Find a calculation of Schrödinger's cat experiment?

[Joe Cool]

Hello,
does someone know where to find a calculation of Schrödinger's cat experiment? (like psi(cat)=|dead> + |alive>...)

 

[Nugatory]

Hello,
does someone know where to find a calculation of Schrödinger's cat experiment? (like psi(cat)=|dead> + |alive>...)

That calculation doesn't exist, because it would require knowing the detailed state of every one of the $10^{25}$ or so particles that make up the cat and the vial-breaking mechanism - completely impractical.

The calculation wasn't really needed because Schrodinger did not come up with this thought experiment to suggest that the the dead/alive superposition would happen. Instead, he was pointing out a problem with the then-current (1930 or thereabouts) understanding of QM: the theory seemed to suggest that we would have a such a superposition although we know perfectly well that we don't. In fact we have a box containing either a dead cat or a live cat and we just don't know which until we look. It took a few more decades to work out the answer to this problem; google for "Quantum decoherence" or give David Lindley's book "Where does the weirdness go?" a try.

 

[vanhees71]

There are tons of articles on Schrödinger-cat experiments (of course with atoms, cavity QED and the like not with real cats), e.g.,

http://tf.boulder.nist.gov/general/pdf/1112.pdf

 

[Lord Jestocost]

The calculation wasn't really needed because Schrodinger did not come up with this thought experiment to suggest that the the dead/alive superposition would happen. Instead, he was pointing out a problem with the then-current (1930 or thereabouts) understanding of QM: the theory seemed to suggest that we would have a such a superposition although we know perfectly well that we don't. In fact we have a box containing either a dead cat or a live cat and we just don't know which until we look. It took a few more decades to work out the answer to this problem; google for "Quantum decoherence" or give David Lindley's book "Where does the weirdness go?" a try.

I imagine that such a description would have raised a smile on Schroedinger’s face. Schroedinger wanted to point out with his cat fable - a little bit ironically: "There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks." Maybe, he was aware that some day something like the “decoherence” stuff would come up.

The elimination of coherence doesn't imply that the quantum ignorance "AND" (snapshot of clouds and fog banks) can be replaced by the classical ignorance "OR" (shaky or out-of-focus photograph) when the quantum mechanical formalism is consequently applied. One has just to accept that in order to avoid mere interpretations on base of personal world views.

 

[ZapperZ]

Hello,
does someone know where to find a calculation of Schrödinger's cat experiment? (like psi(cat)=|dead> + |alive>...)

A.J. Leggett "Testing the limits of quantum mechanics: motivation, state of play, prospects", J. Phys. Condens. Matt., v.14, p.415 (2002).

Zz.

 

[Joe Cool]

Thank you for the answers :-). Have I understand it correctly that (if quantum mechanics is applied to the cat as if it would be single atom with two states) the cat and the atom$$(\frac 1 {\sqrt{2}} |decay> + \frac 1 {\sqrt{2}}|no~decay>) |cat>$$evolves to$$U(\frac 1 {\sqrt{2}} |decay>|cat> + \frac 1 {\sqrt{2}}|no~decay> |cat>)=\frac 1 {\sqrt{2}} |decay>|dead> + \frac 1 {\sqrt{2}}|no~decay> |alive>$$This is not an eigenstate of an "is the cat alive?" operator $(+1)|alive><alive|+(-1)|dead><dead|$ which means that it is not defined if the cat is alive or dead.

It took a few more decades to work out the answer to this problem; google for "Quantum decoherence" or give David Lindley's book "Where does the weirdness go?" a try.

Thanks, I will read about it.

 

[Lord Jestocost]

Some quotes from "The Quantum Measurement Problem" by A. J. Leggett (SCIENCE vol 307, 2005):

• Basically, the quantum measurement paradox is that most interpretations of QM at the microscopic level do not allow definite outcomes to be realized, whereas at the level of our human consciousness it seems a matter of direct experience that such outcomes occur (indeed, it seems so difficult to imagine what it would be like for the world to be otherwise that I suspect that Immanuel Kant, had he had occasion to consider the problem, would have classified our knowledge of this state of affairs as "synthetic a priori").

• It is convenient to classify reactions to this problem into three broad classes, defined by the following three different views on the status of QM: (a) QM is the complete truth about the physical world, at all levels, and describes an external reality. (b) QM is the complete truth (in the sense that it will always give reliable predictions concerning the nature of experiments) but describes no external reality. (c) QM is not the complete truth about the world; at some level between that of the atom and that of human consciousness, other non–quantum mechanical principles intervene.

• Even a decade ago, considerable skepticism existed about the prospect of ever observing quantum superpositions involving more than a few "elementary" particles. However, in the last 5 years progress in this direction has been spectacular, ranging from traditional Young's slits experiments conducted with C₇₀ molecules (~1300 "elementary" particles) to SQUID experiments in which the two superposed states involved ~10¹⁰ electrons behaving differently (1). Thus, the experiments are beginning to impose nontrivial constraints on hypotheses of class (c).

I would say: The "cats" are still very tiny but they are starting to grow.

 

[ZapperZ]

I would say: The "cats" are still very tiny but they are starting to grow.

If you do a search on "Delft or Stony Brook experiments", you'll find that this has been discussed in several threads. These two experiments were the result of Leggett's paper and showed the Schrodinger Cat-states in SQUID experiments measuring the coherence gap.

Zz.

 

[Lord Jestocost]

$$\frac 1 {\sqrt{2}} |decay>|dead> + \frac 1 {\sqrt{2}}|no~decay> |alive>$$

Schrödinger dubbed such a “superposition” an entangled state because the state of the particle (decayed/not decayed) is correlated with the state of the cat. And here is the essence of his cat fable: A “superposition” expresses our “quantum ignorance” about a system. When one accepts that quantum theory is a fundamental and complete physical theory, then one has to accept that this theory contains no “physical” law to convert “quantum ignorance” (superposition) to “classical ignorance” (mixed state).

Thus, what people think about the state of the cat in the box before one opens the box (measurement/observation) doesn’t follow from quantum theory. It is, to my mind, a mere expression of psychological predispositions.

Nevertheless, I would be very cautious when opening the box cause it sometimes pays to listen to Terry Pratchett (in “Lords and Ladies“): “In fact, the mere act of opening the box will determine the state of the cat, although in this case there were three determinate states the cat could be in: these being Alive, Dead, and Bloody Furious.”

 

[Joe Cool]

Nevertheless, I would be very cautious when opening the box ...

:-D

So does the "state" vector represent not a physical state of the cat but our knowledge(or "quantum ignorance") about it? But we are not able to interpretate the superposition just as "with 50% probability the cat is alive", are we?

 

[Lord Jestocost]

So does the "state" vector represent not a physical state of the cat but our knowledge(or "quantum ignorance") about it?

As Schroedinger said to Sommerfeld, 1931: "Quantum mechanics forbids statements about the object. It deals only with the object-subject relation." Or, to quote Wheeler: "No elementary phenomenon is a phenomenon until it is a registered (observed) phenomenon." Thus, we are talking about our knowledge of reality rather than reality itself.

But we are not able to interpretate the superposition just as "with 50% probability the cat is alive", are we?

Quantum theory remains silent when questions regarding "interpretations" come up. Therefore, it rather depends on personal world views how people try to interpret what a superposition might be.

 

[vanhees71]

For me the socalled Schrödinger-cat paradox is ununderstandable since "dead" and "alive" are for sure not pure states in the sense of quantum theory but very coarse-grained macroscopic observables, which are quite classical.

Whenever the preparation of superpositions of eigenstates of microscopic observables are technically feasible, which today is indeed possible for larger and larger objects (see the statement by Leggett quoted somewhere earlier in this thread), the predictions of (minimally interpreted) QT have been seen valid. Although Schrödinger wouldn't like it, Nature seems indeed to behave quite accurately to the outcomes of QT. Nature doesn't care what physicists think it should behave but she just behave as she does, even if the physicist in question is a genius like Schrödinger or Einstein.

 

[Joe Cool]

As Schroedinger said to Sommerfeld, 1931: "Quantum mechanics forbids statements about the object. It deals only with the object-subject relation." Or, to quote Wheeler: "No elementary phenomenon is a phenomenon until it is a registered (observed) phenomenon." Thus, we are talking about our knowledge of reality rather than reality itself.

Nice quotes, they really made the point of the cat story more clear for me, thanks!

For me the socalled Schrödinger-cat paradox is ununderstandable since "dead" and "alive" are for sure not pure states in the sense of quantum theory but very coarse-grained macroscopic observables, which are quite classical.

Maybe future experiments will confirm that macroscopic classical observables can't be described by the concept of pure quantum states, who knows ;-)

 

[AlexCaledin]

Nature doesn't care what physicists think it should behave but she just behave as she does, even if the physicist in question is a genius like Schrödinger

And Nature always knows whether each of her cats is dead or alive, even if Schrödinger thinks it's his cat.

 

[Lord Jestocost]

Maybe future experiments will confirm that macroscopic classical observables can't be described by the concept of pure quantum states, who knows ;-)

You hit the nail on the head. Physics has to rely upon experiments, not personal world views or eager "interpreters". Physics can reveal no "real world" beyond what is observed. Either quantum theory is incomplete or even flawed or "classical reality" is an illusion. No way out!

 

[vanhees71]

Well, I take it as an obvious observational fact that a cat is a macroscopic object and, at least as long as it's living, a far-from-equilibrium system in continuous exchange with the environment. The states "dead" and "alive" are for sure no pure quantum states, and it's impossible to prepare a cat in a pure state, because for that you'd have to prepare about $10^{30}$ compatible observables precisely.

 

[Lord Jestocost]

Well, I take it as an obvious observational fact that a cat is a macroscopic object and, at least as long as it's living, a far-from-equilibrium system in continuous exchange with the environment. The states "dead" and "alive" are for sure no pure quantum states, and it's impossible to prepare a cat in a pure state, because for that you'd have to prepare about $10^{30}$ compatible observables precisely.

There is one equation and one quantity which define quantum theory – the time dependent Schroedinger equation and the associated wave function. What’s now – regarding physics – the qualitative difference whether one sets up a time dependent Schroedinger equation and its associated wave function for 1, 100 or 10³⁰ “particles”.

 

[rude man]

My cat Henry's comment on the above: MEEOW!

 

[Lord Jestocost]

My cat Henry's comment on the above: MEEOW!

Clever cat. Indeed, many err explaining our world.

 

[vanhees71]

There is one equation and one quantity which define quantum theory – the time dependent Schroedinger equation and the associated wave function. What’s now – regarding physics – the qualitative difference whether one sets up a time dependent Schroedinger equation and its associated wave function for 1, 100 or 10³⁰ “particles”.

Well, the difference is that you aren't even able to write down the initial wave function, let alone solve the SG, and fortunately that's also not necessary since coarse grained observables are sufficient to understand the relevant and interesting macroscopic dynamics. Without quantum statistics there's no way to do any kind of condensed-matter physics!

 

[Demystifier]

For me the socalled Schrödinger-cat paradox is ununderstandable since "dead" and "alive" are for sure not pure states in the sense of quantum theory but very coarse-grained macroscopic observables, which are quite classical.

If this is so about the cat, then how about a bacteria? A virus? A DNA molecule? A protein? At which point it becomes understandable to you?

 

[vanhees71]

Well, there have been experiments with superpositions (then even called "Schrödinger cat states"), e.g., using Rydberg states of atoms, and they all behaved as expected according to QT. So there is no problem with "Schrödinger cat states" for systems, where you can prepare them. A real cat is for sure not preparable in a pure state for any experimentally realizable setup.

 

[kbomeisl]

I think Schrödinger's thought experiment assumed the cat was not quantum entangled with the state of the particle in the box. The particle is its own quantum system, the poison gas is its own second macroscopic system in which the quantum characteristics are ignored, and the cat is a macroscopic system whose quantum characteristics are completely ignored. The three are assumed to have no quantum interactions. The whole point of the experiment is the interaction of the quantum scale with the macro-scale, not quantum interactions between these three systems. The quantum characteristics of everything except the particle were assumed irrelevant for the sake of the thought experiment. The particle itself would be measured by a machine in the box, and if it were spin up it released the poison gas, and if it were spin down it didn't release the poison gas. You can use any particle you want prepared in any state you want (I believe Schrodinger used an electron so -1/2 spin=dead cat, 1/2 spin=alive cat, the probabilities of each are the probabilities of the electron being in a state of spin -1/2 and spin 1/2 so it depends on how you prepare the electron or particle system). The important point is that releasing the poison gas was the only interaction the particle has with the other two systems.

The whole point of the thought experiment was that if the electron was in a superposition of up and down spin, the poison gas was in a superposition of released and not released and therefore the cat was in a superposition of dead and not dead. So the wave function of the cat being dead or alive depends on what particle system you are using as the mechanism to release the poison gas. It simply the wave function of whatever particle system you are using as the poison gas release mechanism.

 

[Lord Jestocost]

A real cat is for sure not preparable in a pure state for any experimentally realizable setup.

With all due respect, that’s a very trivial statement. The cat is of course nothing but a symbol of any macroscopic object (or has the cat fable been misunderstood completely?).

Uniqueness of reality would require that - in course of time - any quantum ignorance “AND” should somewhere/sometime be replaced by the classical ignorance “OR”. This, however, cannot be explained within the time dependent Schroedinger equation: the conceptual transition from quantum to classical ignorance has to be put in “by hand”?

 

[vanhees71]

Of course, it's a trivial statement. So?

 

[Lord Jestocost]

Of course, it's a trivial statement. So?

Trivial, because it has nothing to do with the Schroedinger's fundamental "question" which is stiil awaiting a physically consistent "response".

 

[vanhees71]

The response is that whenever QT, including superpositions and entanglement, where experimentally tested, the predictions of QT were found to be correct with an astonishing accuracy. The conclusion is that this fundamental ingredients of QT are to a large extent correct, and no validity limits of QT are yet known (despite the theoretical problem of finding a satisfactory theory for quantum gravitation).

 

[Gigaz]

I propose the following experimental setup: ;)

The box contains the cat, the poison device and life support. We put the box in a tight orbit around a small Black Hole and isolate everything as strongly as possible against the outside world. It's probably a good idea to sedate the cat because it might produce gravitational waves while moving. When the cat has reached the state |dead>+|alive>, a sensor measures the heart beat. If the cat lives, a photon is creates at point A on the device surface and sent away to the physicist. If the cat is dead, the photon is sent out from neighboring point B. During the flight time of the photons, the whole device is steered into the Black Hole so that the state of the cat at the measuring time cannot be recovered. The photon is then taken as one data point. Repeat a few thousand times and there should be an interference pattern. Totally impractical, but not completely impossible, I guess.

 

[Lord Jestocost]

The response is that whenever QT, including superpositions and entanglement, where experimentally tested, the predictions of QT were found to be correct with an astonishing accuracy. The conclusion is that this fundamental ingredients of QT are to a large extent correct, and no validity limits of QT are yet known (despite the theoretical problem of finding a satisfactory theory for quantum gravitation).

I completely agree. To my mind, that’s the reason why quantum theory shouldn’t serve as a playground for “interpretations” other than “agnostic” ones.

 

[Joe Cool]

Well, the difference [between a wave function for 1, 100 or 1030 “particles” ] is that you aren't even able to write down the initial wave function, let alone solve the SG, and fortunately that's also not necessary since coarse grained observables are sufficient to understand the relevant and interesting macroscopic dynamics. Without quantum statistics there's no way to do any kind of condensed-matter physics!

Sorry for asking again, but I sill ponder on that. I haven't learned quantum statistics yet, just undergraduate quantum mechanics. So I would write the state of a (macroscopic) Geiger counter (I understand a cat is not in equilibrium and that makes it more complicated) measuring if the atom decays as $\frac 1{\sqrt 2}|decay~measured>+\frac 1{\sqrt 2} |no~decay~measured>$. I have always read the axiom of quantum mechanics

A physical system is associated with a Hilbert space H. Rays in H are associated with the states of the system.

as every physical system.

 

[vanhees71]

True, but you cannot treat the Geiger counter by solving the Schrödinger wave equation exactly for its $10^{30}$ (or so) constituents, and it is indeed completely sufficient to treat the relevant physics of the Geiger counter in terms of classical physics. It's misleading to write $|\text{decay measured} \rangle$ since the observation that a decay has been registered is not a microscopic but a macrscopic observable. Here lies the key for the understanding that there is no measurement problem, and this was emphasized already by Bohr in the early days of QT.

To understand theoretically, why the classical description of the macroscopic observables of macroscopic systems is a valid approximation of QT, you need quantum statistics or "many-body theory". That's all I'm saying.

 

[Lord Jestocost]

True, but you cannot treat the Geiger counter by solving the Schrödinger wave equation exactly for its $10^{30}$ (or so) constituents, and it is indeed completely sufficient to treat the relevant physics of the Geiger counter in terms of classical physics. It's misleading to write $|\text{decay measured} \rangle$ since the observation that a decay has been registered is not a microscopic but a macrscopic observable. Here lies the key for the understanding that there is no measurement problem, and this was emphasized already by Bohr in the early days of QT.

To understand theoretically, why the classical description of the macroscopic observables of macroscopic systems is a valid approximation of QT, you need quantum statistics or "many-body theory". That's all I'm saying.

There lies no key for the understanding that there is no measurement problem. You are simply making an artificially cut between – whatever you call it – “microscopic” and “macroscopic” observables - quietly and secretly assuming that somewhere a "collapse" or "reduction" occurs.

To say it in terms by Landau and Lifshitz: “Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.”

 

[vanhees71]

Well, LL is right with that, but I don't consider this as a problem, because QT contains classical mechanics as a limiting case. So there is no contradiction between classical and quantum theory, where the classical approximation is valid. Of course, there is no fundamental "cut". QT is more comprehensive than classical theory, and the better our preparation procedures become, thanks to technological progress, the larger systems can be shown to behave according to QT, e.g., there have been double-slit interference demonstrations by Zeilinger's group for fullerene molecules and as well the demonstration that already a pretty small temperature is enough to destroy the quantum interference due to decoherence because of the emission of just a few thermal photons.

 

[Joe Cool]

True, but you cannot treat the Geiger counter by solving the Schrödinger wave equation exactly for its $10^{30}$ (or so) constituents, and it is indeed completely sufficient to treat the relevant physics of the Geiger counter in terms of classical physics. It's misleading to write $|\text{decay measured} \rangle$ since the observation that a decay has been registered is not a microscopic but a macrscopic observable. Here lies the key for the understanding that there is no measurement problem, and this was emphasized already by Bohr in the early days of QT.

Okay, I now get the idea of taking it as an paradox free macroscopic thing. Before talking with you about the classical limit or black holes I surely need to learn more advanced QT first ...

Thanks everyone for taking so much time to explain it to me in detail!

 

[bhobba]

So there is no contradiction between classical and quantum theory, where the classical approximation is valid.

In fact a careful analysis shows the real basis of classical mechanics is QM - but that is a whole new story and another thread if anyone wants to pursue it either here or on the classical physics sub-forum.

Thank
Bill