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Copenhagen interpretation of quantum mechanics

I have considered myself a champion of the Consistent Histories interpretation of quantum mechanics for almost 20 years - Roland Omnes' 1992 article just erased all my doubts about the statement that the foundations of quantum mechanics have been fully understood.

However, I have always realized that the "improvements" that this interpretation brings relatively to the Copenhagen interpretation are very subtle and it has been annoying to see that pretty much everyone misunderstands the basic points of the Copenhagen interpretation.



Bohr, Heisenberg, and Pauli. Am I the only one who thinks that Pauli looks like good soldier Švejk here?

There's a lot of misunderstandings being spread about the Copenhagen interpretation - and I would say that some of them should better be classified as deliberately propagated lies and propaganda. In this text, I would like to clarify some of them.




People in Copenhagen

First, let's ask which people are "fathers" of the Copenhagen interpretation. Well, the name indicates that they have something to do with the Danish capital. Clearly, we mean Niels Bohr - a natural leader of the group - and the people who worked with him and/or visited him in Copenhagen in the mid 1920s - especially Werner Heisenberg.

Max Born is of course a key co-author of the Copenhagen interpretation - after all, he supplied the probabilistic interpretation and received a Nobel prize for that. Wolfgang Pauli has also spent some time over there and he would also sign to the principles of the Copenhagen interpretation. The number of top physicists who may be considered parts of the Copenhagen school of thought is much larger - although some of them worked on topics that were further from the "foundations". Let me mention Lise Meitner and Carl Friedrich von Weizsäcker as two "nuclear" examples. I am also confident that people such as Paul Dirac would essentially subscribe to the Copenhagen interpretation.

There was no fundamental disagreement about the meaning of quantum mechanics among those people. Obviously, many other people such as Albert Einstein, Erwin Schrödinger, or Louis de Broglie didn't ever accept the Copenhagen interpretation but they didn't have any alternative. The Copenhagen interpretation works well and we don't need another hero.



Gabriela Gunčíková - Tina Turner: We Don't Need Another Hero. Czech Slovak Superstar II. Yes, all Czech girls and women between 17 and 71 years of age look just like her.

Subjectivity of the wave function

The first major confusion - or propagandistic distortion - is linked to the interpretation of the wave function. The Copenhagen folks were very carefully applying positivism. That means that they refused to talk about properties of physical systems that can't be measured unless some observations or consistency of the predictions make it necessary to talk about them - which is an attitude that became essential with the birth of quantum mechanics. In this sense, they uniformly rejected the assumption of "realism". If the observations are described by a framework that doesn't contain any "real and objective" things or properties before the measurement but if the measurement may be predicted, then it is how the things should be.

Lots of fringe stuff, garbage, and crackpottery was later written by various people who weren't really part of the Copenhagen school of thought but who found it convenient to abuse the famous brand. That's why one can also hear that the Copenhagen school may (or even must) interpret the wave function as a real wave that collapses much like a skyscraper when it's hit by an aircraft on 9/11.

But nothing like that has ever been a part of the Copenhagen school of thought. If you open any complete enough description of the Copenhagen interpretation or if you look at Bohr's or Heisenberg's own texts, you will invariably see something like the following six principles:

  1. A system is completely described by a wave function ψ, representing an observer's subjective knowledge of the system. (Heisenberg)
  2. The description of nature is essentially probabilistic, with the probability of an event related to the square of the amplitude of the wave function related to it. (The Born rule, after Max Born)
  3. It is not possible to know the value of all the properties of the system at the same time; those properties that are not known with precision must be described by probabilities. (Heisenberg's uncertainty principle)
  4. Matter exhibits a wave–particle duality. An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results, according to the complementarity principle of Niels Bohr.
  5. Measuring devices are essentially classical devices, and measure only classical properties such as position and momentum.
  6. The quantum mechanical description of large systems will closely approximate the classical description. (The correspondence principle of Bohr and Heisenberg)
Note that the very first point says that the wave function is a collection of numbers describing subjective knowledge. That doesn't mean that in practice, everything will be always subjective - or whatever the spiritual people have attributed to quantum mechanics. Of course that constant interactions between parts of the world - and different people - pretty much guarantee that they have to agree about many "objective properties". But as a matter of principle, this rule is important for quantum mechanics and Werner Heisenberg has never left any doubts that this is how one had to interpret it.

Heisenberg would often describe his interpretation of the wave function using a story about a guy who fled the city and we don't know where he is but when they tell us at the airport they saw him 10 minutes ago, our wave function describing his position immediately collapses to a smaller volume, and so on. This "collapse" may occur faster than light because no real object is "collapsing": it's just a state of our knowledge in our brain.

In practice, everyone can use pretty much the same wave function. But in principle, the wave function is subjective. If the observer A looks at a quantum system S in the lab, he will use a wave function where S has a well-defined sharp spin eigenstate as soon as the spin of S is measured by A. However, B who studies the whole system A+S confined in a lab won't "make" any collapse, and he evolves both S and A into linear superpositions until B measures the system. So A and B will have different wave functions during much of the experiment. It's consistent for B to imagine that A had seen a well-defined property of S before it was measured by B - but B won't increase his knowledge in any way by this assumption, so it is useless. If he applied this "collapsed" assumption to purely coherent quantum systems, he would obtain totally wrong predictions.

So the wave function is surely subjective if one wants to obtain a universal description of the world. It's a collection of probability amplitudes that may be combined in various ways, before the squared absolute values of the combinations are interpretated as probabilities. All probabilities of physically meaningful events may be calculated in this way - as the squared absolute value of some linear combination of the probability amplitudes.

No collapse in the principles of the interpretation

A widely propagated myth is that the Copenhagen interpretation is all about the "collapse". However, if you look at the six principles above, there is not even a glimpse of a comment about a "collapse" because it's not needed. The notion of an objective collapse was introduced by John von Neumann in 1932 and he was clearly not a part of the Copenhagen school of thought anymore. Comments that Heisenberg later switched to an "objective wave function" or an "objective collapse" are untrue, and even if these legends were true, these new opinions wouldn't be a part of the Copenhagen interpretation and, more importantly, they wouldn't be valid.

Because the wave function is subjective, see rule 1, everything that happens with the wave function has to be subjective as well.

Consider a cat. You will evolve a wave function and the final state is "0.6 alive + 0.8i dead." (The fact that the actual state is not pure in any useful sense will be discussed later.) When you observe the cat, it's - unexpectedly - alive. Once you know that the cat is alive, it becomes a fact. You have to use this new knowledge in all your subsequent predictions and retrodictions if they're supposed to be any accurate. Or valid, for that matter. I think that the previous statement is totally obvious.

However, people invent lots of nonsensical gibberish in order to obscure what's actually going on even though it is fundamentally very clear.

For example, you will hear all the time that it is so difficult to get a "collapse" and there must be some complicated gadget or mechanism that does so. But if you realize that those things are probability amplitudes rather than potatoes, you must see that absolutely nothing has to be supplemented to "get" the collapse.

The laws of physics predict that with the state above, there is a 36% probability that we will measure the cat to be alive and 64% probability that it is dead. Just to be sure, there is a 0% probability that there will be both an alive cat and a dead cat. The last sentence, while totally obvious, is once again being obscured and crippled pretty much by everyone who has every said that he sees a problem with the Copenhagen interpretation.

Once decoherence eliminates the off-diagonal elements, the density matrix for the cat is
rho = 0.36 |alive> <alive| + 0.64 |dead> <dead|
The diagonal entries of the density matrix, 0.36 and 0.64, are the probabilities that we will get either of the results. But the result "we will have both types of a cat" isn't among the options with nonzero probabilities at all, so it will certainly not occur. Only one of the options with the nonzero entries, "dead" or "alive", will occur, and the probabilities are 64% and 36%, respectively.

So one of them has to occur and the "symmetry" between them surely has to be broken. That's what the formulae imply. There is no possible answer of the form "half-dead, half-alive", so the latter result can't be observed. If you used another basis where a "half-dead, half-alive" state would be one of the basis vectors, the off-diagonal elements of the density matrix wouldn't ever go to zero and the interpretation of the diagonal elements of the density matrix as "probabilities" would be illegitimate.

In this sense, while the density matrix (also a subjective tool to predict and explain!) isn't an "observable", it's still true that you may compute its eigenvalues - the allowed probabilities of observable microstates - and the corresponding eigenstates of the density matrix are those that can actually be observed with well-defined probabilities. If a state is not an eigenstate of the density matrix, it's not possible to imagine that this state will be realized after a measurement. The Hamiltonian evolves the density matrix and dictates which states are "observable" in the classical sense.

So when you realize that the numbers 36% and 64% are not piles of potatoes but probabilities, it's very clear that once we learn that the cat is alive, even though the chance was just 36%, the number 64% has to be replaced by 0% while 36% jumps to 100%. We know that the cat is alive so for us, it's a fact. It's nonsensical to try to "preserve" the dead option because the dead option was not realized.

It makes no sense to claim that it's "predetermined" that the cat would be seen as alive. The free-will theorem, among other, morally equivalent results, shows that the actual decision whether the cat is seen alive or dead has to be made at the very point of the spacetime where the event (measurement) takes place; it can't be a functional of the data (any data) in the past light cone.

One more comment, linked to the recent discussions about Everett's "many words", is the following. People often say that is unfair that some terms disappear. And they say that the disappeared terms should exist in separate worlds, and all this crap. But it is the very meaning of probability that only one of the options occurs and the others just disappear once we know that they were not realized. The actual event we were trying to predict breaks any "democracy" between the different possible results that were "on equal footing" prior to the actual event.

Even more importantly, people often say that "rho = rho1 + rho2" decomposition of the density matrix means that there are "two worlds", one that is described by "rho1" AND one that is described by "rho2". (Similarly for "psi", but for "rho", the comments are more clear.) But this is a complete misunderstanding of what the density matrix and addition means. The density matrix is an operator encoding probabilities - its eigenvalues are predicted probabilities. And we're just adding probabilities, not potatoes.

What does it mean to add probabilities? It doesn't mean "AND" at all! If "P(A and B)" vanishes i.e. if A and B are mutually exclusive, then "P(A) + P(B) = P(A or B)". You see the "or" word over there? It's always "OR"! So whenever you ADD terms in the density matrix - and similarly the state vector - it always means "OR" rather than "AND"! So the existence of the two terms can in no way imply that there should exist both options or both worlds. It's complete rubbish. People who say that the two terms in the wave function imply that there have to exist two real objects somewhere in the "multiverse" are making an error that is as childish as confusing addition and multiplication. Literally. They're just confused about basic school-level maths. And you can't just confuse the words "and" and/or "or" because if you do, your whole framework for logic becomes flawed.

Let me rephrase a point of the "Lumo interpretation" clearly: you describe the system by a density matrix evolving according to the right equation and it is always legitimate to imagine that the world collapsed to an eigenstate of the density matrix and the probabilities of different eigenstates are given by the corresponding eigenvalues of the density matrix. Incidentally, this also works for pure states for which "rho = psi.psi". In that case, "psi" is the only eigenstate of "rho" with the eigenvalue of "1", so you may collapse into "psi" with 100% probability which leaves "rho" completely unchanged. ;-) The only illegitimate thing to imagine is that the world has collapsed into a state which is not an eigenstate of "rho".

Uncertainty principle

The third rule is the uncertainty principle. It means that generic pairs of properties, such as "x" and "p" or "J_x" and "J_z" - but pairs of projection operators describing various Yes/No properties of a system could be even better examples - can't have well-defined values at the same moment. Especially John von Neumann, before he began to say silly things, liked to emphasize that the nonzero commutators and the Heisenberg uncertainty principle is the actual main difference between classical physics and quantum physics. If the commutators were zero, the evolution of the density matrix would be equivalent to the evolution of the classical probabilistic distribution on the phase space.



Because the projection operators P and Q corresponding to two Yes/No questions about a physical system typically don't commute with one another, it is totally illegitimate to imagine that during its evolution, a quantum system had well-defined both Yes/No properties, P and Q. It just couldn't have had because P and Q don't commute and they don't usually have any common eigenvectors (unless there are eigenvectors of the commutator [P,Q] with the vanishing eigenvalue).

This main principle is the main "underlying reason" why the GHZM experiment or Hardy's experiment produce results that are totally incompatible with the classical or pre-classical reasoning. The classical reasoning is wrong, the quantum reasoning is right - and the nonzero commutators in the real world are the main reason why the classical reasoning can't agree with the observations.

Complementarity principle

Bohr's favorite principle shows that the systems exhibit both particle-like and wave-like properties, but the more clearly you can observe the latter, the more obscure has to be the latter, and vice versa. In some sense, this principle just follows from the uncertainty principle for "x" and "p" because particle-like behavior occurs for measurable states that are close to eigenstates of "x" while the wave-like properties are seen when the measurable states are close to eigenstates of "p" - with long enough wavelength (more precisely, long enough "any" features of the wave) so that the wave properties may be seen.

In some sense, the complementarity principle is totally uncontroversial - unless you are Shahriar Afshar, of course. ;-) So I won't spend more time with that.

Measurement devices are classical

The fifth rule says that the measurement devices follow the rules of classical physics. This is another source of misunderstandings.

The Copenhagen school surely didn't want to say that quantum mechanics couldn't be applied to large systems. Indeed, many people from the Copenhagen school were key researchers who helped to show that quantum mechanics works for large systems including molecules, bubble chambers, solids, and anything else you can think of.

Instead, this rule was meant as a phenomenological rule. If you measure something, you may assume that the apparatus behaves as a classical object. So in particular, you may assume that these classical objects - especially your brain, but you don't have to go up to your brain - won't ever evolve into unnatural superpositions of macroscopically distinct states.

Is that true? Is that a sign of a problem of the Copenhagen interpretation?

It is surely true. It's how the world works. However, one may also say that this was a point in which the Copenhagen interpretation was incomplete. They didn't quite understand decoherence - or at least, Bohr who probably "morally" understood what was going on failed in his attempts to comprehensibly and quantitatively describe what he "knew".

However, once we understand decoherence, we should view it as an explicit proof of this fifth principle of the Copenhagen interpretation. Decoherence shows that the states of macroscopic (or otherwise classical-like) objects whose probabilities are well-defined are exactly those that we could identify with the "classical states" - they're eigenstates of the density matrix. The corresponding eigenvalues - diagonal entries of the density matrix in the right basis - are the predicted probabilities.

Because the calculus of decoherence wasn't fully developed in the 1920s, the Copenhagen school couldn't have exactly identified the point at which the classical logic becomes applicable for large enough objects. However, they were saying that there is such a boundary at which the quantum subtleties may be forgotten for certain purposes and they were damn right. There is such a (fuzzy) boundary and we may calculate it with the decoherence calculus today. The loss of the information about the relative phase of the probability amplitudes between several basis vectors is the only new "thing" that occurs near the boundary.

Again, this point was the only principle of the Copenhagen interpretation that was arguably "incomplete" but their proposition was definitely right! To make this point complete, one didn't have to add anything new to quantum mechanics or distort it in any way. One only needs to make the right calculation of the evolution of the density matrix in a complicated setup. In that way, one proves that they always treated the measuring devices in the right way even though they couldn't fully formulate a fully quantum proof why it was the right way.

Correspondence principle

Both Bohr and Heisenberg also emphasized the correspondence principle, e.g. that the quantum equations reduce to the classical ones in the appropriate limit. If you study e.g. the evolution of the expectation value of "x" and "p", they will evolve according to the classical equations. It's the Ehrenfest theorem.

As discussed in the previous point, it is not enough to show that the world of classical perceptions will occur in the limit as well: we also need to know that the right "basis" will become relevant in the classical limit. Nevertheless, with the extra additions that had been demonstrated in recent decades, I mean decoherence, we know that it is true that the classical perception and choice of states does occur in the appropriate classical limit.

So all the points of the Copenhagen interpretation were right and they were only incomplete because they didn't explicitly calculate where the classical-quantum boundary really is. I need to emphasize that this boundary is fuzzy. And this boundary doesn't mean that quantum mechanical laws ever break down. They never break down. What's true is that for large enough systems, one may use - and should use - the approximate classical scheme (the word "approximate" sounds too scary but in reality, these approximations are super excellent for all practical and even most of the impractical purposes) of asking questions because one may show that it becomes legitimate.

Well, I surely don't expect that people will stop being hysterically angry about the Copenhagen interpretation and its alleged flaws - which don't exist. But at least, I would like to see the chronic Copenhagen haters to acknowledge that the Copenhagen interpretation says what it says and that it clearly doesn't have any demonstrable flaws. It has no internal inconsistencies and it is not in contradiction to any observation done as of today.

And that's the memo.

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reader Heikki said...

I had thought that in the Copenhagen interpretation any 'conscious' observer that makes a measurement causes the wave function to 'collapse', if we are allowed to use that word. In the Motl interpretation the only observer is myself. What happens to the world after I die and can no longer make observations?


reader Luboš Motl said...

Hi, first of all, if there's only one observer, it's me and not you. The latter is just a dull bound state of protons and electrons, get used to it. ;-)

Second and more seriously, quantum mechanics allows any observer, whether it's me, you, or someone who looks nothing like us, to calculate predictions using the very same maths. There is nothing anthropocentric or spiritual about it. So sorry to tell you but nothing special happens when you die.

Moreover, all the past results of well-defined observations are objective for all observers as soon as they become able to measure them i.e. see the results.

But of course, a conscious observer is needed if something is to be aware of and conscious about the events' confirming the laws of quantum mechanics.

It's wrong, wrong, wrong, that there is any "cause" of a "collapse". There is no collapse and your talking about collapse here shows that either you failed to have read my text, or you're incapable to learn basic things.

The "collapse" of the wave function is just the process in which it becomes known which outcome actually took place, so of course the probabilities have to be adjusted to 100% for this outcome and 0% for all others. This is no objective process which is why one doesn't have to - and one shouldn't - look for a "cause".

The fact that the probability (and probability amplitude) of mutually exclusive alternatives drop to zero once it's known that one "non-alternative" outcome took place is a tautology and tautologies are not "caused" by anyone. Tautologies are just true *regardless* of anyone.


reader Ulla said...

Lubos just realized the fact that selves has their own lightcones of reality and perceptions. So consciousness is different too, and the observer-effect. The collapse gives the information and is a result of the measurement. Information is not material, not bound to material patterns etc. So what is it?

The contenta is that he discovered the Zero Energy Ontology :) Maybe also the p-adic algebra :) Isn't that ironic?

Looks like he finally has began to THINK. I am sure he is good in the thinking process.

If he only stopped insulting people.

Ulla.


reader harting said...

My undergraduate quantum text was by Amit Goswami.

Chapter 24 was "The Unfinished Chapter: The Meaning and Interpretation of Quantum Mechanics
Appendix. The Delta Function"

I couldn't believe the spiritual BS he was pushing as science. My professor told me to ignore that chapter.

I later read other crap by Goswami claiming that "out of body" experiences and other "psychic" phenomenon could be proved by quantum mechanics.

I'm still pissed that my professor used that book when many better texts existed. I have had to use these other texts to really gain an understanding of the subject.


reader harting said...

My undergraduate Quantum text was by Amit Goswami. It had a chapter 24 about "interpretations".

I couldn't believe the spiritual crap he was trying to push as science.

Later I read other stuff by him like The Self-Aware Universe: How Consciousness Creates the Material World, which said idiotically unscientific things like

"...psychic phenomena, such as distant viewing and out-of-body experiences, are examples of the nonlocal operation of consciousness . . . . Quantum mechanics undergirds such a theory by providing crucial support for the case of nonlocality of consciousness.

Your explanation of the Copenhagen interpretation leaves out all of the crap that others have tried to inject.

As for particle wave duality have you read New Foundations of Quantum Mechanics by Alfred Lande?


reader e said...

Is it even useful to introduce the phrases "observer" and "measurement"? Is the situation more that we have system A and system B, and "prior" to "collapse" of B's wave function from A's perspective the information exchange from B to A is such that the evolution of A's state is independent of any fixed state of B, and that "collapse" simply indicates that the information exchange is such that A's future evolution takes a path the details of which differentiate between previously indiscriminable states of B?

Even if that's totally off base, introducing "observerhood" and "measurementness" seems just to complicate a straightforward account of the physical interactions between entities.

I guess I see the "observer" and "measurement" concepts as way too anthropocentric to begin with.


reader Luboš Motl said...

Dear e, I also have doubts about the overall usefulness of the concepts such as "observer" etc. It's related to the following points:

Dear Harting, many people inject lots of spiritualism in these discussions - and that's "both sides". Those who love supernatural nonsense claim that quantum mechanics supports it because what they really care about is to support their superstitions. That's wrong, of course.

Those who are most concerned with the fights with the previous groups are afraid that quantum mechanics or any piece of science would use words that look similar to the previous group, so they hysterically react if they hear that something is subjective and so on.

Of course, both groups are just wrong. Quantum mechanics doesn't allow any truly supernatural phenomena to occur. What the "subjective" nature of wave function means that wave function is really a gadget that collects all the information about the system that may be known to an observer, and that may be used to predict the future.

The only subjective thing about it is that at different points of space and time, i.e. at locations of different observers, different things about the past are known and used as the input - initial wave function - while the rest is to be calculated probabilistically.

The subjectivity just means that the "collapse" is nothing else than the fact that some previously unknown, speculated about outcomes become known, and for different observers, that occurs at different moments, so there's no paradox in the fact that different observers see the probability waves "collapse" in different order and differently. Of course, when the history contains a large collection of decohered outcomes, all observers may ultimately reconstruct the same macroscopic past. It's just the "intermediate state of affairs" before the individual events that are "fuzzy" - in Feynman's approach, one has to sum over all histories how to get from A to B so everything that happens between A and B are just "intermediate results" that don't have any objective properties.


reader TLP said...

@Luboš, for extra visibility and clarification, you should make a YouTube video explaining the double-slit experiment, which is what most people associate with "quantum strangeness".


reader الرجل ذبح بعضهم البعض ولكن الخيول باهظة الثمن said...

if there's only one observer, it's God and God it's me?

The subjectivity just means that the "collapse" is nothing else than the fact that some previously unknown, speculated about outcomes become known, and for different observers, that occurs at different moments, so there's no paradox in the fact that different observers see the probability waves "collapse"
in different order and differently....replication


Of course, when the history contains a large collection of decohered outcomes, all observers may ultimately reconstruct the same macroscopic past.

It's just the "intermediate state of affairs" before the individual events that are "fuzzy" - in Feynman's approach, one has to sum over all histories how to get from A to B so everything that happens between A and B are just "intermediate results" that don't have any objective properties....

fuzzy indeed


reader Poeteye said...

COPENHAGEN INTERPRETATION
Values we define
by measurement cause collapse
of the probable.

COPENHAGEN LIMITATION
Values we define
by capacity cause collapse
of what’s possible.

COPENHAGEN EXTRAPOLATION
Evaluation
by expectation causes
potential collapse.