Nov 16 2009

Chopra Mangles Quantum Mechanics – Again

Deepak Chopra has made a career out of misunderstanding quantum mechanics (QM) – and through his popularity, confusing the public. Like many others, he has found a superficial way in which to interpret quantum mechanics to make is seem as if it is congruent with Eastern metaphysics.

And now he has done it again, in that anti-science rag the Huffington Post. Chopra goes beyond the typical New Age distortion of QM, which is basically the claim that QM is really weird, therefore magic is real. Chopra assumes some very specific, and common, misinterpretations of QM. He writes:

Quantum physics tells us that objects exist in a suspended physical state until observed, when they collapse to just one outcome — we don’t know what happens until we investigate, and our investigation influences that reality. Whether or not certain events may have happened some time ago, may not actually be determined until some time in your future — it may actually be contingent upon actions that have not yet taken place.

Chopra is referring to the wave-particle duality of matter, quantum entanglement, and the uncertainty principle – but he gets them profoundly wrong. First he makes the common mistake of interpreting the collapse of the wave function as being dependent on an observer, which is false. QM states that light, electrons, and all fundamental particles exist not as  discrete point particles, but spread out like a wave. We can only describe the probability that they will be in a specific place at any moment, and that probability is the wave function. Particles, when free from interactions with other matter, actually behave like waves (see the double slit experiments).

But when a particle (whether of photon of light or an electron) interacts with other stuff it is no longer spread out but collapses down to a point particle. This is the wave-particle duality of matter. The collapse to a particle, however, is not dependent on any observer – just interaction with other stuff. No observer is necessary. When a photon from the sun strikes the earth and its energy is absorbed by a leaf on a tree in the middle of the jungle, it collapses to a particle. The same is true when it strikes a dead rocky asteroid out in space. Consciousness, and even life, is not necessary.

Next Chopra mangles quantum entanglement:

Scientists in France shot particles of light “photons” into a measuring apparatus, and showed that what they did — now, in the present — could retroactively change something that had already happened in the past. As the photons passed a fork in the apparatus, they had to decide whether to behave like particles or waves when they hit a beam splitter. Later on — well after the photons passed the fork — the experimenter could randomly switch a second beam splitter on and off electronically. It turns out that what the observer decided at that point, determined what the particle actually did at the fork in the past.

Yes, these experiments are fascinating. But they do not describe the future affecting the past, as Chopra misinterprets. They describe what is known as quantum entanglement. When particles are paired or linked in some way – for example when they are created by the same process – some of their properties are linked, even while still being in a wave of probability. For example, if one particle is spin up, the other will be spin down, even though their spin is not determined until much later, and even if the particles are separated by millions of light years at the time of the collapse of the wave function.

Physicists do not pretend to understand the fundamental nature of quantum entanglement – that is a Nobel prize waiting to be won. But it does not represent the future affecting the past. Nor does it represent faster than light, or instantaneous communication. Experiments have been done showing that it is impossible to transmit information faster than light using quantum entanglement. Information is not going faster than light, or into the past.

Chopra is using a common trick of the pseudoscientist – exploit cutting edge science, which the public is not likely to understand, and pretend as if there is proof where there is uncertainty. Take some interesting experiments, then leap way ahead to conclusions that serve their metaphysical purposes, but which are not settled science.

In short – beware of anyone pretending to understand the ultimate implications of QM and that it supports their far out philosophy.

And here is Chopra’s woo philosophy:

It was only with the advent of quantum physics that scientists began to consider again the old question of the possibility of comprehending the world as a form of mind.

Indeed, the quantum theory implies that consciousness must exist, and that the content of the mind is the ultimate reality

The universe is a mind, and consciousness is the ultimate reality. Not surprisingly, a very Eastern philosophy, packaged nicely  for a Western audience.

Another false underlying assumption of Chopra, which he does not state expressly, is that all of this quantum weirdness (whatever its implications) applies to the macroscopic world. This is true, in a way (depending on how you look at it), but highly misleading. All objects, no matter how large, also are waves and particles. However, the wavelength of matter (the degree to which it is spread out, rather than having a definite position) decreases with mass and velocity. This is defined by the de Broglie equation:

λ = h/mv

Where λ is the wavelength, h is Planck’s constant, m is mass, and v is velocity. What this means is that when you start to get larger than a large molecule, the quantum wavelength shrinks to insignificance. When you get to macroscopic objects, the wavelength is orders of magnitude less than the size of elementary particles. So you have a wavelength, but it is so close to zero it can be treated as zero – which is another way of saying you do not behave like a quantum object, but like an object in classical physics.

Chopra, however, thinks that QM applies to everything equally. He writes:

If we do not look at it, the moon is gone. In this world, only an act of observation can confer shape and form to reality — to a dandelion in a meadow, or a seed pod, or the sun or wind or rain. Anyway, it’s amazing, and even your dog can do it too.

He is doubly wrong – not only is consciousness (an observer) not the thing that collapses wave functions, but QM effects do not apply to dandelions or moons (try shooting dandelions through a double slit experiment).

Another reason QM effects are not seen in the macroscopic world in which Chopra’s brain exists is decoherence. Remember that weird quantum entanglement I mentioned? Well, this can be observed only in carefully contrived experimental situations. It exists in nature, of course, but tends to be fleeting, because of decoherence. Whenever either paired particle interacts with other stuff in the universe, it becomes less entangled with its original partner, until they are decohered – their properties are no longer linked. It is hard to keep particles from decohering – in fact this is a stumbling block to the development of quantum computers that exploit QM effects. The particles keep decohering and losing their properties which can be used to store information.

For any macroscopic object, all the particles in that object are interacting with each other and decohering all the time. Again – the weird quantum world collapses to a classical physical world when you scale up to dandelions and people.

Chopra really needs to have a conversation with a real quantum physicist. You would think that before someone makes a career out of promoting a specific scientific interpretation to the public they would make sure they got the science right. But I suspect Chopra doesn’t care about getting the science right. He seems to be working backwards from his metaphysics, and then happily misinterpreting QM to suit his needs.

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32 responses so far

32 Responses to “Chopra Mangles Quantum Mechanics – Again”

  1. mannik5000on 16 Nov 2009 at 11:33 am

    “Professor Farnsworth: As Deepak Chopra taught us, quantum physics means anything can happen at any time for no reason. Also, eat plenty of oatmeal and animals never had a war. Who’s the real animals?”
    -Professor Hubert J Farnsworth

  2. Eternally Learningon 16 Nov 2009 at 11:45 am

    Is it me or does it seem like Chopra is just trying really hard to answer that age old question, “If a tree falls in a forest, and no one is around to hear it, does it still make a sound?”

  3. Adam_Yon 16 Nov 2009 at 11:47 am

    I have one minor little nitpick. The wavefunction is not a probability function. The square of the wavefunction is what you are actually referring to.

  4. daedalus2uon 16 Nov 2009 at 12:59 pm

    Eternally Learning, I don’t think so, I think that Chopra is trying to get the answer to a different question, “what can I say that will get all these new-age woo-believers to give me lots of money”.

    So far he has been quite successful with his approach, but it isn’t “science” or “medicine”, it is entertainment dressed up as pseudoscience quackery.

  5. thequarkon 16 Nov 2009 at 1:01 pm

    I have a couple of comments regarding some of the interpretation you present.

    My main issue is with the claim:

    “But when a particle (whether of photon of light or an electron) interacts with other stuff it is no longer spread out but collapses down to a point particle. This is the wave-particle duality of matter.”

    “Particles” are (usually) described as a discrete number of plain waves. The wave particle duality comes from the fact that the waves only act in certain “quanta” that we call particles. Interactions don’t in general collapse the particle down to a localized object (unless the interaction is very local). On the other hand, an “observer” can be a very localized interaction (for instance see my own blog-post: http://bit.ly/12T7FJ with a very localized interaction with a spin-1/2 particle which effectively “collapses” the wave function without need for consciousness). I feel it’s a little cartoony of a picture to have waves that magically become point particles in order to interact and somewhat misleading — in reality it’s just the quantum field doing its thing and interacting.

    I can see where this comes from though, with the double slit experiment the end reaction localizes the electron to a specific position on the screen since interaction with the screen is necessarily local in nature (though if you got close enough, there’d still be a spread for each “dot” resulting both from the experiment and from some quantum effects… i.e. the impossibility of making an interaction that only takes place at one point). But the electron has already interacted with other stuff. When it got to the two-slits before hitting the last screen, it had to pass through *both* slits which can be accomplished by interacting with the sheet with the slits in it (as if it were an infinite barrier/potential with two little slits). If it had become a point particle for the interaction with the first screen, it would have necessarily gone through only one of the two slits when in fact it’s more appropriate to say it collapsed down to two point particles (one for each slit… well really even that’s a simplification) and passed through to hit the final screen with the appropriate interference pattern.

    As another example, if you have two particles interacting via the Coulomb force (1/r^2), to get the result, you have to integrate over all of space.. So if one particle is particularly massive (a proton for instance) and you shoot an electron at it, you have to add up the effects of an electron interacting at each point in space to get a final answer for where the electron “could” end up.

    Unrelated and just to throw a wrinkle in things.. given lambda = h/mv … If I’m massive, what happens if I stand very, very still (v=0)? :-)

    The explanation of decoherence was very nice though, and appropriate for explaining the classical limit.

    All and all a good article taking on the quacks misrepresenting quantum mechanics for more than we physicists use it for. It’s so aggravating seeing these people using quantum mechanics for their quackery and metaphysical claims.

  6. Ribozymeon 16 Nov 2009 at 2:32 pm

    As Richard Feynman said: “If you think you understand quantum mechanics, you don’t understand quantum mechanics”.

  7. taustinon 16 Nov 2009 at 2:51 pm

    “Chopra really needs to have a conversation with a real quantum physicist.”

    What makes you think he would understand a single word past “Hello, my name is . . .”?

  8. Watcheron 16 Nov 2009 at 3:07 pm

    He could probably understand it given enough time. Would he be willing to change his mind, is what I would like to know :)

  9. Steven Novellaon 16 Nov 2009 at 3:23 pm

    It’s not like he hasn’t been criticized about this for years. Either he doesn’t ever read the commentary of others on what he writes, he is too profoundly confused to be moved by criticism, or he doesn’t care.

  10. Watcheron 16 Nov 2009 at 4:03 pm

    Yeah, I guess any way you look at it he’s a willful woo-pusher by this point without regard to the facts. Any of those three reasons boils down to gross negligence in some manner.

  11. gr8googlymooglyon 16 Nov 2009 at 5:39 pm

    Eternally Learning: “Is it me or does it seem like Chopra is just trying really hard to answer that age old question, “If a tree falls in a forest, and no one is around to hear it, does it still make a sound?”.

    I think that question has been answered. I wonder if Mr. Chopra’s profound insights can address the most important existential question of all time, which is:

    If a man says something in the woods, and there is no woman around to hear him, is he still wrong?

  12. haggholmon 16 Nov 2009 at 5:51 pm

    Ignorance is not a sin, unless it is wilful. When it comes to the Chopras and Ray Comforts of the world — the high-profile peddlers of idiocy — it’s so obvious that they have received criticism and correction and simply choose to ignore it because they can further their own agendas better by ignoring facts.

    Greedy liars, plain and simple.

  13. YairRon 16 Nov 2009 at 6:39 pm

    I agree that Chopra is wrong to use science to justify his metaphysics, but we must not do that same. Strictly speaking, science does not show his core position – the importance of consciousness – to be false. It is a valid, albeit philosophically bankrupt and arrogant, interpretation that is consistent with physics.

    As far as quantum mechanics (QM) is concerned, the problem is that you can always “delay” the point where you decide collapse has happened. For example, I work in the field of quantum open systems, where I talk about a large system with a huge chunk of it (the “environment”) “measuring” a smaller part (the “open system”), thereby leading to its “collapse” (well, a series of partial collapses). But it is possible to delay the collapse, conditioning it on the experimental measurement of the system – so that the collapse happens when the environment is measured, not when the environment does the measuring.

    The science is compatible with consciousness causing collapse. It is also compatible with hairy bipeds causing collapse, with macroscopic systems causing collapse, and so on – it’s even largely compatible with European/American people causing collapse, although lately some Asians and Indians got into the fray. There is no good evidence that Inuits cause collapse, to my knowledge!

    Incidentally, as noted above, “interaction” is one of the few criteria that don’t work. Try “interaction that preserves which-path-information” instead, this kinda works – but watch out for those quantum eraser effects.

    The choice of which criterion to use is a philosophical one. For myself, I choose not to apply any criterion – no criterion makes philosophical sense, they’re all at about the same level of absurdity as claiming only old white males cause collapse. I advocate the multiple worlds interpretation.

    A somewhat related point – decoherence does not lead to the classical world. It leads to loss of entanglement, but not to the realization of one option. This is only done by collapse (in the Copenhagen interpretation), decoherence only chooses the basis and thus sets the probabilities for the collapse that follows.

  14. Bronze Dogon 16 Nov 2009 at 6:53 pm

    Thought I’d link to a little entry of mine somewhat related to the topic. I’m a layman when it comes to QM, but I think I have some inkling of how they misinterpret something that’s actually unsurprising if you think about it.

  15. daijiyobuon 16 Nov 2009 at 8:26 pm

    Per: “he seems to be working backwards from his metaphysics, and then happily misinterpreting QM to suit his needs.”

    Yes, through the lens of bias delights do entertain.

    -r.c.

  16. sonicon 16 Nov 2009 at 9:28 pm

    While I agree that Chopra’s statements seem to ignore decoherence, your analysis is lacking for the following reasons–

    Chopra is refering to the ‘measurement problem’ and he is specifically refering to the difficulties in explaining the results of the ‘quantum eraser’ experiment (and even worse the ‘delayed choice quantum eraser’) without including some notion that the experimenter has changed the past. This problem is well described by Paul Davies in his book “About Time”.
    (Paul Davies also describes the experimenter as ‘changing the past’ BTW)
    This is the experiment that superseeds the double slit experiment as it shows the ‘reasons’ for the results described by Feynman are not valid.
    http://arxiv.org/abs/quant-ph/9903047
    (Your use of the double slit is passe).

    You claim that the ‘measurement problem’ has been solved (no observer is needed)
    A more accurate take on the situation–

    http://plato.stanford.edu/entries/qm-decoherence/

    “Unfortunately, naive claims of the kind above (that decoherence solves the measurement problem) are still somewhat part of the
    ‘folklore’ of decoherence, and deservedly attract the wrath of physicists (e.g.,
    Pearle 1997) and philosophers (e.g., Bub 1999, Chap. 8) alike.”

    (from earlier in that article)

    “The fact that interference is typically very well suppressed between localised
    states of macroscopic objects suggests that it is relevant to why macroscopic
    objects in fact appear to us to be in localised states. A stronger claim is that
    decoherence is not only relevant to this question but by itself already provides
    the complete answer. In the special case of measuring apparatus, it would
    explain why we never observe an apparatus pointing, say, to two different
    results, i.e., decoherence would provide a solution to the measurement problem.
    As pointed out by many authors, however (recently e.g., Adler 2003; Zeh 1995,
    pp. 14-15), this claim is not tenable…”

    There are those who think that decoherence and the MWI do give us an alternative to the notion of the collapse of the wave function. But as is pointed out recently
    http://arxiv.org/PS_cache/gr-qc/pdf/9703/9703089v1.pdf

    “…no plausible set of axioms exists for an MWI that describes known physics.”

    (In general I find the orthodox interpretation of von Neumann-Heisenberg to be of value)

  17. Michael Varneyon 16 Nov 2009 at 9:52 pm

    “Particles, when free from interactions with other matter, actually behave like waves (see the double slit experiments).

    But when a particle (whether of photon of light or an electron) interacts with other stuff it is no longer spread out but collapses down to a point particle. ”

    Really?
    Not really.

    Actually, particles in any context, in any given potential are dual in nature. If it were otherwise, the Schrodinger equation would be nonsense as a method of describing the universe.

    Also, when a photon of light interacts with other “stuff” it does not collapse to a point. Its state function collapses to a specific eigenstate that allows us to determine the most probably equation of state. It is still spread out to some extent, Heisenbergs uncertainty principle demands it, for if the state function were to allow the particle to collapse to a point, its momentum would diverge to infinity.

    I could go on more about de-coherence length, macroscopic coherence lengths in Bose Einstein Condensates and the like, but I don’t want to digress too far from the enjoyable fact that you kicked Chopras crackpot ass but good.

    Regards,

    Mike

  18. eiskrystalon 17 Nov 2009 at 4:31 am

    Chopra is refering to the ‘measurement problem’ and he is specifically refering to the difficulties in explaining the results of the ‘quantum eraser’ experiment

    No, Chopra is randomly spouting gibberish with some fancy words to sound good to new agers. To take him seriously as anything but a fraud is a waste of time.

    He should be used only as an example of how not to do something.

  19. Emmanuel J. Karavousanoson 17 Nov 2009 at 9:34 am

    Quantum mechnics may be fine as a study, but does not provide an easy, simple way for people to reach for and understand consciousness. It has often been said that we overlook the obvious. There is a litany of quotes that provide a basis for analyzing things that are already obvious and known to us. Alfred North Whitehead wrote in his book, Science and the Modern World, “Familiar things happen and mankind does not bother about them. It requires a very unusual mind to undertake the analysis of the obvious.” It was Hegel who gave us these words: “Because it’s familiar, a thing remains unknown.” Reinforcing this is psychologist Gustav Ichheiser who said, “Nothing evades our attention as persistently as that which is taken for granted.” Supporting Ichheiser Aldous Huxley wrote, “Most human beings have an almost infinite capacity for taking things for granted. There are a number of others but these suffice to make the point and provide the solid foundaiton to analyze familiar, obvious and things already known to us.

    As young children we learn many things which we simply take for granted and subsequently ignore. One of these is the fact that we think and that we are conscious beings. We become aware of it, but only on the surface — superficially. How over have we heard, “I heard that but I didn’t really realize it.” This is what happens occasionally with some of the many things we learned as young children. Benjamin Franklin, for example, looked at lightning. Everyone knew of lighting, but it was Franklin who analyzed something he knew from childhood. He realized there is something here that could possibly be harnessed. In time electricity was born. Madame Curie discovered radium and as a result of serendipitous happenings, applications for this discovery has resulted in the healing of disease and medical equipment to help in this fight. Animals can never make the discoveries humans make because they only live in the so-called present. They do not analyze familiar, obvious and known things. But humans can. We must remember that the mind is a paradox. We can’t have peace of mind when we think and we can’t learn unless we think.

    In the realm of consciousness, the same holds true as in science. Jesus, it is said, upon gaining the holy spirit — what we today know to be the mystical experience, the onset to the mystical state — realized his thoughts were “temptations”. He was tempted by hunger, by the thought of suicide (which he wanted to throw himself from a high place) and the thought of wanting to own all the kingdoms of the world. Well, it was the gift of insight that made him see all his thoughts for what thoughts are. He called them temptations. Buddha saw them as attachments. Hindu mystics see thoughts as cravings. In today’s world we can see thoughts as neuroses or hangups. The only way — let this be stressed — the only way we can gain a knowledge of the higher consciousness that is within us is through analysis of our thoughts. In this regard, Buddhism has a head start in knowing to look inward, however Buddhism lacks the basis, the evidence and the logic so one can have the incentive to reach for and come to know higher consciousness — ultimate reality.

    Because the study of consciousness is so, so important, let it be stressed once again and in no uncertain terms why it is thaat the mystical experience occurs: it takes place when one analyzes things that were generally learned in a most cursory and hasty way — just as lighting always was — taken for granted and ignored. Realizing this alone could be the inchoate step for each of us to finally have a reason to reach for and attain the higher state of mind. We can believe that Moses, Jesus, Mohammad, Buddha and Confucius would all approve.
    By analyzing familiar, obvious and known things, insight is triggered and the FULL realization of what we already know, is gained where before it was known only on the surface.

    The answer to the consciousness problem has been called an irreducible complexity. Once experienced, it is seen as irreducible simplicity. It is where we come to know the self, the universe and as a result,the meaning of oneness and God. God becomes that attainable state of mind where one becomes one with the universe. Philo of Alexandria some 2100 years ago said that God is ultimate reality.

    What is vitally important is the inclusion of faith. Why? Because with faith we can stick to the analysis of familiar, obvious and known things. This is where science and religion come together and are not opposing forces as some contend. Why? Because questions are the realm of science. Faith is, of course, the realm of religion. If we have faith in the analysis of our thoughts, our thinking, our mind, insight, insight will occur in time. We know we have thoughts. We know we think. We know our mind. Yet, we know these only on the surface and not in the intuitive way that we should. We should recognize that the answer to the consciousness question is not an intellectual one. It is one of insight. It is one of realization.
    It is one of a sudden discernment. It is precisely like an idea that suddenly enters the mind. The fact is that each individual on the face of the planet must gain this gift for himself or herself.

    Emmanuel J. Karavousanos
    EKaravousa@aol.com

  20. Doctor Evidenceon 17 Nov 2009 at 1:00 pm

    “If we do not look at it, the moon is gone.”

    I bet that Mr. Chopra does a lot of ‘looking’ at his bank account-

  21. OnceWasLoston 17 Nov 2009 at 9:20 pm

    Thank you for writing this, Dr. Novella. I used to be in to that new age/QM mumbo jumbo.. I’d read and read and read the likes of Evan Harris Walker, Lama Surya Das, etc., thinking that these guys knew what they were talking about and if I could just wrap my mind around their obscure descriptions, then I would get it too.

    Fortunately, I’ve found my way back into the light of actual, real, science. If i was still in that mode of thinking, I’m sure I’d be into Deepak Chopra as well. Hell, for all the time he spends taking up valuable airtime on my precious PBS, I might have been tuning in every time his little infomercials came on.

    Anyway, thank you kindly and keep up the great work. =) As an aside, the weekly releases of SGU have become one of the things I look forward to most each week, so a big kudos to everyone involved with the show.

  22. lizkaton 18 Nov 2009 at 11:55 am

    However Steve might not be aware of quantum biology.

  23. Old Coyoteon 19 Nov 2009 at 9:39 am

    Julia Sweeney said it best: “Deepak Chopra is full of shit!”

  24. Paradymon 19 Nov 2009 at 10:37 am

    Quote Of The Week:

    “(try shooting dandelions through a double slit experiment)”

    Can’t stop chuckling. Still wiping coffee spray off monitor.

    Trying to figure out what would be the best tool for this. Musket?

  25. lizkaton 20 Nov 2009 at 3:18 pm

    You might want to look at recent quantum biology research. It turns out that qm is not irrelevant in living systems.

  26. Steven Novellaon 20 Nov 2009 at 5:25 pm

    Quantum biology is still highly speculative – very far from settled science. And even if there are quantum effects at the level of organelles (so far, all that has been possibly detected) that would not rescue any of Chopra’s woo from being total nonsense.

  27. lizkaton 21 Nov 2009 at 8:26 pm

    “What this means is that when you start to get larger than a large molecule … you do not behave like a quantum object, but like an object in classical physics.”

    Chopra’s woo may be nonsense, but quantum biology may suggest that this statement of yours could turn out to be wrong. If qm is shown to be relevant to some life processes, it may be relevant to others, even to all. Chopra is speculating far ahead of the evidence, but so is anyone who takes the opposite view. We do not know at this time.

  28. Doctor Evidenceon 22 Nov 2009 at 2:42 am

    “Chopra is speculating far ahead of the evidence, but so is anyone who takes the opposite view.”

    attempted Tu quoque fallacy. however, rejecting conclusions based on far-ahead speculation is not taking an ‘opposite’ view, it is taking a skeptical view.

    “We do not know at this time.”

    from the tone of your comment, it sounds to me like maybe you do think that you know, but that you are being coy.

  29. lizkaton 22 Nov 2009 at 9:28 am

    “from the tone of your comment, it sounds to me like maybe you do think that you know”

    I said I don’t know and I mean I don’t know. If we do not have scientific evidence for or against something, then, if we are skeptics, we admit we do not know. If Chopra is wrong about this, that does not mean the opposite view — that qm has no relevance to biological processes — must be correct. What I said does not resemble a tu quoque fallacy. We do not know how quantum biology will evolve and if we are truly scientific skeptics we should wait for the evidence before taking a definite stand.

  30. sonicon 22 Nov 2009 at 9:41 pm

    lizkat-Doctor Evidence–

    You both seemed to have missed this-

    http://cerncourier.com/cws/article/cern/30130

    (the discovery (or the possible discovery) that photosynthesis uses quantum coherence to achieve the high levels of effeciency achieved.))

    Quantum biology begins!

  31. lizkaton 23 Nov 2009 at 7:43 pm

    No, I didn’t miss it. That’s why I mentioned quantum biology.

  32. Aquilaon 29 Nov 2009 at 10:02 am

    When the philosophy of quantum mechanics gives us such a stupid thing as the Copenhagen interpretetion, it is easy for Kapra end the like to derive anything.

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