, be farther from anything else than the age of the universe itself requires considerable explanation. If folks in a distant galaxy 10 BLY out measured the distance to us, they would get the same result we do, when we measure in their direction.
Yes, space has expanded since the “Big Bang”…of course, to the extent that a tiny quantum entity now has an observed radius of 13.7 Billion light years.
I like the way Lawrence takes an idea, works it carefully and then, after all that work, disgards the idea as conceptually inconsistent, or lacking in some way. He does not feel his reputation as a scientist and teacher is in any way encumbered by engaging in the scientific process. He and Ned Wright… in fact John and Sam are searching. Our search may be mathematical and emperical. Our search may be conceptual and of course our search may be philosophical.
However, in science we expect to make mistakes. I told my students I would give them exta credit if they followed their instructor carefully and were able to point out errors. I gave a lot of extra credit! However, the students also paid closer attention to what was going on, and most of all, I taught them the spirit in which scientific investigation should proceed.