Appropriate Concepts in the Formulation of Quantum Mechanics

[dx]

It is clear that the concepts of position and momentum are idealizations from our macroscopic experience which are not appropriate in the quantum domain. Yet, in the presentations of elementary quantum mechanics I've seen so far, they are still used in a fundamental way. But obviously, they will be nothing like the position and momentum we know and therefore will seem "counterintuitive". What I'm trying to get at is that this counterintuitiveness is just an illusion, because we have no intuition about the quantum world that it can be "counter" to. This arises from the bad decision use the same names of "position" and "momentum" for something completely different and more fundamental.

The standard presentations give the idea that objects in the quantum domain still have the properties of position and momentum except that they are nothing like the ones we know. I think this is a bad way of saying it and leads to a lot of confusion. A better way is to say that position and momentum are approximate concepts valid in the large scale which arise from more fundamental concepts/concept ( which shouldn't be called by the same names ) valid at all scales, as far as we know.

An analogy with special relativity make this clearer. Before SR, the concepts of absolute spatial and temporal seperations were used in the description of motion. What special relativity taught us was that these concepts are valid at low velocities but are just approximations to a more fundamental entitiy called the spacetime interval. But we still sometimes think about special relativity by retaining our our old concepts and using the "counterintuitve" rules such as the lorentz transformations.

The language of events and invariant spacetime intervals is clearly superior to the language of lengths, time intervals and transformations. What I want to know is if there is an analogous viewpoint in Quantum Mechanics which uses more appropriate concepts for its formulation? If there is, I would be grateful if you can provide some references. FYI, I'm a first year undergraduate, so if you think it will be too technical for me at this stage, you can tell me that.

 

[lightarrow]

It is clear that the concepts of position and momentum are idealizations from our macroscopic experience which are not appropriate in the quantum domain. Yet, in the presentations of elementary quantum mechanics I've seen so far, they are still used in a fundamental way. But obviously, they will be nothing like the position and momentum we know and therefore will seem "counterintuitive". What I'm trying to get at is that this counterintuitiveness is just an illusion, because we have no intuition about the quantum world that it can be "counter" to. This arises from the bad decision use the same names of "position" and "momentum" for something completely different and more fundamental.

The standard presentations give the idea that objects in the quantum domain still have the properties of position and momentum except that they are nothing like the ones we know. I think this is a bad way of saying it and leads to a lot of confusion. A better way is to say that position and momentum are approximate concepts valid in the large scale which arise from more fundamental concepts/concept ( which shouldn't be called by the same names ) valid at all scales, as far as we know.

I think exactly the same, but this idea has some problems (see down).

An analogy with special relativity make this clearer. Before SR, the concepts of absolute spatial and temporal seperations were used in the description of motion. What special relativity taught us was that these concepts are valid at low velocities but are just approximations to a more fundamental entitiy called the spacetime interval. But we still sometimes think about special relativity by retaining our our old concepts and using the "counterintuitve" rules such as the lorentz transformations.

I believe that example is not well chosen, since space and time still exist even at high speeds; it's only that they are not as fundamental as interval (because they're not invariant). Instead, in QM, concepts like position can not exist at all (which is the position of a plane wave?), as you pointed out.

The language of events and invariant spacetime intervals is clearly superior to the language of lengths, time intervals and transformations. What I want to know is if there is an analogous viewpoint in Quantum Mechanics which uses more appropriate concepts for its formulation? If there is, I would be grateful if you can provide some references. FYI, I'm a first year undergraduate, so if you think it will be too technical for me at this stage, you can tell me that.

I don't think we know other concepts on which to refer state vectors, other than space, time, momentum, ecc., because these are the quantities we can measure.

 

[dx]

I believe that example is not well chosen, since space and time still exist even at high speeds; it's only that they are not as fundamental as interval (because they're not invariant)

I was referring to the concepts of absolue space and time seperations. space and time seperations do indeed exist between events, but in relativity, there is no absolute space interval or absolute time interval between events.

We can think of this as a sort of relativistic comlementarity. depending on relative velocities, the separation between two events can manifest itself in a more spacelike way or in a more timelike way, or equally. The space and time intervals are complementary components of the more fundamental spacetime interval.

What about complementarity in quantum mechanics? Wave-like and particle-like behavior are complementary. Is there a unifying analogue of the spacetime interval in this case?

I also have another question about quantum mechanical amplitudes. In the double-slit experiment, we have an amplitude that the electron will go to any particular point on the screen. But actually, the amplitude is only for the event that the electron will interact with a particular point of the screen and produce a dot on that screen. And this is actually the amplitude that we detect the dot at that particular point when we shine photons on it. This presupposes that we can treat the screen classically and that looking at the screen won't disturb the location of the dot. So whenever we give the amplitude as a function of time and position, we are actually presupposing that there is some substratum that can be treated classically, and the interaction of the particle with the substratum can be precisely located without disturbing it. Am I wrong?

 

[Fra]

I'm not entirely sure I got your main point, but I think you were basically reflecting over the choice of variables, and parametrization in quantum mechanics?

That's an interesting reflection. In relativity one has a global observer-invariant view, which is observer invariant. And to get the view of a particular observer, in a particular frame of reference one has to define the reference frame.

The question is what is more fundamental, the observer-invariant view that relates the views of different observers, or the observer specific view?

This is IMO one of the clashes between QM and classical mechanics, and I this may also relate to various interpretations.

When we are going to "quantize" this, what set of variables are we to choose? If we consider that quantum mechanics supposedly delas with the observers information about it's environment, I am leaning towards that the quantization should be done in the observer view, not the birds view. But then of course the procedure may not be observer-invariant, but the question is if this is a problem?

Maybe the loss of that invariance need to recovered with another symmetry, possibly even an emergent one, rather than fundamentally exact. Who knows?

Although SR and QM are considered reasonably unified, that's still just a special case and GR remains. So perhaps there is not yet and universally accepted answer to your question?

/Fredrik

 

[Fra]

FYI, I'm a first year undergraduate, so if you think it will be too technical for me at this stage, you can tell me that.

While I think that while (in general) some things may be technically difficult - such as requiring a lot of background formalisms to be described, some things are more conceptually difficult, and from my own experience in the first QM course, far from all in the class acquired much of a deeper vision. I think it's too much to ask for someone coming from classical mechanics to see all issues at once. From what I recall, most of the focus was the conceptual step to go from classical deterministic thinking, to probabilistic thinking. ( The objections I personally have on the determinism of the probability didn't become clear until later. It was first when I thought I have found the beauty of the probability formalism, and done away with classical thinking, that I noticed that this still doesn't make complete sense, and it was not as beautiful anymore. )

/Fredrik

 

[dx]

I'm not entirely sure I got your main point, but I think you were basically reflecting over the choice of variables, and parametrization in quantum mechanics?

Although SR and QM are considered reasonably unified, that's still just a special case and GR remains. So perhaps there is not yet and universally accepted answer to your question?

/Fredrik

I think you misunderstood me. I was not talking about the relationship between relativity and quantum mechanics, or its unification with general relativity. Let me try to explain again.

In quantum mechanics, we have the concepts of momentum and position. They don't mean the same thing that they meant in classical mechanics. This is because we have found that the exact measurement of position is impossible even in principle. There is no such thing as a classical-position. It is an approximation to another more fundamental concept which again we have chosen to call position. The same goes for momentum. These quantum mechanical versions of position and momentum form a conjugate pair. Heisenberg's uncertainty relation connects the uncertainties in these two variables.

There is a similar "complementarity" between space-interval and time-interval in relativity, i.e. one depends on the other. They complement each other in the sense that a particular combination of them called the spacetime interval is invariant. These "conjugate" quantities can be different for different observers but there is something more fundamental that is the same. Is there some similar concept for conjugate pairs in quantum mechanics and for wave-particle duality in general? I'm not saying there should be, I'm just wondering if there is.

 

[vanesch]

In quantum mechanics, we have the concepts of momentum and position. They don't mean the same thing that they meant in classical mechanics. This is because we have found that the exact measurement of position is impossible even in principle. There is no such thing as a classical-position. It is an approximation to another more fundamental concept which again we have chosen to call position. The same goes for momentum. These quantum mechanical versions of position and momentum form a conjugate pair. Heisenberg's uncertainty relation connects the uncertainties in these two variables.

Personally, I wouldn't say that the concept of a precise position, or a precise momentum, is meaningless in quantum mechanics, or that these concepts are altered in some way. With position, we still mean "point in Euclidean space" in quantum mechanics. However, a big difference between quantum mechanics and classical mechanics is this: systems are in superpositions of classical states in quantum mechanics, while they can only be in a single classical state in, eh, classical mechanics. This means that in classical mechanics, a point particle has a single precise position, while in quantum mechanics, it will be in a superposition of position states.

Another difference between quantum mechanics and classical mechanics is that position and momentum are independent parts of the dynamical state in classical mechanics, while momentum states are superpositions of position states and vice versa in quantum mechanics. It is because of *this* property that we cannot simultaneously measure position and momentum, because a single value for one automatically means: a whole superposition for the other.

But the concepts themselves, of position (and correspondingly precise position states), and of momentum (and correspondingly precise momentum states) are not altered by quantum theory.

 

[lightarrow]

Personally, I wouldn't say that the concept of a precise position, or a precise momentum, is meaningless in quantum mechanics, or that these concepts are altered in some way. With position, we still mean "point in Euclidean space" in quantum mechanics. However, a big difference between quantum mechanics and classical mechanics is this: systems are in superpositions of classical states in quantum mechanics, while they can only be in a single classical state in, eh, classical mechanics. This means that in classical mechanics, a point particle has a single precise position, while in quantum mechanics, it will be in a superposition of position states.

Another difference between quantum mechanics and classical mechanics is that position and momentum are independent parts of the dynamical state in classical mechanics, while momentum states are superpositions of position states and vice versa in quantum mechanics. It is because of *this* property that we cannot simultaneously measure position and momentum, because a single value for one automatically means: a whole superposition for the other.

But the concepts themselves, of position (and correspondingly precise position states), and of momentum (and correspondingly precise momentum states) are not altered by quantum theory.

Yes, they're not altered but nontheless they seems too "stretched" concepts in some cases, for example the position of a plane wave. A particle like an electron has a wavefunction described by a more or less spatially localized wavepacket; but a wavepacket is the "result" of superposition of plane waves; more massive objects (and so classical objects) are described by more localized wavepackets so one could think that concepts as position are something that arises from other fundamental quantum concepts and not the other way round.

 

[vanesch]

Yes, they're not altered but nontheless they seems too "stretched" concepts in some cases, for example the position of a plane wave.

A plane wave doesn't have *A* position, but is the superposition of many many position states. You can say, alternatively, that a plane wave is a pure momentum state.

Again, quantum theory distinguishes itself from classical theory in that a quantum object can be (will be) in a superposition of different "classical" states. This means that one cannot, at that moment, assign ONE SINGLE classical state to such a quantum object. In this case, the single classical state is "position", and a quantum particle can be in a superposition of position states. That means then that we cannot assign a single position to that particle (but rather, "many in parallel", although the right word is not "parallel" but "in superposition"). This doesn't mean that the concept of "position" itself has become fuzzy ! It simply means that our system (particle) is in a superposition of positions.

 

[Fra]

I take it this is patly a philosophical question, so here goes some reflections without claim that it's standard reflections.

There is a similar "complementarity" between space-interval and time-interval in relativity, i.e. one depends on the other. They complement each other in the sense that a particular combination of them called the spacetime interval is invariant. These "conjugate" quantities can be different for different observers but there is something more fundamental that is the same. Is there some similar concept for conjugate pairs in quantum mechanics and for wave-particle duality in general? I'm not saying there should be, I'm just wondering if there is.

It's true that momentum is differently defined in QM.

In QM, the notion of adding information about position and information about momentum has a special meaning. The reason one can not have arbitrary information about both in combination is because there by definition of momentum exists a relation between the two. This is of course also expressed via the commutator of q and p.

In QM, the information about the system by postulate makes up a linear vector space. And the information is abstractly represented by a vector. Maybe this state vector is the "invariant" object you are looking for? Different questions/measurements is represented by projecting the state vector onto other vectors. But once a measurement is actually made, the state vector itself changes. This is why it seems like the state vector itself, really isn't objective after all, but these discussions relates partly also to the interpretational issues of QM.

IMO, the plausability in non-commuting observables, is that in general there is no a priori reason to assume that information is independent of each other, I find it more "natural" to say that "you don't know". If you add two chunks of "information" it may well be that they are in contradiction, and then a rule for "making the addition" is needed so as to come out with a single consistent "piece of information" (a new state vector in QM). And it should not be unexpected that the order of addition matters.

I may misundestand you still, but maybe you are after some idea that there must only be one reality, and how can the different views, giving different information, be understood in a general way?

Maybe a difference is that in classical mechanics knowledge (answers) has some absolute meaning in some sense, although still relational, like in relativity.

In quantum mechanics, knowledge (~answers) depends on the questions you ask - ie your choice of measurement. But still, in a certain sense there is a objective background info in QM, that is supposedly contained in the state vector. So that different questions giving different possible answers, still relates back to the one and same state vector. But there is a difference between considering different possible expected answers, and actually actually firing a question, because once it's fired and you get back the answer your state of information is by definition perturbed.

Apart from this, maybe there is no such similar concept, or I don't understand your question.

I see two reflections of objective/invariant and subjective in QM, and I'm not sure if they touch your issue, this is my personal thinking so don't take my word for it...

1) The state vector itself is relative to the questioner, or to the questioners information or knowledge. And thus is not _known_ to be observer invariant IMO in the general sense. But that doesn't prevent us from guessing an beeing lucky. And after some training it may not be a conincidence that we keep getting lucky ;)

2) The possible answers, any given questioner can fire, yields a spectrum of possible answers. And the possible expected answers to the possible questions is invariant with respect to different measurements, in the sense that they relate to the statevector. But as soon as a measurement is actually made, the questioners state (his information) is changed, to technically thes recently "informed" questioner is different that the old uninformed one :)

This I mean in the sense of Zurek: What the observer knows is inseparable from what the observer is. The physical nature of the observer, is nothing but a manifestation or encoding of his information about his environment.

/Fredrik

 

[vanesch]

In QM, the information about the system by postulate makes up a linear vector space. And the information is abstractly represented by a vector. Maybe this state vector is the "invariant" object you are looking for? Different questions/measurements is represented by projecting the state vector onto other vectors.

I would indeed argue that the statevector is the unifying concept behind the "miriads of different superpositions" that one can consider.

But once a measurement is actually made, the state vector itself changes. This is why it seems like the state vector itself, really isn't objective after all, but these discussions relates partly also to the interpretational issues of QM.

Well, it is not so surprising that an interaction with a system changes its state ; but here we enter interpretational issues, and while some think "random projection", others (me included) prefer to think "entanglement".

What is correct, however, is that we can, FAPP (for all practical purposes) now consider the vector to be projected on the outcome of the measurement, and that all further results obtained from there onward will be correct.

 

[dx]

Personally, I wouldn't say that the concept of a precise position, or a precise momentum, is meaningless in quantum mechanics, or that these concepts are altered in some way. With position, we still mean "point in Euclidean space" in quantum mechanics.

This is exactly what I'm questioning. We are trying to extend our classical concepts to the quantum domain, just like we extended the classical concepts of space separation and time separation between events into the relativistic domain with the lorentz transform. But then, in the case of relativity, we found a more satisfactory viewpoint using the concept of spacetime interval and its invariance. The two viewpoints are equivalent, but one is more satisfactory than the other. In a similar way, we need a more fundamental concept to describe quantum phenomenon without an unnatural extention and modification of the concepts of momentum and position. These latter must emerge from the more fundamental concept as approximations. They should not be used in its formulation.

However, a big difference between quantum mechanics and classical mechanics is this: systems are in superpositions of classical states in quantum mechanics, while they can only be in a single classical state in, eh, classical mechanics. This means that in classical mechanics, a point particle has a single precise position, while in quantum mechanics, it will be in a superposition of position states.

Doesnt this seem unnatural? The classical states of exact position and exact momentum are approximations to something deeper, yet we try to describe the thing that is deeper using these very concepts in an altered form.

Another difference between quantum mechanics and classical mechanics is that position and momentum are independent parts of the dynamical state in classical mechanics, while momentum states are superpositions of position states and vice versa in quantum mechanics. It is because of *this* property that we cannot simultaneously measure position and momentum, because a single value for one automatically means: a whole superposition for the other.

The most satisfactory understanding of why we can't measure simultaneously position and momentum is by noting that any attempt at measurement of one affects the value of the other. It is impossible in principle. This hints at a deeper level where the concepts themselves have no meaning. We have made great theoretical progress in the past by abolishing such things. The ether, absolute time, absolute space ...

 

[Fra]

Well, it is not so surprising that an interaction with a system changes its state ; but here we enter interpretational issues, and while some think "random projection", others (me included) prefer to think "entanglement".

Well, it is not so surprising that an interaction with a system changes its state ; but here we enter interpretational issues, and while some think "random projection", others (me included) prefer to think "entanglement".

What is correct, however, is that we can, FAPP (for all practical purposes) now consider the vector to be projected on the outcome of the measurement, and that all further results obtained from there onward will be correct.

FAPP I agree :)

/Fredrik

 

[lightarrow]

In a similar way, we need a more fundamental concept to describe quantum phenomenon without an unnatural extention and modification of the concepts of momentum and position. These latter must emerge from the more fundamental concept as approximations. They should not be used in its formulation.

The classical states of exact position and exact momentum are approximations to something deeper, yet we try to describe the thing that is deeper using these very concepts in an altered form.

I have the same idea. Instead of describing a quantum object as a superposition of classical states, it would be more intuitive for me to describe classical states as superpositions of quantum states (described in a some, more fundamental, way).
Regards.

 

[dx]

it would be more intuitive for me to describe classical states as superpositions of quantum states (described in a some, more fundamental, way).
Regards.

I wouldn't use the word superpositions, but yes, that's along the lines of what I was trying to say.

 

[dx]

In QM, the information about the system by postulate makes up a linear vector space. And the information is abstractly represented by a vector. Maybe this state vector is the "invariant" object you are looking for? Different questions/measurements is represented by projecting the state vector onto other vectors. But once a measurement is actually made, the state vector itself changes. This is why it seems like the state vector itself, really isn't objective after all, but these discussions relates partly also to the interpretational issues of QM.
/Fredrik

Yes Fra, I think this is what I was looking for. I don't quite understand what exactly a state vector is because I haven't done formal quantum mechanics, but its nice to know that something like that exists. Thanks for the help.

 

[Fra]

You will form your own understanding when you take the first QM course. As is clear from the many threads on here, different people interpret things differently.

I'd say that the first conceptual difference to face is that

(1) in classical mechanics the state of the systems, is objective and evolves deterministically. Given initial conditions all future states are determined.

(2) In quantum mechanics, the new state of the system we consider, is really better interpreted as the state of our(the observers) information about the underlying system. And the evolution of the state vector is still deterministic in quantum mechanics! It's just that this describes the "selfevolution" of our state of information about the system - in between measurements. Given initial information, all future states of information are determined, or rather, all future spectrum of answers for possible questions are determined. BUT, as soon as a mesaurement is MADE, then you have reset your "initial state" to the updated information.

Also the observer in QM, does not refer to humans or consicoussness. It just notes that information is relative. In principle one can imagine the observer to be an electron, observing the atom nucleus and the innner shells. This is the deeper stuff that you probably were after. But IMHO, there is still some lacking bits to make this entirely consistent in the general case. But I don't think that will become apparent during the first QM course either. It didn't for me at least.

The indeterminism of the underlying system, althought the state vector evolves deterministically, is because the information we have about the system is incomplete. The fact that we have information about someting, does not mean we know everything. And this is in QM not just a practical matter, it's because questions in general doesn't commute. A way of interpreting that is to say that the two non-commuting questions are in part contradiction. ie you are trying to do two partly competing things at once.

But why nature behaves like this, is still given different interpretations and somewhat open. And although the calculational scheme of QM is overly successful, there are problems in the big picture.

/Fredrik

 

[vanesch]

I have the same idea. Instead of describing a quantum object as a superposition of classical states, it would be more intuitive for me to describe classical states as superpositions of quantum states (described in a some, more fundamental, way).
Regards.

I'm affraid that that's missing the fundamental idea of quantum theory entirely. The whole idea of quantum theory is that "things we observe as unique, can be in superpositions of several of them": it is the famous superposition principle.
If we observe things to have a unique position, then quantum theory tells us that it can be in superpositions of different positions. If we can observe spin to be up or down (along the z-axis), then quantum theory tells us it can be in superpositions of up and down. And that to each different complex combination of them, corresponds a different physical state, which can, in principle, be distinguished from another one by a DIFFERENT measurement.

THIS is the basic idea of quantum theory: what seems to be unique, can in fact occur in superpositions, and each of those is a different physical state (distinguishable observationally from any other, at least in principle).

Note that I leave in the middle whether this is ultimately TRUE or not. I'm just saying that this is the basic principle behind quantum theory.

So if you apply this to classical mechanics, where we have point particles that can be at specific positions, then the quantum version of it requires us of course to consider superpositions of those positions: the famous "wavefunction" which gives us the complex weight of each combination of positions.

And now comes something important: while time evolution in classical mechanics is given by a second-order differential equation (Newton's equation), and we need hence, next to the actual "positions" of particles, also their "momentum" as part of the dynamical prescription, it turns out that time evolution in quantum theory is given by a first-order equation (the Schroedinger equation). The quantum state is hence sufficient, and we need no "momentum of quantum state" or something.

So this means that the quantum state (the specific superpositions of position!) must include in it the "dynamics", and hence whatever will be reduced to classical momentum. It is from this property that results that one cannot have a quantum state of a particle with a single position AND momentum ! Because that state with the given position would be one unique state, and there is no way to "include" different possible values of momenta. The "dynamics" follows namely from the exact superpositions of positions (the wavefunction). As such, in order to be able to have something which looks like any form of "momentum", one needs the superposition of many position states. But at that point, we cannot assign a single, unique position anymore to this state.

It is from this property that follows the complementarity of position and momentum: that momentum must be coded in the superpositions of positions, simply because quantum time evolution is first order in time, and classical dynamics is second order in time. The second "initial condition" which is free in classical dynamics must be coded in superpositions in the first-order initial condition (= the initial wavefunction) in quantum dynamics.

So yes, in a way, you can say that "classical momentum" is something that emerges from the dynamics of the superposition of position states in quantum theory.

 

[lightarrow]

I'm affraid that that's missing the fundamental idea of quantum theory entirely. The whole idea of quantum theory is that "things we observe as unique, can be in superpositions of several of them": it is the famous superposition principle.

Yes, but we usually describe a wave packet as a superposition of many "pure" sinusoidal waves so it would be more intuitive for me to see a pure sinusoidal wave as the fundamental object and the wave packet as the derived one.

An object's classical behaviour, at least in the sense of having precise values of position and momentum, should be, in my idea, the result of superposition of all particles wavefunctions' in a wave packet which has a precise group velocity, which is well spatially localized (resembling a Dirac's delta), and with very little spreading during its evolution (something like a "soliton"?). Don't know how this could be true, however.

If we observe things to have a unique position, then quantum theory tells us that it can be in superpositions of different positions. If we can observe spin to be up or down (along the z-axis), then quantum theory tells us it can be in superpositions of up and down. And that to each different complex combination of them, corresponds a different physical state, which can, in principle, be distinguished from another one by a DIFFERENT measurement.

THIS is the basic idea of quantum theory: what seems to be unique, can in fact occur in superpositions, and each of those is a different physical state (distinguishable observationally from any other, at least in principle).

Note that I leave in the middle whether this is ultimately TRUE or not. I'm just saying that this is the basic principle behind quantum theory.

So if you apply this to classical mechanics, where we have point particles that can be at specific positions, then the quantum version of it requires us of course to consider superpositions of those positions: the famous "wavefunction" which gives us the complex weight of each combination of positions.

And now comes something important: while time evolution in classical mechanics is given by a second-order differential equation (Newton's equation), and we need hence, next to the actual "positions" of particles, also their "momentum" as part of the dynamical prescription, it turns out that time evolution in quantum theory is given by a first-order equation (the Schroedinger equation). The quantum state is hence sufficient, and we need no "momentum of quantum state" or something.

So this means that the quantum state (the specific superpositions of position!) must include in it the "dynamics", and hence whatever will be reduced to classical momentum. It is from this property that results that one cannot have a quantum state of a particle with a single position AND momentum ! Because that state with the given position would be one unique state, and there is no way to "include" different possible values of momenta. The "dynamics" follows namely from the exact superpositions of positions (the wavefunction). As such, in order to be able to have something which looks like any form of "momentum", one needs the superposition of many position states. But at that point, we cannot assign a single, unique position anymore to this state.

It is from this property that follows the complementarity of position and momentum: that momentum must be coded in the superpositions of positions, simply because quantum time evolution is first order in time, and classical dynamics is second order in time. The second "initial condition" which is free in classical dynamics must be coded in superpositions in the first-order initial condition (= the initial wavefunction) in quantum dynamics.

So yes, in a way, you can say that "classical momentum" is something that emerges from the dynamics of the superposition of position states in quantum theory.

 

[Fra]

I think of the quantum vs classical world as the classical world beeing an ontological construct, and the quantum world describing the epistemological view. The state of information we have, does project several possible ontologies, all consistent with our state of information.

Regardless of mentioned problems, possible objections(I have many, as I'm sure others have) and various things to fix in quantum theory to make it even better, IMO the one most beautiful idea that is an excellent expression of something very close to a "scientific ideal" is that

Whatever anything "is" or "is not", this information is something that has to be acquired, and acquired by something.

This something is the physical observer.
This acquisition (information transfer) process is the physical measurement.

The idea is that the constraint of the mesurement process, does limit what one can meaningfully deduce about anything. The idea is that quantum mechanics deals with what we can measure. To say that x is this or that, makes little sense unless it can actually be measured.

This is indeed a very sound ideal and IMHO a gigantic philsophical improvement over classical ideals.

But still, closer analysis at the terms shows imperfections, some of which are ignored in QM as we currently know it... the real interesting thing is wether these issues are unrelated to the current problems in physics, QG and such, or not. And here there is disagreement too as it seems. I am one who think the last word in QM isn't on the table yet.

/Fredrik

 

[vanesch]

Yes, but we usually describe a wave packet as a superposition of many "pure" sinusoidal waves so it would be more intuitive for me to see a pure sinusoidal wave as the fundamental object and the wave packet as the derived one.

And why don't you see a wavepacket as a superposition of delta-functions ?

An object's classical behaviour, at least in the sense of having precise values of position and momentum, should be, in my idea, the result of superposition of all particles wavefunctions' in a wave packet which has a precise group velocity, which is well spatially localized (resembling a Dirac's delta), and with very little spreading during its evolution (something like a "soliton"?). Don't know how this could be true, however.

Well, it was one of the first ideas, and it is usually specified as an elementary exercise in quantum mechanics: consider a gaussian wavepacket and let it evolve under the unitary evolution of a free particle... you'll be disappointed by the result...

 

[siddharth]

It is clear that the concepts of position and momentum are idealizations from our macroscopic experience which are not appropriate in the quantum domain. Yet, in the presentations of elementary quantum mechanics I've seen so far, they are still used in a fundamental way. But obviously, they will be nothing like the position and momentum we know and therefore will seem "counterintuitive"

The standard presentations give the idea that objects in the quantum domain still have the properties of position and momentum except that they are nothing like the ones we know. I think this is a bad way of saying it and leads to a lot of confusion. A better way is to say that position and momentum are approximate concepts valid in the large scale which arise from more fundamental concepts/concept ( which shouldn't be called by the same names ) valid at all scales, as far as we know.

This is mainly expanding a small part of what Fra and Vanesch mentioned in their posts, but a comparison of the postulates of CM and QM should help in making clear the similarities and differences in concepts, including position & momentum, used in each formulation.

(i) Classical Mechanics

Here, a system is described by its dynamical variables,

$$q_1, q_2 , ...$$

$$p_1,p_2,...$$

which are canonically conjugate pairs and satisfy the poisson brackets algebra

$$\{q_i,p_j\} = \delta_{ij}$$
$$\{q_i,q_j\} = \{p_i,p_j\} = 0$$

The "dynamics" of the system comes in while describing the time evolution of these variables, by the existence of the Hamiltonian of the system [itex]H(p,q)[/tex] such that

$$\dot{q_i} = \frac{\partial H}{\partial p_i}, \dot{p_i} = - \frac{\partial H}{\partial q_i}$$

Along with the initial state of the system(ie, the initial values of p's and q's), this will describe how the variables evolve with time.

(ii) Quantum mechanics

In QM, the state of the system is described by a state vector (ex, $$\left| \psi(t) \right>$$) in a Hilbert space.

The dynamical variables (ie, p's and q's) which represent the generalized coordinates and momenta in classical mechanics, are "replaced" by self adjoint operators in QM (which act on the state vectors, and physically measurable quantities are related to the expectation values of these operators).

Similarly, the analogue to the possion bracket algebra in classical mechanics are the commutator brackets relations of operators in QM.

Finally, the time evolution of the state vector is given by Schrodinger equation,

$$i \hbar \frac{d}{dt} \left| \psi(t) \right> = H \left| \psi(t) \right>$$

where $$H$$ is now an operator in QM.

This is a first order differential equation. Together with the initial condition (ie, the initial state $$\left| \psi(0) \right>$$) this will describe how the state vector evolves with time.

If you'd like a reference, try "Principles of Quantum Mechanics" by Shankar, where he presents a detailed discussion on the postulates of QM, and how they are related to the classical formulation.

 

[lightarrow]

And why don't you see a wavepacket as a superposition of delta-functions ?

Because those delta functions refers to states which have exact momentum but completely indeterminate position or the other way around. Classical states are different because they have (almost) exact momentum and position.

Well, it was one of the first ideas, and it is usually specified as an elementary exercise in quantum mechanics: consider a gaussian wavepacket and let it evolve under the unitary evolution of a free particle... you'll be disappointed by the result...

More than disappointed I was amazed (it was 1988)
However a gaussian wavepacket can't obviously be the solution; the quantum description of a classical object like a 1 cm steel ball must be different.

 

[muppet]

This is exactly what I'm questioning. We are trying to extend our classical concepts to the quantum domain, just like we extended the classical concepts of space separation and time separation between events into the relativistic domain with the lorentz transform.
...
Doesnt this seem unnatural? The classical states of exact position and exact momentum are approximations to something deeper, yet we try to describe the thing that is deeper using these very concepts in an altered form.

This is Bohr's reasoning. His belief was that nature's ontology was unknowable. He argued that all quantum theory had to do was provide predictions for measurements of these concepts from classical physics, and that we cannot remove these concepts from physics; but that this did not mean that these concepts had a meaning in the microscopic domain.

The most satisfactory understanding of why we can't measure simultaneously position and momentum is by noting that any attempt at measurement of one affects the value of the other. It is impossible in principle. This hints at a deeper level where the concepts themselves have no meaning.

The first part of this isn't quite true. Heisenberg preferred to call the uncertainty principle the "principle of indeterminacy". He believed that reality does not posess well-defined values of a quantity prior to a measurement of that quantity. He considered that all that existed prior to measurement was a "field of potentialities" that somehow condensed into a particle (say) upon measurement in accordance with the probabilistic laws of QM.

 

[vanesch]

Because those delta functions refers to states which have exact momentum but completely indeterminate position or the other way around. Classical states are different because they have (almost) exact momentum and position.

I just mentionned delta-functions because you saw as "fundamental" components "sine waves" (eg. momentum states), and I wanted to show you that there's nothing fundamental to sine waves, and that you can have your wavepacket also as a superposition of deltafunctions (position states), or Haar functions, or whatever... and vice versa! That you can see deltafunctions, or sine functions ... as superpositions of wavepackets.

That's exactly what it means, to be a vector in Hilbert space: you can expand it in any basis you like, and it can be a basis vector itself if you like.

The "resolution" to the riddle of the expansion of the gaussian wave packet is of course that in reality, particles are not free, and they interact with their environment (charged particles can't avoid interacting with the EM field for instance). Under this interaction with the environment, one can obtain "coherent states" which remain relatively stable in spatial as well as momentum spread - it is one of the results of decoherence theory.

 

[lightarrow]

The "resolution" to the riddle of the expansion of the gaussian wave packet is of course that in reality, particles are not free, and they interact with their environment (charged particles can't avoid interacting with the EM field for instance). Under this interaction with the environment, one can obtain "coherent states" which remain relatively stable in spatial as well as momentum spread - it is one of the results of decoherence theory.

It's the same for an object like the one I wrote (a 1 cm steel ball)?

 

[reilly]

The real reason that we end up with non-commuting variables in QM, q and p, is due to various experiments -- particularly the ins and outs of atomic spectra, and the Davisson Germer experiment. Can you imagine what it must have been like to observe that electrons actually diffracted while going through a crystal? QM was born to explain such things.

So, two threads of ideas, grew in response to Nature's weirdness. One that started with E&M waves, and then the whole Hamiltonian-Contact Transformation(HCT) approach of classical mechanics ultimately steered the founders toward modern QM. When formulated with appropriate contact transformations, classical mechanics has an uncanny similarity with QM and unitary transformations, including a first order DE for evolution in time. E&M waves brought the idea of superposition, and a fundamental explanation of diffraction and interference. And, in a sense the killer app., both approaches provide a similar picture of optic's eikonal rays and mechanic's rays associated with Hamilton's Principle Function. That is, Hamilton demonstrated a very close relationship between optical rays and mechanical paths. See Lanczos for an extensive discussion of these similarities.

QM was created largely with an intuitive approach, and from suggestions from classical physics. Axioms and rigor have come after the fact. The actual development was very untidy. If there is some more fundamental theory, it has yet to be found. We use "p and q" because nobody has come up with anything better.

Note: we routinely use superposition in classical mechanics. Certainly superposition plays a huge role in small oscillation problems. And, for that matter, I can always work with Newton's Eq. in a Fourier transformed space -- works for small oscillations, but would be tedious for other types of problems.

Regards,
Reilly Atkinson

 

[lightarrow]

"We use "p and q" because nobody has come up with anything better."

Is it possible to use Lorentz interval s and phase f of the wave, at least in one dimension x, only, to write x and t as functions of s and f (which are invariant)? Naively I would write something like:

df = kdx - wdt; w = frequency
ds^2 = (cdt)^2 - (dx)^2

and then find dx and dt as functions of ds and df.

 

[vanesch]

It's the same for an object like the one I wrote (a 1 cm steel ball)?

A 1 cm steel ball is hardly to be described by a 1-particle wavefunction. Well, the COG can be eventually considered that way, but the geometrical extension (1 cm) has nothing to do with a wave packet spread of about 1 cm! It is 1 cm big, because consisting of many many atoms, which each have their quantum description.

There's a world of difference between a STRUCTURE of many particles, which ends up having a spatial occupation of 1 cm, and a SINGLE point particle having a wavepacket of 1 cm wide.

 

[lightarrow]

A 1 cm steel ball is hardly to be described by a 1-particle wavefunction. Well, the COG can be eventually considered that way, but the geometrical extension (1 cm) has nothing to do with a wave packet spread of about 1 cm! It is 1 cm big, because consisting of many many atoms, which each have their quantum description.

There's a world of difference between a STRUCTURE of many particles, which ends up having a spatial occupation of 1 cm, and a SINGLE point particle having a wavepacket of 1 cm wide.

Sorry, I was probably not clear, I didn't mean to compare the ball's dimension with the wavepacket's spreading; I simply intended to ask how to describe the wavepacket of a macroscopic classical object like that ball; I didn't intentionally use the word "macroscopic" or "classical" just because there exists many macroscopic systems which have a quantistic behaviour and microscopic systems which have a classical one.

 

[reilly]

It should be clear that the atoms/molecules in the ball are in a bound state. If you use a spherical well coinciding with the ball, then you are virtually home free in determining the wave function -- forget atom-atom interactions. A crude model to be sure, but it will capture the basic structure of the interior and provide a physically reasonable wave function.
Regards,
Reilly Atkinson

Sorry, I was probably not clear, I didn't mean to compare the ball's dimension with the wavepacket's spreading; I simply intended to ask how to describe the wavepacket of a macroscopic classical object like that ball; I didn't intentionally use the word "macroscopic" or "classical" just because there exists many macroscopic systems which have a quantistic behaviour and microscopic systems which have a classical one.