is frequency/wavelength of a photon quantized?
Daniel
hi,
i know that photons are quantized, as e=hf, what i wonder is whether
the frequency is quantized. is there anything that prevents one photon
to differ from another photon by an extremely small increase in f?
thanks
dan
Uncle Al
Any frequency you like - but remember to look long enough to make
the measurement meaningful. Frequency and time are conjugate.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
Igor
ensa...@yahoo.com (Daniel) wrote in message news:<ba566c17.04072...@posting.google.com>...
We usually speak about quantization of dynamic observables such as
energy or momentum. These quantities are related to frequency and
wavelength, respectively, of the wave function. Technically, there
are no real limitations on these wave properties, provided there is a
system having the proper energy level differentials to produce them.
Daniel
>
> We usually speak about quantization of dynamic observables such as
> energy or momentum. These quantities are related to frequency and
> wavelength, respectively, of the wave function. Technically, there
> are no real limitations on these wave properties, provided there is a
> system having the proper energy level differentials to produce them.
so if wavelength/frequency is not quantized, but infinitely
continuous, then why should space and time be quantized, as in most
systems of quantum gravity such as string/loops.
John Chunko
ensa...@yahoo.com (Daniel) wrote in message news:<ba566c17.0407...@posting.google.com>...
> >
> > We usually speak about quantization of dynamic observables such as
> > energy or momentum. These quantities are related to frequency and
> > wavelength, respectively, of the wave function. Technically, there
> > are no real limitations on these wave properties, provided there is a
> > system having the proper energy level differentials to produce them.
>
If you have a wavefunction that is bound by some potential, for
example, then that wavefunction's wavelength is quantized in a manner
that depends upon the potential and the system's boundary conditions -
this is what gives you energy levels. In general, a wavefunction does
has limitations placed upon its wave properties in a system posessing
energy levels.
> so if wavelength/frequency is not quantized, but infinitely
> continuous, then why should space and time be quantized, as in most
> systems of quantum gravity such as string/loops.
First of all, wavelength can be quantized (see above).
Now, if you begin with the premise that spacetime is quantized
then there exists a fundamental length, the Planck length or some
small multiple thereof. You can then argue that wavelength is
quantized in a manner analagous to that which one encounters in
statistical mechanics. For example, phonons in a crystal lattice have
their minimum wavelengths set by the spacings between individual atoms
in that lattice. Phonon wavelengths smaller than this minimum value
cannot physically exist. Now, wavefunctions, both in bound and unbound
states, have a minimum wavelength that is set by the quantized
spacetime length scale for a similar reason. Another way of saying
this is that since spacetime, and hence length, is quantized, you
can't have a length between wavecrests that is smaller than the
fundamental length. So, even for wavefunctions that are unbound and
have no restrictions on their wave properties, they will still have a
wavelength that comes in multiples of the fundamental length -
wavelength is quantized on a fundamental scale. (Wavelength appears
continuous because the Planck scale is so fantastically small - glass
appears smooth because we can't observe the atoms that make up the
glass on the length scale at we exist.)
Now, let's argue that spacetime IS continuous, without apealing
to quantized spacetime at the outset. As you decrease the wavelength
of a wavefunction you increase its energy. Energy causes spacetime
curvature. A point is then reached where, for a particle, its Compton
wavelength becomes smaller than its Schwarzchild radius and a black
hole is formed. From Heisenberg's Uncertainty Principle, vacuum
fluctuations occur throughout spacetime all the time. The energies of
the particles that are created can have energies that cause the
formation of black holes with length scales on order of the Planck
length for a very short time. This is a simplified way of describing
the generation of Wheeler's spacetime foam. Since we start with the
assumption that spacetime is continuous, we find that when quantum
effects are considered, spacetime becomes "foamy" and, when all the
dust clears, the foam sets the length scale at which classical physics
breaks down and spacetime must be considered as being discrete.
Anyway, hope this helps!
John
jdff
<SNIP>
>> If you have a wavefunction that is bound by some potential,
for
> example, then that wavefunction's wavelength is quantized in a manner
> that depends upon the potential and the system's boundary conditions -
> this is what gives you energy levels. In general, a wavefunction does
> has limitations placed upon its wave properties in a system posessing
> energy levels.
Definitely true.
<SNIP>
You can then argue that wavelength is
> quantized in a manner analagous to that which one encounters in
> statistical mechanics. For example, phonons in a crystal lattice have
> their minimum wavelengths set by the spacings between individual atoms
> in that lattice. Phonon wavelengths smaller than this minimum value
> cannot physically exist. Now, wavefunctions, both in bound and unbound
> states, have a minimum wavelength that is set by the quantized
> spacetime length scale for a similar reason.
Also definitely true
> So, even for wavefunctions that are unbound and
> have no restrictions on their wave properties, they will still have a
> wavelength that comes in multiples of the fundamental length -
> wavelength is quantized on a fundamental scale.
Whoah there - is this true? I don't see why a statement like X>Y at
all implies that X=nY. It MAY be true, and if so I think it is both
non-trivial and new physics. For example, it is true for phonons in a
perfectly crystalline lattice - because there are impedance
discontinuities at the crystal boundary, and the crystal is a finite
integer number of atoms wide. But to say that Planck-scale space-time
quantisation looks crystalline in some sense is a leap of faith, no?
For example I don't think anyone knows enough to say that measuring
the distance between two points will always give an observable which
is an integer number of Planck distances. Although it would be a very
fundamental property if it were true - it would tell us that distance
metrics were equivalent to counting edges of graphs in some sense.
What is much more certainly true is that for any "object" including a
photon we have some knowledge about where it is (at worst "somewhere
in the observable universe"). Therefore, we have some boundary
conditions and hence in principle a wavefunction with some spatial
extent, hence quantisation of momentum, hence quantisation of
wavelength, as you stated at the top of the post
Daniel
i am aware that a sufficiently energic photon will become a black
hole, which will rapidly dissipate due to hawking radiation,
what i wonder is that for any value of wavelength, lambda "y", say
650nm, is there any law in physics that restricts the next possible
wavelength after lambda? for example, 650.1nm, 650.01nm,
650.0000000000001nm, etc.
i wonder if we live in a quantum world, whether the next smallest
wavelength after 650nm is a multiple of planck's constant.
FrediFizzx
news:ba566c17.0408...@posting.google.com...
Nobody knows for sure yet if spacetime is quantized. I believe it is. If
it is, current theory is thinking the smallest increment in length is the
Planck length. Which is mighty tiny. You would have to add a whole bunch
of zeros to the right of your decimal point to your last number for
wavelength increment. But Planck length may not be valid. We have some 11?
orders of magnitude smaller to go to get down to Planck length
experimentally. It will be quite awhile before that happens.
FrediFizzx
Kefka G
>hi,
>i am aware that a sufficiently energic photon will become a black
>hole, which will rapidly dissipate due to hawking radiation,
>
Careful - by a Lorentz transformation we can move into a frame where a given
photon has any energy we desire. Therefore by your statement EVERY photon is a
black hole. But this is wrong, at least by my crude analysis. I'll argue by
analogy: suppose we're dealing with a massive glob of matter, which is spread
out enough so that it is not a black hole. Now suppose we switch reference
frame to one where it is moving with almost the speed of light. In particular,
we make sure that the total energy (kinetic + rest) in the new frame is enough
so that we "would" have a black hole if we naively used E=mc^2 and calculated
the Schwarzschild radius using the usual formula. Will we really have a black
hole, though? No, because that would mean that in one reference frame there
are no outgoing null geodesics that reach infinity from the glob, yet in
another there are. The question "can light escape from the glob to infinity?"
can only have one answer, regardless of reference frame or coordinate system,
and hence we conclude that speeding an object up enough will not make it into a
black hole.
The extension to the massless case proceeds similarly - we know there's no
black hole in one frame (i.e. a frame where the frequency, and hence the
energy, is miniscule), so there can't be one in any other. This is, to be
fair, a whole messy mismatch of special relativistic and general relativistic
reasoning, but I think the conclusion should hold.
We can, however, have a black hole formed out of two photons, because in that
case there is a center of momentum frame where we can define the "rest mass" of
the system. Keep in mind, however, that this is really a quantum problem, so
you should really deal with it as a quantum gravity thing. To reason sloppily
and semiclassically, though, remember that you can't really localize a photon
within any distance narrower than its wavelength - you may want to, for fun (I
say this because the end result is probably meaningless in light of quantum
gravity considerations), figure out the wavelength/energy necessary in the rest
frame of two photons to create a black hole under the assumption that we can
localize them to one wavelength. What is the resulting mass of the black hole?
>what i wonder is that for any value of wavelength, lambda "y", say
>650nm, is there any law in physics that restricts the next possible
>wavelength after lambda? for example, 650.1nm, 650.01nm,
>650.0000000000001nm, etc.
>
>i wonder if we live in a quantum world, whether the next smallest
>wavelength after 650nm is a multiple of planck's constant.
Don't know about this, sorry.
-Eric
Igor
ensa...@yahoo.com (Daniel) wrote in message news:<ba566c17.0407...@posting.google.com>...
And just how does non-quantization of frequency and wavelength affect
any possible quantization of spacetime? Frankly, I don't see how one
should affect the other. Spacetime quantization is not really a done
deal, anyway. Remember that, as of right now, these are simply
theories with no real empirical support. Also remember that energy
and momentum quantization occurs despite the fact that frequency and
wavelength are essentially still continuous quantities. Same could be
true for spacetime. Or not.
arivero
> so if wavelength/frequency is not quantized, but infinitely
> continuous, then why should space and time be quantized, as in most
> systems of quantum gravity such as string/loops.
An excelent point!
In quantum mechanics you have an undeterminacy principle for x and p,
and then
a quantization principle for its product, the angular momentum.
In quantum gravity you could expect undeterminacy principles for pair
or coordinates, sat x and t, and then a quantization principle for its
area.
Univ Zaragoza, PhD Science------------------------------------------------------------------------
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John Chunko
ensa...@yahoo.com (Daniel) wrote in message news:<ba566c17.0407...@posting.google.com>...
> >
> > We usually speak about quantization of dynamic observables such as
> > energy or momentum. These quantities are related to frequency and
> > wavelength, respectively, of the wave function. Technically, there
> > are no real limitations on these wave properties, provided there is a
> > system having the proper energy level differentials to produce them.
>
If you have a wavefunction that is bound by some potential, for
example, then that wavefunction's wavelength is quantized in a manner
that depends upon the potential and the system's boundary conditions -
this is what gives you energy levels. In general, a wavefunction does
has limitations placed upon its wave properties in a system posessing
energy levels.
> so if wavelength/frequency is not quantized, but infinitely
> continuous, then why should space and time be quantized, as in most
> systems of quantum gravity such as string/loops.
First of all, wavelength can be quantized (see above).
Now, if you begin with the premise that spacetime is quantized
then there exists a fundamental length, the Planck length or some
small multiple thereof. You can then argue that wavelength is
quantized in a manner analagous to that which one encounters in
statistical mechanics. For example, phonons in a crystal lattice have
their minimum wavelengths set by the spacings between individual atoms
in that lattice. Phonon wavelengths smaller than this minimum value
cannot physically exist. Now, wavefunctions, both in bound and unbound
states, have a minimum wavelength that is set by the quantized
spacetime length scale for a similar reason. Another way of saying
this is that since spacetime, and hence length, is quantized, you
can't have a length between wavecrests that is smaller than the
fundamental length. So, even for wavefunctions that are unbound and
have no restrictions on their wave properties, they will still have a
wavelength that comes in multiples of the fundamental length -
John Chunko
jdff...@hotmail.com (jdff) wrote in message news:<3f96fbb1.04080...@posting.google.com>...
> > quantized in a manner analagous to that which one encounters in
> > statistical mechanics. For example, phonons in a crystal lattice have
> > their minimum wavelengths set by the spacings between individual atoms
> > in that lattice. Phonon wavelengths smaller than this minimum value
> > cannot physically exist. Now, wavefunctions, both in bound and unbound
> > states, have a minimum wavelength that is set by the quantized
> > spacetime length scale for a similar reason.
>
> Also definitely true
> > So, even for wavefunctions that are unbound and
> > have no restrictions on their wave properties, they will still have a
> > wavelength that comes in multiples of the fundamental length -
> > wavelength is quantized on a fundamental scale.
>
> Whoah there - is this true? I don't see why a statement like X>Y at
> all implies that X=nY. It MAY be true, and if so I think it is both
> non-trivial and new physics.
True. However, consider the case where Y0 is the eigenvalue of
some operator in the ground state of some system. If the spectrum of
the operator consists of a number of eigenvalues bound from below by
Y0, then X=nY0, where X is an element of the spectrum and n=1,2,...
So, in my posting above I might not have been entirely clear when I
stated that wavelength comes in multiples of some fundamental length.
Hope this removes that ambiguity.
> For example, it is true for phonons in a
> perfectly crystalline lattice - because there are impedance
> discontinuities at the crystal boundary, and the crystal is a finite
> integer number of atoms wide. But to say that Planck-scale space-time
> quantisation looks crystalline in some sense is a leap of faith, no?
>
Since I was arguing by analogy it might seem that I was saying
that Planck-scale spacetime takes on a crystaline structure, but
that's not what I was getting at. In fact, spacetime cannot take on a
crystaline structure in the above sense - you would get a breaking of
Lorentz invariance since you're in effect setting up a background
reference frame defined by the "spacetime lattice".
> For example I don't think anyone knows enough to say that measuring
> the distance between two points will always give an observable which
> is an integer number of Planck distances. Although it would be a very
> fundamental property if it were true - it would tell us that distance
> metrics were equivalent to counting edges of graphs in some sense.
It's nice that you brought this point up since it's the general
approach one uses in loop quantum gravity. Leaving the problem of
defining observables in grantum gravity aside one can address the two
points raised in this post - "spacetime lattices" and a fundamental
length. I'll just briefly run through how the process of quantizing a
spacetime manifold works as motivation for the argument of fundamental
length...
Fix a three-dimensional manifold, M, equipped with an SU(2)
connection, A, and a densitized inverse triad, E; this allows you to
define the Ashtekar variables for M. You can now define the loop
observables for the theory as the trace of the holonomy of the
connection. These loop observables coordinatize the phase space of the
theory and allow you to define the kinematics of the theory as the
unitary representation of the loop algebra on some Hilbert space H.
You can now work out the cylindrical states as functionals, f(A), of
the amplitude of A; define an inner product between the functionals to
get to a Hilbert space. Now, pick some graph, Y. The parallel
transport operator of A along Y is U, and the cylindrical states
become f(U(A)). Any two cylindrical functions have the same graph.
Define the inner product between f(U(A))'s and complete the space of
linear combinations in the Hilbert norm to get to H. You now have an
unconstrained quantum state space for loop gravity.
Get an orthonormal basis of H. Define a graph, G, with an
irreducible rep. of SU(2) for each link and an intertwiner, v, for
each node; this gives you a spin network. Basis states of H can be
constructed from spin network states. You can now define quantum
operators for length, area, etc., as linear operators on H. Now, apply
constraints since H is a huge non-separable space. The diffeomorphism
constraint reduces H to Hdiff = H/Diff(M) and the states of Hdiff can
be labeled by s-knots. The Hamiltonian constraint yields a Hamiltonian
operator and governs the time evolution of the spin network states.
So, you now have a theory with quantum states of gravity given by spin
networks (or s-knots), and well-defined operators for observables and
time-evolution. What can we now say about "spacetime lattices" and
fundamental lengths?
Let's address the question of a fundamental length first. One can
define a length operator, L, acting on the spin network states. One
can show (Thiemann, gr-qc/9606092) that the spectrum of L goes as
lp*F(j_i) where lp is the Planck length and F(j_i) is a function that
depends upon the spin-labelling of the edges of the spin network
states. Therefore, one immediately sees that there does exist a
fundamental length that is some multiple of the Planck length when
F(i_j) is a minimum (a minimum of F(i_j) exists and is well defined
because of the spin-labelling).
Now, what about the concept of "spacetime lattices". The
diffeomorphism constraint helps to remove this problem for us. The
spin network states are abstract, non-embedded, colored graphs that
define M. Each edge carries a quantum of area and each node carries a
quantum of volume. The diffeomorphism constraint tells us that Hdiff
is composed of sets of equivalence classes of spin network states,
where the equivalence classes are defined by spin network states under
the action of Diff(M). The spin networks of these classes are all
topologically identical. Therefore, M is not built up from quanta of
area and volume that form a "lattice" structure.
Hope this helps!
John
Urs Schreiber
news:1ecfc875.04081...@posting.google.com...
[on LQG]
> The diffeomorphism constraint reduces H to Hdiff = H/Diff(M)
This is not true in ordinary path-integral quantization of 2-d gravity
coupled to matter. I am still wondering why this postulate is assumed to
make sense in higher dimensions.