photon wave-functions?
(Greg Weeks)
assumed that the photon has a wave-function. Even in free field theory, I
don't believe this is true.
Certainly, there are momentum eigenstates. But to obtain position
eigenstates, you need a natural way to fix the phases of the momentum
eigenstates; and I don't believe there is one.
There are two *classical* fields associated with a single photon state |s>.
But I don't see how they are useful. (And I don't see how a classical
field can be viewed as a wave-function anyway.) One is the expectation
value of the A-field:
<s| A(x) |s>
But this is just zero (assuming free fields). The other is obtained by
replacing a(k) in the plane-wave decomposition of the classical field with:
<k|s>
I don't see that this classical field *means* anything physically.
What, then, is a photon's wave-function?
Regards,
Greg
john baez
(Greg Weeks) <we...@orpheus.dtc.hp.com> wrote:
>In the discussion single-photon wavetrains, it seems to be generally
>assumed that the photon has a wave-function. Even in free field theory, I
>don't believe this is true.
Education is a process of telling a carefully chosen sequence of lies
in which the amount of deliberate deception gradually tends towards zero.
There is a limit to how much truth someone can absorb all at once without
their brain turning to jelly!
Oz - or whoever originally asked the question - seems to be wondering
something like "what's the shape of the wavefunction of a photon of a
given energy?" Of course they're not phrasing it that way, but that's
my desperate attempt to translate it into something I can understand.
Now you're right, it's a bit of a pity that they chose a *photon* as
the particle to ask about in this question. Massless particles are a
nuisance because the Newton-Wigner localization breaks down. Gauge
bosons are a nuisance because it's harder to separate out the physical
degrees of freedom in a gauge theory. So even *ignoring* the extra
subtleties when we take interactions into account and drop the pleasant
fictions of free field theory and Fock space, we have some serious
issues to deal with in a complete answer to this question!
But if someone asks the question "what's the shape of the wavefunction
of a photon of a given energy?" and you start talking to them about
Newton-Wigner localization, gauge-invariance, and Fock space, their
brain is going to turn to jelly! They're going to walk away in a daze
having learned nothing. They'll probably be shocked that such a simple
question elicited such a complicated bunch of mumbo-jumbo. They may
become politicians and cut funding for physics.
So you have to tell them something helpful even if it's oversimplified.
First and foremost, it seems to me, you have to disabuse of them of the
assumption that the wavefunction of a particle has some fixed "wavetrain
with finitely many wiggles" shape that depends solely on the energy of the
particle. When one starts out learning physics, one tends to think of
a particle as a little tennis ball or something, perhaps with some wiggly
waves thrown in for good measure. The idea that it's just a "field mode"
doesn't come easily! Usually one absorbs this slowly and painfully by
solving Schrodinger's equation with all sorts of different boundary conditions
and potentials, learning all sorts of different orthonormal bases for the
space of states, and eventually realizing that the choice of basis is just
a matter of convenience. The idea that a particle is just a solution of a
partial differential equation and that there are *lots* of solutions having
the same expectation value of energy, or even the same eigenvalue - that
doesn't come easily! So, somehow you have to broach these issues.
Thus I'm reluctant to talk about the issues you're raising now. They're
too fancy for this conversation. I'll just whisper to you the approach
I'm implicitly taking towards this question:
>What, then, is a photon's wave-function?
I'm taking it to be a solution of Maxwell's equations, either described
using the vector potential in some fixed gauge, or perhaps even better
for the present purposes, using the electric and magnetic fields. I bet
people who do quantum optics do something like this when they talk about
the wavefunction of a photon, and I don't think it's so bad, despite the
objections you note.
(Greg Weeks)
: Thus I'm reluctant to talk about the issues you're raising now. They're
: too fancy for this conversation. I'll just whisper to you the approach
: I'm implicitly taking towards this question:
There is no difficulty talking about the wave-function of an electron (in
nonrelativistic quantum mechanics). If the original question can be
rephrased in those terms, then fine. If not, I highly recommend
*conceptual* accuracy, even for beginners.
: >What, then, is a photon's wave-function?
: I'm taking it to be a solution of Maxwell's equations, either described
: using the vector potential in some fixed gauge, or perhaps even better
: for the present purposes, using the electric and magnetic fields. I bet
: people who do quantum optics do something like this when they talk about
: the wavefunction of a photon, and I don't think it's so bad, despite the
: objections you note.
It sounds like your wave-function is the the classical field obtained by
replacing the "a(k)" in the plane-wave decomposition of the classical field
solution with <k|s>. Is that right?
I vaguely recall that the same classical field was used in Dirac's quantum
mechanics text. [Oh why did I throw away all my physics books?] But the
question remains: What does this classical field have to do with anything?
It isn't a wave-function, and it may or may not have anything to do with
the interference pattern in a two-slit experiment.
Greg
Dr Paul Kinsler
: [...]
> It sounds like your wave-function is the the classical field obtained by
> replacing the "a(k)" in the plane-wave decomposition of the classical field
> solution with <k|s>. Is that right?
> I vaguely recall that the same classical field was used in Dirac's quantum
> mechanics text. [Oh why did I throw away all my physics books?] But the
> question remains: What does this classical field have to do with anything?
> It isn't a wave-function, and it may or may not have anything to do with
> the interference pattern in a two-slit experiment.
OK, so the classical field mode we quantise the photons inside isn't a
wave function. But what do we need the wave function of the photon for
anyway? Surely, it's its EM field that we use when interacting it with
things, I've never come across a need to know it's wfn.
--
------------------------------+------------------------------
Dr. Paul Kinsler
Institute of Microwaves and Photonics
University of Leeds (ph) +44-113-2332089
Leeds LS2 9JT (fax)+44-113-2332032
United Kingdom P.Ki...@ee.leeds.ac.uk
WEB: http://www.ee.leeds.ac.uk/staff/pk/P.Kinsler.html
john baez
(Greg Weeks) <we...@orpheus.dtc.hp.com> wrote:
>It sounds like your wave-function is the the classical field obtained by
>replacing the "a(k)" in the plane-wave decomposition of the classical field
>solution with <k|s>. Is that right?
I don't quite understand what you're suggesting, since I don't know what all
those letters stand for. So I'll just say what my photon wavefunction is:
it's a solution of the classical source-free Maxwell equations.
>I vaguely recall that the same classical field was used in Dirac's quantum
>mechanics text.
Quite possibly; it's pretty standard stuff.
>But the question remains: What does this classical field have to do with
>anything?
Suppose we obtain a free quantum field theory by quantizing a linear
classical field theory. Then the single-particle states of the free
quantum field theory are precisely the normalizable solutions of the
classical wave equations. So in particular, the state of a single free
photon may be described by giving a normalizable solution of the
classical source-free Maxwell equations. People often describe the
state of a photon this way and call it the "photon wavefunction".
As I said before, you can describe a solution of Maxwell's equation
either using the vector potential in a fixed gauge or using the
electromagnetic field. Depending on what you are doing, one or the
other may be more convenient, but they contain the same information.
The textbooks on quantum optics that I've been reading lately seem
to use the vector potential a lot. That's also what quantum field
theorists mainly use.
>It isn't a wave-function, and it may or may not have anything to do with
>the interference pattern in a two-slit experiment.
Normalizable solutions of the classical source-free Maxwell equations
are in 1-1 correspondence with photon states --- so I feel free to call
them "photon wavefunctions". So do lots of other people. People routinely
use them to compute interference patterns.
Perhaps you are puzzled because there is no "position representation"
of the quantum state of a massless particle analogous to the famous
wavefunction psi(x) appearing in the Schrodinger equation for a
*massive* particle. That's true, but that's exactly why we use other
representations of photon states, like the vector potential or
electromagnetic field.
(Greg Weeks)
: OK, so the classical field mode we quantise the photons inside isn't a
: anyway? Surely, it's its EM field that we use when interacting it with
: things, I've never come across a need to know it's wfn.
Yes, but suppose you write down the plane-wave decomposition of the
classical field and replace the unknowns "a(k)" with <k|s>. (Here, |k> is
a plane-wave photon state and |s> is the single-photon state of interest.)
In what sense is this field the EM field of the photon state |s>? I know
of no such sense. After all, <s|E|s> = <s|B|s> = 0. Why should this
classical field of ours have anything to do with how a photon is detected
when it passes through a pair of slits?
The states with fairly well-defined E and B fields are the coherent states,
with gazillions of photons. Not single-photon states.
Regards,
Greg
(Greg Weeks)
: Normalizable solutions of the classical source-free Maxwell equations
: are in 1-1 correspondence with photon states
Yes. But, personally, I've never been able to understand this as more than
a coincidence. I mean, the classical and quantum equations look the same,
and so do their explicit solutions. All you need to make a connection,
then, is some quantities in the quantum theory with the same covariance
properties as the classical a(k). The momentum-space wave-functions of the
single-particle states fill the bill. (Those were the <k|s>'s I was
talking about.)
But I don't see what this -- to me artificially cooked up -- classical
solution has to do with physics. For example, if this solution is
rigorously localized in space, then does a photon detector outside of the
localization region have a chance of detecting the photon? My guess is
"yes" and that there is no sharp cutoff for how distant the detector must
be before it stops detecting. So, even without the two-slits, this
"wave-function" doesn't tell me a whole lot about where the photon is
likely to be detected. Why, then, should I expect its interference pattern
to produce an interference pattern of detected photons?
And does the suggested interpretation work for free massive uncharged
scalar bosons? At low energies, these single-particle states have
well-defined position-space wave-functions. Do these agree with the
associated classical solutions to the Klein-Gordon equation? Well, no they
don't. The classical solution is approximately proportional to the *real
part* of the position-space wave-function. Well, that's *something*, but I
don't know how much.
Hmm. With free massive *charged* scalar bosons you can do better... But
photons are not massive or charged. Well, I'm meandering here. Sadly, I
just don't see what seems obvious to others. This situation is rarely
resolvable (short of taking a class).
Greg
Haruspex77
John Baez wrote:
>But if someone asks the question "what's the shape of the wavefunction
>of a photon of a given energy?" and you start talking to them about
>Newton-Wigner localization, gauge-invariance, and Fock space, their
>brain is going to turn to jelly!
>...They may
>become politicians and cut funding for physics.
John is right, but I wonder if he understands why. Fun as the math is
(for some of us) the underlying assumptions have to match some reality
we understand before it makes any sense in human terms.
Even simple things like adding integers need to be understood that way.
I didn't understand that until I learned set theory, and got a good
explanation of 1+1=2. My kids got the "New Math" (of the 1970's) in
school and were taught it right away. Unfortunately, that curriculum
didn't include *why* they needed set theory. Any mathematical
conclusions are only as good as the match between the assumptions and
reality. 1+1=2 doesn't work well for mercury blobs or rabbits.
Then, after sounding like an evil shaman who lies to his students to
guide them toward the truth, he gets down to saying:
[lines reformatted]
>I'm taking it to be a solution of Maxwell's equations, either
>described using the vector potential in some fixed gauge, or perhaps
>even better for the present purposes, using the electric and magnetic
>fields. I bet people who do quantum optics do something like this
>when they talk about the wavefunction of a photon, and I don't think
>it's so bad, despite the objections you note.
My first reaction was that John was dead wrong, mixing apples [classical
electromagnetic waves] with oranges [quantum wavefunctions]. I
visualize wavefunctions as having values of (sqrt of) probability and
arguments of whatever the dimensions of the problem are, and they only
happen to be wavelike in that common solutions are sinusoidal.
Electromagnetic waves are pretty tangible things in comparison.
And they are very classical, not quantum like at all.
The wiggle that Oz mentioned was obviously what a charge or magnet
feels when that tangible wave goes past. The mystery is the
quantization into a photon, not the wave itself. In the first example
in my "MWI Photons" posting, the wavefunction only varies with energy
and doesn't "wave" at all.
But I wonder, John, if you aren't revealing something by your comment
that has been nibbling at the edge of my brain. In that first example,
one wiggle (more or less) was clearly identified as a photon. In the
second, I hadn't calculated the size of the photons, and wondered if
they might not be identified with a half cycle of the waving charge
(between zero acceleration points)by having the same energy. That would
simplify things, but I am not sure of the implications for the random
emission/arrival times.
What I think is needed to keep physics going is a reasonable sounding
description of the physical assumptions behind quantum mechanics.
Something on the same level as general relativity described as curved
space-time that has a clean mapping to the math, even if it is hard to
visualize.
--
Haruspex (Remove the extra x to reply)
Boris Borcic
>
> Education is a process of telling a carefully chosen sequence of lies
> in which the amount of deliberate deception gradually tends towards zero.
[...]
> First and foremost, it seems to me, you have to disabuse of them of the
> assumption that the wavefunction of a particle has some fixed "wavetrain
> with finitely many wiggles" shape that depends solely on the energy of the
> particle.
Equating lies to wiggles, this suggests in self-referential manner,
that the freedom in the choice of the sequence of lies used to
transmit intellectual light, might be severely underestimated.
After all, it seems clear that all but the best teachers, tend
to reproduce in their teaching their own path to enlightenment.
Not only that, but the actual responsibility of teaching segments
of the train of lies is split between different teachers,
schools, and grades, which artificially freezes the sequence.
--
A question about the "disabusing" paragraph above. My naive
view of photons follows from their initial definition by the
photoelectric effect, e.g. as units of exchange between light
and matter. Now the precision you give here affords two views,
one is that, whatever their shape, photons absorbed correspond
to photons emitted, one by one.
The other is that photons only exist as units of exchange,
and that the conservation only applies between the sum of the
vawefunctions of the photons emitted and the sum of those
absorbed, but not piecewise. Plus : "background radiation" or
"fluctuations of the void" allows for a relative independance
of shape to apply even in the case the lightwave's total
energy is (nearly) that of a single photon, by permitting
"accounts" not to "balance to the last cent", so to say.
A few of the remarks you made (e.g. "Beethowen shaped photons"),
appear to support the first view, while my tendency would
be to believe in the second view. Which this correct ? Or is my
language so confused that there is no clear minimum of
deception between the two views ? (I might be making a
stupid confusion thinking of "photon wavefunction" as
a summative component of the lightwave, for instance,
although your statement "it is a solution of Maxwell's
equation" suggests it).
If the second view is (approximately) correct, second
question: how wrong would I be, thinking of QFT (of
which I don't understand the first word) as precisely
that which is necessary to fit the assumption that
the (relatively unconstrained) relation of shape
between wavefunction of photon emitted and photon absorbed,
does not depend on the total energy of the lightwave, even
in the low limit where that energy is about that of a single
photon (Plus similar requirements as applied to other
types of forces/fields/particles) ?
Boris Borcic
Dr Paul Kinsler
> Hmm. With free massive *charged* scalar bosons you can do better... But
> photons are not massive or charged. Well, I'm meandering here. Sadly, I
> just don't see what seems obvious to others. This situation is rarely
> resolvable (short of taking a class).
"The quantum theory of light", R Louden, (Oxford Uni Press, 1983)
Chapter 4.
john baez
(Greg Weeks) <we...@orpheus.dtc.hp.com> wrote:
>john baez (ba...@galaxy.ucr.edu) wrote:
>: Normalizable solutions of the classical source-free Maxwell equations
>: are in 1-1 correspondence with photon states
>
>Yes. But, personally, I've never been able to understand this as more than
>a coincidence.
It's not a coincidence - in any free quantum field theory the single-particle
Hilbert space is the same as the corresponding classical phase space, and
this is the basis for relating the classical and the quantum theories. I
explained "from classical to quantum" and "from quantum back to classical"
in another post - but I expect you already know this. So my point here is
simply that we really do take full advantage of this relationship between
the classical Maxwell equations and their quantized version when we solve
problems in quantum optics or quantum electrodynamics.
It's possible that your prolonged exposure to axiomatic quantum field theory
has left you with little confidence in the things most physicists take for
granted. That's why they sometimes call this subject "destructive field
theory". While this skepticism is good, it's also good to see how everyday
physicists, oblivious to these issues, happily make successful use of many
ideas which axiomatic quantum field theory brands as oversimplifications.
>[...] this "wave-function" doesn't tell me a whole lot about where the
>photon is likely to be detected.
Yes it does. Briefly, the photon is detected when it interacts with a
charged particle. The coupling of photons to charged matter is given
by the operator j.A. To work with this, we expand the operator A into
creation and annihilation operators which create and annihilate photons
corresponding to particular modes of the A field. These modes correspond
to solutions of the classical Maxwell equations, written in terms of the
A field. So there is a real relation between the "wave-function" A and
where we detect the photon - even though the quantity |A(x)|^2 does not
represent a "probability density for photons" in the same way Schrodinger's
famous |psi(x)|^2 represents a probability density.
Oz
<harus...@aol.comx> writes
>The wiggle that Oz mentioned was obviously what a charge or magnet
>feels when that tangible wave goes past.
Possible, but not obligatory. Generally (in my usual incompetantly
ignorant way) I view it as a cyclical state of being. Actually I
visualise it as the passage of the thing (photon/electron/neutrino,
whatever) roughly round a helical path with axis 'time' and a particular
plane through the axis as 'space' and it's existance being it's vector
resolved in the plane. Since I have never had the time or opportunity to
examine in detail how valid this visualisation is, I take it under
caution. It sort of crudely patterns interference, SR and some other
effects.
Ooops, sorry for butting in .... carry on ...
--
Oz
(Greg Weeks)
: (Greg Weeks) <we...@orpheus.dtc.hp.com> wrote:
: >Yes. But, personally, I've never been able to understand this as more than
: >a coincidence.
: ... So my point here is
: simply that we really do take full advantage of this relationship between
: the classical Maxwell equations and their quantized version when we solve
: problems in quantum optics or quantum electrodynamics.
That is what is being asserted, yes. I must say, though, that neither
collision theory nor the study of equilibrium states in QED relies on this
relationship.
: >[...] this "wave-function" doesn't tell me a whole lot about where the
: >photon is likely to be detected.
: Yes it does. Briefly, the photon is detected when it interacts with a
: charged particle. The coupling of photons to charged matter is given
: by the operator j.A. To work with this, we expand the operator A into
: creation and annihilation operators which create and annihilate photons
: corresponding to particular modes of the A field. These modes correspond
: to solutions of the classical Maxwell equations, written in terms of the
: A field. So there is a real relation between the "wave-function" A and
: where we detect the photon - even though the quantity |A(x)|^2 does not
: represent a "probability density for photons" in the same way Schrodinger's
: famous |psi(x)|^2 represents a probability density.
Since silence gives consent, I suppose I should state that I don't follow
this argument.
Regards,
Greg
Dr Paul Kinsler
Haruspex77 <harus...@aol.comx> wrote:
> I read into what you are saying that the only thing that the quantum
> Calculations are going to do beyond that is give the "number of photons"
> (the mean of a poisson distribution?)in a particular mode. This sounds
> suspiciously like the probability of something interacting with a field
> which has a particular shape someplace (E as a function of time).
> That doesn't give me warm fuzzy feelings that the EM waves are really
> quantized, just that their interactions with matter are.
You can measure all sorts of things other than the photon number
of a mode. You can interfere the mode with a strong coherent
state and measure the amplitude of the mode by counting the
photons from the interference pattern. You can measure the
sqyueezing (two photon correlations) by using a two-photon
absorbtion process as your counter. You can measure photon
bunching and antibunching by interfering the mode with time-
delayed bits of itself. Etc...
Haruspex77
ee...@eensgi4.leeds.ac.uk (Dr Paul Kinsler) wrote:
Haruspex77 <harus...@aol.comx> wrote:
>> I read into what you are saying that the only thing that the quantum
>> Calculations are going to do beyond that is give the "number of photons"
>
>... You can interfere the mode with a strong coherent
>state and measure the amplitude of the mode by counting the
>photons from the interference pattern.
And this interference will be something different than what you would
calculate as the interference of the classical EM waves? Under what kinds
of conditions do you get different results? Clearly, the answer is in
different terms (photon number instead of field strength), but where does it
predict different
phenonema?
> You can measure the
>sqyueezing (two photon correlations) by using a two-photon
>absorbtion process as your counter.
This, again, is calculating an interaction with matter. We do agree that
matter interactions have to be quantized, but that involves the "probability
wave" of the matter as well as the EM wave.
I thought though,from a posting by John Baez, that squeezing had more to do
with ways of preparing photons photons (eg. my MWI Interpretation posting),
than with correlations.
> You can measure photon
>bunching and antibunching by interfering the mode with time-
>delayed bits of itself. Etc...
Now John's description of this as altering the time correlation of photon
interactions sounds really cool, and like it might have valuable applications
in reducing the signal to noise ration in comunications applications. I would
be interested in seeing some of the papers on it.
But it also sounds like the solitons we can create in an optic fiber
(and occasionally form in rivers and canals). Are they related?
I can see that quantum optics is a convenient way to get EM fields into
compatible terms for further calculations, but what different results does it
predict than just staying classical until something with rest mass is
involved?
:
Florian Dufey
problem in how to extract the probability of finding a photon at a given
position from its "wavefunction".
In this context An article in the new edition of Physical Review A seems
to be quite interesting:
Margaret Hawton
"Photon position operator with commuting components"
Phys. Rev. A 59(2), p. 954, (Feb. 1999)
The constructed Operator seems to fulfill all the requirements, e.g.
Poincare covariance.
The new idea seemingly is the inclusion of a term which takes into
account the Berry-phase of the photon.
Florian
-----------------------------------------------
From:
Florian Dufey
Technical University of Munich
Physics Department T38
James-Franck-Strasse
D-85748 Garching
Germany
phone: 0049/89/289-13768
fax: 0049/89/289-12444
e-mail: du...@jupiter.t30.physik.tu-muenchen.de
------------------------------------------------
john baez
>In this context an article in the new edition of Physical Review A seems
>to be quite interesting:
>
>Margaret Hawton
>"Photon position operator with commuting components"
>Phys. Rev. A 59(2), p. 954, (Feb. 1999)
>
>The constructed operator seems to fulfill all the requirements, e.g.
>Poincare covariance.
I haven't read this article, but I should point out that there are
dozens of papers in the literature proposing photon position operators,
dozens more attacking those proposals, and various "no-go" theorems
saying that a photon position operator can't have all the desirable
properties one could imagine. So one would have to read a fair amount
of this literature before understanding the full story. As a friend of
mine would say, "much ink has been spilt" over this subject. It's a
bit like the arrow of time or the interpretation of quantum mechanics,
though on a lesser scale - a nexus of controversy and confusion.
(Greg Weeks)
: The states with fairly well-defined E and B fields are the coherent states,
: with gazillions of photons. Not single-photon states.
A couple of people pointed out that a coherent state could have (an
expectation value) of any number of photons, large or small. True. But if
you hold the E-B field fixed and let h --> 0, the number operator diverges
like 1/h. In the classical limit, you have gazillions of photons.
So, I believe that if the values of the well-defined E and B fields are
significantly greater than the uncertainties in the E and B fields, then
there are lots of photons. Something like that.
Greg
super...@my-dejanews.com
> You prepare the photons with the right correlations. Note in passing,
> that "squeezing" and (anti)bunching are not exactly the same thing.
>
> > Now John's description of this as altering the time correlation of photon
> > interactions sounds really cool, and like it might have valuable
applications
> > in reducing the signal to noise ration in comunications applications.
> > I would be interested in seeing some of the papers on it.
>
> Find a QO text --
> Quantum Optics, Walls & Milburn, (Springer 1994)
>
> > But it also sounds like the solitons we can create in an optic fiber
> > (and occasionally form in rivers and canals). Are they related?
>
> There was a good Nature article on quantum solitons a few years back,
> I don't have the ref to hand but one of the authors was P. D. Drummond.
It just so happens that this week "Physics World" arrived. A very nice article
about solitons and shot noise in optical communications. AFAICR they talked
about being able to squeeze the information packet down to just 30% more than
the theoretical limit.
NJH.
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