What is light a wave in according to quantum mechanics?

[LeonhardEuler]

I have only taken an introductory course on QM so far, and a lot of it doesn't make sense to me. One thing that confuses me is what light is. In classical physics it was a wave consisting of oscilating electric and magnetic fields. In QM, I keep hearing about "wave-particle duality". My question is this: when light is a wave, what is it a wave in.

Here is why this confuses me: Correct me if I'm wrong, but I don't think it makes sense to speak of a "force" in QM, at least not one resembling a classical force because there really is no position in QM, so there can't be a second time derivative of position, and what I always (incorrectly?) took as the definition of force was $\sum\vec{F}=m\vec{a}$. If there is no force there can be no force field. If there is no force field, then there is no wave in the force field. Usually in QM equations you have a "potential" instead of a force. I was thinking maybe light was a wave in electric and magnetic potential, but then I remembered Faraday's and Ampere's law. $\oint\vec{E}\cdot ds$ and $\oint\vec{B}\cdot ds$ are not necessarily zero around closed loops, so you can't speak of a potential without speaking about a particular path! And yet, light must clearly have some electromagnetic nature because it can push charges around in a circuit, for example in a radio (I'm using the word "light" in a more general sense than visible light). So what is light a wave in, if not $\vec{E}$ and $\vec{B}$ fields. And if it is not a wave in these how does it move charges. And if it is a wave in these, how does it get around the force problem? Thank you for reading all this!

 

[EddieZ]

I'll take a crack at this, but I'm sure you'll get some better answers soon ;)

The question of what is light a wave in is a good question. First I'll address the classical view. The electric and magnetic fields of a light wave are what is waving so to speak. If you look at a picture of an electromagnetic wave, you will see a depiction of two sine waves, out of phase with each other, one for the magnetic field and one for the electric field. The plane of oscillation of each of the waves at each point is always perpendicular. What is actually happening is that there is an oscillation of energy between the electric field and the magnetic field. At the point in the trajectory where the magnetic field is at a maximum the electric field is zero (essentially), and when the electric field is at a peak, the magnetic field is zero. In between these points, the electric field is waning while the magnetic field is increasing (and so on). So, the energy in the electromagnetic field is bouncing back and forth between being fully expressed in a magnetic field and an electric field, and as each works its way through a cycle, it generates the other, and this process repeats itself at whatever the frequency of the wave (light) is.

I'll invite someone more knowledgeable to comment on how the magnetic and electric potentials fit into the process.

Regarding the wave-particle duality, I'm sure there are a number of threads on that in this forum already, so I won't expand on it here, unless someone would like to.

 

[neurocomp2003]

what is a E & B field =]

 

[Pengwuino]

what is a E & B field =]

Electrical and Magnetic fields

 

[neurocomp2003]

mmm ok so if i typed it out you would have replied E and B fields?

 

[Pengwuino]

mmm ok so if i typed it out you would have replied E and B fields?

yes. I think M already has a designation... I think that's why we use B.

 

[CarlB]

Here is why this confuses me: Correct me if I'm wrong, but I don't think it makes sense to speak of a "force" in QM, at least not one resembling a classical force because there really is no position in QM, so there can't be a second time derivative of position, and what I always (incorrectly?) took as the definition of force was $\sum\vec{F}=m\vec{a}$. If there is no force there can be no force field.

It's true that quantum mechanics deals with a different representation of the object, one that doesn't require a particular position. But even with classical mechanics objects that are described without a specific position can nevertheless have forces applied to them. For example, a gas doesn't have a particular position, but gases exert forces have masses and can have forces exerted upon them.

Usually in QM equations you have a "potential" instead of a force. I was thinking maybe light was a wave in electric and magnetic potential, but then I remembered Faraday's and Ampere's law. $\oint\vec{E}\cdot ds$ and $\oint\vec{B}\cdot ds$ are not necessarily zero around closed loops, so you can't speak of a potential without speaking about a particular path!

First, for scalars, one can take the gradient of the potentials used in QM to get a force. However, this is not done for the same reason that the corresponding classical situation also uses potential energy instead of forces. It's an easier problem to solve with potential energy, especially if you can write the potential energy as a scalar. That is, the force is a vector so it is naturally more complicated to work with than a scalar.

For the case of photons, one can analyze a problem in several different ways. Since my education is in elementary particles, I naturally prefer to analyze photons as elementary particles. In the elementary particle point of view, there is no electricity and magnetism, just elementary particle interactions.

In elementary particles, light takes the form of photons. These are excitations of the vacuum that are like any other particle. Forces are conveyed between particles by the exchange of other particles. In the case of electrons interacting by E&M, the force is mediated by the exchange of photons. Uh, photons can also have forces applied to them, for example by the exchange of electron / positron pairs or the like.


As an example of the simplicity of the elementary particle version of physics, as opposed to the condensed matter method of dealing with E&M, consider the Aharonov-Bohm effect. In QM, this is a mysterious thing wherein an electron is influenced even though it travels only in regions with zero magnetic and electric fields. The electron's wave function is modified and that changes its interference with itself. A big mystery and a subtle part of QM.

Looked at from the elementary particle interaction point of view, the electron travels through regions that are stuffed to near overflowing with photons, they just happen to cancel their interactions in a way that eliminates classical effects but leaves quantum effects. [Classically, we say that the magnetic field is "shielded", but that does not mean that no photons are exchanged in the elementary particle sense. Photons are excitations of the electric and magnetic potentials, not the E&M fields themselves. Instead, "shielded" only means that the photons are arranged so that there is no net classical effect. In the A-B effect, the E&M fields are shielded, but their potentials are not.] The quantum effect, naturally, changes the phase of the electron and hence the Aharonov-Bohm effect. No mystery at all.

Carl

 

[Nicky]

[...] I don't think it makes sense to speak of a "force" in QM, at least not one resembling a classical force because there really is no position in QM, so there can't be a second time derivative of position, and what I always (incorrectly?) took as the definition of force was $\sum\vec{F}=m\vec{a}$. [...]

My opinion is that you have hit upon one aspect of the so-called "macro-objectification problem", which is still an active area of debate and research in physics. Macroscopic objects have a definite position, thus we can easily speak of forces and the E and B fields of the Maxwell equations. However, microsopic systems don't have definite position, only wavefunctions. In the wavefunction picture, the E or B you observe depends on how the system is measured, so it is hard to talk about the physical reality of E or B independent of the measurement.

The unsolved problem is, how do you choose which systems are microscopic and which are macroscopic? That choice is left rather fuzzy in the standard quantum mechanics formalism, though it has been good enough so far to produce calculations that agree amazingly well with experiment. Maybe that is because most of the things we know how to study with current technology are either very microscopic (molecules, atoms, particles) or very macroscopic (metal spheres, insulating cylinders, etc.)

Here is one review of the macro-objectification problem that I am trying to read: http://arxiv.org/quant-ph/0302164

Caveat: I am still learning quantum mechanics, so the above views may be quite naive. Those of you who are more experience in the field, please correct any gross misunderstandings.

 

[neurocomp2003]

what are fields? mathematical object or have physicality?

 

[εllipse]

My question is this: when light is a wave, what is it a wave in.

The other replies haven't specifically addressed this, so I thought I would. The wave in a wave-particle duality is a wave of prabability. According to the Copenhagen interpretation, when a photon isn't being measured it exists as a wave, and when it is measured it is a particle. The wave determines the probability of where the photon could be at any given time. In this interpretation, there isn't a physical material (or field) that "waves". What waves is the probability of where the photon will be when observed.

 

[nabodit]

The other replies haven't specifically addressed this, so I thought I would. The wave in a wave-particle duality is a wave of prabability. According to the Copenhagen interpretation, when a photon isn't being measured it exists as a wave, and when it is measured it is a particle. The wave determines the probability of where the photon could be at any given time. In this interpretation, there isn't a physical material (or field) that "waves". What waves is the probability of where the photon will be when observed.

is that supposed to mean that a photon comes to existence out of probability only when measured. thus a photon doesn't exist until u want it to do?
some one please explain me.

 

[selfAdjoint]

is that supposed to mean that a photon comes to existence out of probability only when measured. thus a photon doesn't exist until u want it to do?
some one please explain me.

Well, that's what happens if you push the formalism to be "a description of reality"; "The only photons that exist are the ones we see." But if you interpret the formalism to define what we can know about reality, then it becomes "The only photons we are concerned with are the ones we can see."

 

[Antiphon]

what are fields? mathematical object or have physicality?

I think mathematical object is closer to the truth.

is that supposed to mean that a photon comes to existence out of probability only when measured. thus a photon doesn't exist until u want it to do?
some one please explain me.

Something exists objectively that will manifest various aspects of photonic
behavior when you interact with it.

As a mere Human Being, you cannot not "create" things merely by willing them to exist.

 

[LeonhardEuler]

The other replies haven't specifically addressed this, so I thought I would. The wave in a wave-particle duality is a wave of prabability. According to the Copenhagen interpretation, when a photon isn't being measured it exists as a wave, and when it is measured it is a particle. The wave determines the probability of where the photon could be at any given time. In this interpretation, there isn't a physical material (or field) that "waves". What waves is the probability of where the photon will be when observed.

If light is a wave in the probability of its own position, then by what mechanism does it interact with charges? Is there also some electromagnetic characteristic to it?

 

[LeonhardEuler]

First, for scalars, one can take the gradient of the potentials used in QM to get a force. However, this is not done for the same reason that the corresponding classical situation also uses potential energy instead of forces. It's an easier problem to solve with potential energy, especially if you can write the potential energy as a scalar. That is, the force is a vector so it is naturally more complicated to work with than a scalar.
Carl

You could take the gradient of a potential, but what would it mean? For example, if you took the gradient of the electric potential due to the nucleus of an atom and got a vector field, what would this vector field mean for an electron around the nucleus? Since the electron does not have a position, none of those vectors is really any more meaningful to the electron than another. You can't say the force on the electron is such and such, because the force varies over space. You could define force as the gradient of the potential, but would it mean anything?

 

[CarlB]

Since the electron does not have a position, none of those vectors is really any more meaningful to the electron than another. You can't say the force on the electron is such and such, because the force varies over space. You could define force as the gradient of the potential, but would it mean anything?

First, I hope I never said that one could so define "the force on the electron", but instead simply said that by using the gradient this is how one can define the gradient. Second, I can't understand your logic here. It seems like you're saying that since electrons don't have specific positions, they cannot understand forces. It seems to me that the same argument would also apply to potentials, which also depend on position.

Third, classical mechanics already faced the problem of dealing with extended objects. It's called "continuum mechanics" and it includes the concept of forces being applied to objects that are not defined as having a single specific position. Naturally, we assume that what is really going on is a collection of atoms, but that was not what was assumed in the original classical work on the subject.

A good example of a classical object with an extended position and a force field or potential is the thermodynamics of a gas in the presence of a gravitational potential. This was worked out in the mid 19th century without recourse to atoms. As late as the 1890s, Mach, who was so influential on relativity and QM, was still saying "I do not believe in atoms". He felt that atoms should be disregarded since they were not (at that time) observable.

I recall a conversation I had with a professor of physics many years ago, when I was in my 3rd year of graduate school. I told him that I didn't understand what a force really was. He (jokingly) suggested that perhaps I shouldn't be studying physics. My guess is that you feel much the same way.

For a long time it has been the attitude of most physicists to, as Feynman said, "shut up and calculate". I believe that there are better ways of explaining the nature of forces, but here is not the place to discuss them. When discussing physics with people who don't understand the standard manner of looking at it, we should stick to the standard interpretations. Anything else is like giving condoms to kindergartners.

Carl

 

[LeonhardEuler]

Second, I can't understand your logic here. It seems like you're saying that since electrons don't have specific positions, they cannot understand forces. It seems to me that the same argument would also apply to potentials, which also depend on position.
Carl

What I'm saying is that the force has no meaning. Shrodinger's equation gives meaning to potential because it describes its effect on the wave function. A particle with no definite position and one that is spread out in space are two different things. It is easy to define what a force is on an object that is spread out in space. For example, if you have an electric force that is acting over a body spread over a volume V, with charge density $\rho$ then the force on the whole body will be
$$\iiint \vec{E}(x,y,z)\rho (x,y,z)dxdydz$$
But if you have an object with no definite position this doesn't work. I understand that you can compute the gradient of the potential, but does it have any physical significance?

 

[Crosson]

then by what mechanism does it {light} interact with charges? Is there also some electromagnetic characteristic to it?

Photons are the carriers of the electromagnetic force. Light doesn't interact with charges, it is more correct to say that charges interact through light.

 

[CarlB]

I understand that you can compute the gradient of the potential, but does it have any physical significance?

The Schroedinger's equation is of the form:

$$\frac{d}{dt} \psi(x,t) = \grad \psi + V \psi$$

where I hope you will forgive me for typing from memory and not including $$\hbar, m, i$$. This is the form that enters into the Copenhagen interpretation and sure enough, it uses a potential, not a force.

In the Bohmian interpretation of QM (which is the subject of a group at the University of Munich:
http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/bmstartE.htm )
one reinterprets the wave function as a "guiding" wave for the particle, and gives the particle a particular position. In order to do this, one has to add a "quantum potential", which accounts for the nonlocal effects. In other words, in Bohmian mechanics, which has exactly the same results as standard quantum mechanics (but which has not yet been made to reproduce QFT, to the best of my knowledge), the electron does possesses a particular position. Thus in Bohmian mechanics, the force on the electron does make sense according to your requirements, and naturally enough, the calculated force on the electron works out to be exactly the gradient of the potential.

I should note that the above link, which gives the modern version of Bohmian mechanics for multiple particles, does not explicitly show that the force is given by the gradient of the potential. To see that it is, you can look in the first few pages of chapter 3 of Hiley and Bohm's book on the subject:


The above book uses a substitution (also found in Chapter I of Messiah in the discussion of the BKW approximation if I recall) for the complex valued wave function in terms of its magnitude and phase. I can't recall if I'm using Hiley's or Messiah's terms, but the substitution is something like this (ignoring h-bar):

$$\psi(x,t) = R(x,t) e^{iS(x,t)}$$

where psi is the usual complex valued wave function, R squared gives the probability density and S gives something that looks like a potential for a velocity field. In other words, $$\grad S(x,t)$$ is a velocity field. The electron paths used in Bohmian mechanics take their velocities from this field. In the mathematical sense they are orbits in this field.

Another interesting ontological reinterpretation of quantum mechanics is a statistical one. It's called the "stochastic interpretation" and it is orginally due to Nelson and was extended by Fenyes. In it, one substitutes:

$$\psi(x,t) = e^{U(x,t)} e^{iS(x,t)}$$

I believe that the stochastic interpretation has a smaller following than Bohmian mechanics, but I don't know why this would be the case; I prefer the stochastic mechanical interpretation. For examples of stochastic interpretation papers see:
http://www.iop.org/EJ/abstract/0305-4470/17/1/015
http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:quant-ph/0112063
http://arxiv.org/PS_cache/quant-ph/pdf/0112/0112063.pdf

This substitution has the advantage of giving a natural statistical mechanics interpretation of QM. That is, instead of the probability density being the ontological object, one instead has a field of energies, with h-bar corresponding to a sort of temperature.

By the way, I managed to survive many years of grad school without any of these alternative interpetations even coming up in conversation, so they really are not standard theories. But what I'm getting at with these alternative interpretations of quantum mechanics is that it is not all that obvious which of the things are fundamental and which are just the results of mathematical manipulation. I believe that finding fundamental (or ontological) variables in quantum mechanics is like trying to perform celestial navigation in a house of mirrors. Uh, that doesn't stop me from trying.

I've been interested in the details of these alternative interpretations, and the difference between potential energies and velocities for some time. For the appearance of the force in the Bohmian mechanics, if you can't get Bohm and Hiley's book, then see equation (20) from my obsolete write-up:
http://brannenworks.com/MsrEther.html

I'm now working on a paper that allows one to calculate the masses of the leptons based on principles that are essentially identical to those used in stochastic mechanics, but which are based on Clifford algebraic numbers instead of the usual complex numbers. It turns out that this makes the whole concept of exponentials more mathematically entertaining. Of course the standard model puts the masses of the leptons in as adjustable constants. The derivation is a thing of great beauty in that it really makes a good case for considering statistical mechanics as the underlying theory for quantum mechanics, at least in my opinion. For the arithmetic involved in the formula (but not the rather difficult derivation) see page 8 of:
http://arxiv.org/abs/hep-ph/0505220

Carl

 

[Eppur si muove]

A force in itself has no particular meaning but is rather defined by its effects. Thus what is usually meant by the magnitude of a force is really a rate, a change of momentum with time (or a derivative of momentum with respect to time). Hence, it is irrelevant whether particle-wave's position is fixed or not - as long as it undergoes a change in momentum, there is a force acting on the particle-wave.

 

[reilly]

LeonhardEuler --

I suggest that you take a peek at basic probability; it does what you say can't be done re things that do not have a definite position.

Checkout the Heisenberg equations of motion for a charged particle in an electromagnetic field; they involve the Lorentz force -- as an operator, that can be defined, in fact, over all space. Force is alive in QM, but not so well, if well=useful.

Review the mechanics of assemblages of particles, particularly with respect to forces.

Regards,
Reilly Atkinson

 

[LeonhardEuler]

Thank you everyone for your posts, I think I am coming closer to understanding this. Eppur si muove brought up an iteresting point about defining force as the derivative of momentum. This seems to be a way around the problems that came along with my assumption, which I doubted from the beginning anyway: "I always assumed (Incorrectly?) that F=ma was the definition of force". It gets around this problem because momentum is not defined as mass times the derivative of position, which would be meaningless, but is defined in terms of the properties of the wavefunction. However, when it comes to reilly's claim, I am troubled.

they involve the Lorentz force -- as an operator, that can be defined, in fact, over all space

I do not see how a force can be defined over all space. A force has been defined as the derivative of the momentum, so in order for a force to exist, so must a momentum. How can the force then be defined over empty space? This is not like an electric field where the definition is the force that a particle would feel if it were in that position. And I can't assume that he is talking about something like this kind of field of the force a particle would feel in a certain position because, as pointed out earlier, a quantum particle does not have a definite position [unless we are talking about the Bohmian interpretation, as CarlB pointed out], and so the force it would feel in a definite position is meaningless. It doesn't seem legal to do an integral simmilar to the one for continuum mechanics, except with density replaced by probability density, because (I think) the the interpertation is that the particle is always here or there, not part here and part there. (I'm not suggesting that I believe these conclusions to be true, I'm just stating my thinking so that it can be critiqued)

Thank you CarlB for your post, which I found particularly informative. Earlier you mentioned that a force could be defined as the gradient of a potential. Now that the Bohmian interpertation gives a method for computing with such a field it seems less unimportant, even if the interpertation is non-standard. Above I was talking about a definition of force as the derivative of the momentum. Are these two definitions equivalent to one another? It seems like it would be a useful relation to have that the gradient of the potential is the derivative of the momentum because, if your only looking for the momentum, it reduces the partial differential equation of Shrodinger's equation to an ordinary one.

One last question about the electromagnetic nature of light. Crosson suggested that photons carry the EM force between particles. This is very interesting and confusing to me. Does this mean that it communicates changes in the EM force between locations, as in classical mechnics(except with the word photon replaced with EM waves), or is this actually to say that in order, for example, for a nucleus to exert a potential on an electron it must shoot a constant stream of photons? I am much more inclined to believe that it is the first case because of numerous problems I see with the second, but I have worries about the first, too. If photons are discrete particles, how can they communicate this change to every point in space when a charge moves? wouldn't this require infinitely many photons? Thanks everybody. Unfortunately I have to go to bed now, so I won't be able to reply for a long time, but it gives you more time to think, which is good!

 

[CarlB]

I do not see how a force can be defined over all space. A force has been defined as the derivative of the momentum, so in order for a force to exist, so must a momentum. How can the force then be defined over empty space?

I agree that in a certain way, a force cannot exist without the thing that it is applied to. It doesn't bother me to call the thing a force even when it's not applied to an object, but if it does bother you, than you can call it something else, for example, a "potential force". To produce such a "potential force" field, one can imagine, for example, physics being conducted in the midst of a moving fluid. The fluid induces a force on anything you put in it. One can imagine the force as existing even without an object indicating it.

There is another reason for preferring a force field over a potential but it's kind of subtle. Potentials are always relative. That means that if you take a situation and change the potential everywhere by increasing it by some constant, the result is equivalent to the original situation. This means that if potentials are the ontological reality, then they must have built into them some sort of arbitrariness. But arbitrariness reeks of mathematical artifact.

One can take a quantum mechanical problem defined with a potential energy and redefine the potential energy by adjusting it up or down by a constant. While the result will give the same calculations for all observable measurements, it is not the same situation. In particular, changing the overall potential has the effect of modifying the energies of the particles, and this changes the frequencies of their wave functions. The transformation of a problem by changing the basis of the potential energy amounts to a sort of "gauge transformation", and it is discussed in section 2.6 of Sakurai's classic undergraduate text on QM:
http://theory.itp.ucsb.edu/~doug/phys215/ss/


By the way, I should note that Sakurai's book includes a discussion of the interaction between gravity and quantum mechanics that is in violent disagreement with my understanding of QM and relativity in that it attributes to quantum mechanics, rather than standard relativity theory, the concept that gravitational potential will effect clocks. That is, his book claims a QM effect by performing a calculation that ignores relativistic effects. But that's another story.

So if one desires a potential energy to be the ontological element of reality, then one must define the zero potential. This is kind of ugly. By contrast, if one assumes instead that the force fields are the ontological item, then forces can be defined in a purely local manner that is immune to the gauge transformations associated with changes in potential.

Also, I should mention that the option of making arbitrary potential energy changes is present only in the nonrelativistic quantum theory. In relativity, there is a natural definition for the potential energy of the object and so one can't consider these sorts of gauge changes.

It's kind of interesting that the gauge freedom that is present in non relativistic quantum mechanics as described by Sakurai disappears when one converts to a relativistic theory. This suggests that the gauge freedom that is present in E&M and in the gauge theories of particle theory will also disappear when put into a more unified form.

Thank you CarlB for your post, which I found particularly informative. Earlier you mentioned that a force could be defined as the gradient of a potential. Now that the Bohmian interpertation gives a method for computing with such a field it seems less unimportant, even if the interpertation is non-standard. Above I was talking about a definition of force as the derivative of the momentum. Are these two definitions equivalent to one another?

I guess they're sort of equivalent, but...

It seems like it would be a useful relation to have that the gradient of the potential is the derivative of the momentum because, if you're only looking for the momentum, it reduces the partial differential equation of Shrodinger's equation to an ordinary one.

Unfortunately, this won't be true because even in the absence of any potential, a particle will still possesses local momentum that can change with time. For example, if one begins with a particle localized in free space around the origin (say by a Gaussian), and if one examines the momentum density at some point away from the origin, one will find that the momentum is directed away from the origin. Another way of putting this is to note that free particles tend to spread out.

The conversion to ordinary DEs is especially interesting in the case of the Dirac equation, but that's a subject for another day. Oh what the heck.

First, note that modern particle theories assume that mass is a derived attribute of particles. That is, it takes left and right handed particles to make a single massive particle, and if you turn off the mass interaction (i.e. the Higgs), you end up with massless Dirac equations for the two halves. But if the massless Dirac equation:

$$\gamma^\mu \partial_\mu \psi = 0$$

is true, then so is:

$$\gamma^\mu \partial_\mu e^\psi = 0$$

In other words, modern particle theory can only distinguish between fields and their exponents (or logarithms) by considering the probability interpretation of the field. That is, $$|\psi|^2$$ is assumed to be a probability density, which makes it impossible for $$|e^\psi|^2$$ to also be such. On the other hands, statistical mechanics tells us that probabilities should be given by

$$p = e^{-\frac{kE}{T}}$$

This all suggests that the natural choice for the ontological element corresponding to an electron wave function should be $$ln(\psi)$$ rather than the usual $$\psi$$. That is, the ontological function should be something that acts like an energy which exponentiates to a probability rather than a probability itself. This makes Plank's constant into a sort of Boltzmann constant. And since the massless Dirac equation (which defines a natural equation for a wave function satisfying relativity) is satisfied by both $$\psi$$ and $$\pm ln(\psi)$$, one should assume that it is $$-ln(\psi)$$ that is the ontological object.

Stochastic mechanics is related to what I am talking about here, but I do not think that they put it in the terms I have.

Carl

 

[reilly]

LeonardEuler writes:

I do not see how a force can be defined over all space. A force has been defined as the derivative of the momentum, so in order for a force to exist, so must a momentum. How can the force then be defined over empty space? This is not like an electric field where the definition is the force that a particle would feel if it were in that position. And I can't assume that he is talking about something like this kind of field of the force a particle would feel in a certain position because, as pointed out earlier, a quantum particle does not have a definite position [unless we are talking about the Bohmian interpretation, as CarlB pointed out], and so the force it would feel in a definite position is meaningless. It doesn't seem legal to do an integral simmilar to the one for continuum mechanics, except with density replaced by probability density, because (I think) the the interpertation is that the particle is always here or there, not part here and part there. (I'm not suggesting that I believe these conclusions to be true, I'm just stating my thinking so that it can be critiqued)
>>>>>>>>>>>>>>>>
Your concerns deal with the basics of physics -- forces, fields, mass and probabilities, and so on. These are admirably discussed in many textbooks and treatises. Before you can "get" physics you need to have the basic concepts in mind. For example: from the beginning: we do things like, W(x) defines a potential, over all space -- maybe 1/r --. Then W is, in fact a field. force is -GRAD W(x), again a field. Note that the Lorentz force includes the electric force, qE, q is charge, E is the electric field.

Where is it written that to have force, you must have momentum? Is this why you incorrectly state that force is defined by the derivative of the momentum?

Where is it written that the mass density of continuum is not the same, generically, as a probability density. (In fact, as measure theory indicates, mass density, charge density, probability density are fundamentally the same concept. This has been common knowledge for over a century in math and physics.)

Read.

Regards,
Reilly Atkinson

 

[LeonhardEuler]

I found Sakurai's book for $10. I hope that reading it will clear up many of the confusions I still have. I'm not asking anybody to respond to these questions if the answers are long, but I would just like to state what my difficulties are in case any of them can easily be cleared up quickly. First of all, which equations from classical mechanics are still valid and which are not? For example, the potential from a point charge appears to be computed from Gauss's Law. But are all of Maxwell's equations still valid? For example, is $\oint \vec{E} \cdot ds$ still equal to the derivative of the magnetic flux? If this is the case, how does one define an electric potential when there is path dependance? How exactly do photons interact with electric charges? Does a wave of electric and magnetic force still accompany the photon. What effect exactly does a force have upon a particle? How do photons bring information about changes in electric forces to all points in space when they are only discrete particles that go in some direction? Hopefully the book will answer the questions soon, but it is not coming for 12 days and I may have to wait into the fall semester when I take a more advanced linear algebra course to make progress in it, so if any of these questions has a quick answer it would be helpful, but if not I can wait. Thanks for all of your time.

 

[CarlB]

I found Sakurai's book for $10. I hope that reading it will clear up many of the confusions I still have.

I doubt that Sakurai will clear up much of the confusion you have. In fact, I don't think that you're actually "confused". What physics largely consists of is "shut up and calculate". The questions you're asking are ontological in nature and are not addressed much in a standard physics education.

First of all, which equations from classical mechanics are still valid and which are not?

By "valid", I suppose that you mean are exact and accurate in terms of an ontological standpoint. I would think that there are no classical equations that survive the transformation unaltered.

For example, the potential from a point charge appears to be computed from Gauss's Law. But are all of Maxwell's equations still valid? For example, is $\oint \vec{E} \cdot ds$ still equal to the derivative of the magnetic flux?

I believe that at the level of Sakurai, these will still be true. However, QFT treats photons themselves as particles so Maxwell's equations become only an approximation that is the result of huge numbers of photons interacting with charged particles.

If this is the case, how does one define an electric potential when there is path dependance? How exactly do photons interact with electric charges? Does a wave of electric and magnetic force still accompany the photon.

If I recall, Sakurai will treat E&M as a force or potential that is applied to the various particles. This approach will be abandoned when one looks at things from a field theory point of view. In field theory, charged particles emit and absorb photons. By the use of an "action", one associates a complex number with each possible way that this sort of thing can happen. For any particular interaction, for example, electron A emits photon B with energy C and polarization D at point E, which photon is absorbed by electron F at point G, there is associated a complex number. One doesn't require integration to find this complex number.

But when one let's the energy C, point E and point G vary over the possibilities, one ends up with a lot of different contributions to the action. One adds up these possibilites by setting up an integral. The integral is represented by a "Feynman diagram". Different Feynman diagrams imply different relationships between the various parts. For example, the above situation used only a single photon. A situation that involved exchanging two photons would correspond to a different Feynman diagram and a different integral. With two photons, there would be a lot more degrees of freedom, so the integral will be longer and more difficult to solve.

For an excellent introduction to QED, written at a very basic introductory level but containing a very useful explanation even for graduate students, see Feynman's remarkable book:
QED, The Strange Theory of Light and Matter, Richard P. Feynman


The above is available for $11 on Amazon and should be in the library of every physicist, if only for lending to friends.

How do photons bring information about changes in electric forces to all points in space when they are only discrete particles that go in some direction?

This cuts to the issue of wave / particle duality. From the point of view of the wave, the photon is not necessarily traveling in a specific direction. One can only assign a particular direction to a particular photon after the measurement (if then).

Hopefully the book will answer the questions soon, but it is not coming for 12 days and I may have to wait into the fall semester when I take a more advanced linear algebra course to make progress in it, so if any of these questions has a quick answer it would be helpful, but if not I can wait. Thanks for all of your time.

Take a look at the Feynman book, which avoids any use of mathematics whatsoever, yet explains better than I how QED works. You will find it a lot easier reading, but with better explanations. And hey, congratulations on getting Sakurai so cheap!

Carl