Electrons as quanta
Lester Welch
that we can consider the number of electrons going through a "double-
slit" (Young's experiment) and hitting a position sensitive screen to
be one-at-a-time. The pattern of "hits" on the screen will be - after
a while - an interference pattern with a high central peak at y_0
(defined as y=0) and lesser peaks on either side at y_n (n=1,2...)
In comparing a single electron that hits at y_0 with one that hits at
y_1 (say) the classical kinematics are different. At y_1, clearly the
electron had a "velocity" (hence momentum) in the y direction.
Can one, classically speak of the cause of this difference?
Scattering off of the slits? If so, why are the y-momenta quantized
so as to peak at the different y_n? Does the screen containing the
slits received an equal opposite momenta so as to conserve momenta?
Maybe this question is completely analogous to asking why there is a
interference pattern at all and, thus, the root of QM.
Uncle Al
>
> Imagine a coherent source of electrons.
Electrons are fermions and charged. Coherence in time and coherence
in space for a beam of charged fermions would require further
explanation.
> The intensity is small enough
> that we can consider the number of electrons going through a "double-
> slit" (Young's experiment) and hitting a position sensitive screen to
> be one-at-a-time.
A one-at-a-time coherent beam of charged fermions? What could that
mean?
> The pattern of "hits" on the screen will be - after
> a while - an interference pattern with a high central peak at y_0
> (defined as y=0) and lesser peaks on either side at y_n (n=1,2...)
>
> In comparing a single electron that hits at y_0 with one that hits at
> y_1 (say) the classical kinematics are different. At y_1, clearly the
> electron had a "velocity" (hence momentum) in the y direction.
>
> Can one, classically speak of the cause of this difference?
> Scattering off of the slits? If so, why are the y-momenta quantized
> so as to peak at the different y_n? Does the screen containing the
> slits received an equal opposite momenta so as to conserve momenta?
> Maybe this question is completely analogous to asking why there is a
> interference pattern at all and, thus, the root of QM.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
Igor Khavkine
If the electrons are only classical and scatter off the slits, is
there an interference pattern?
No.
Is momentum conserved?
Yes. The joint electron + slit + screen system, if can be considered
as isolated, conserves momentum. Therefore, its overall state (if it
has definite momentum) is composed of a superposition of only states
of the same definite momentum. Therefore, a measurement, which
projects (or decoheres) onto a state where the electron has made a
definite mark on the screen, will conserve momentum and result in the
slit + screen subsystem absorbing whatever momentum the electron
appears to have lost in the course of the experiment.
What is the origin of the interference pattern in the electron double
slit experiment?
This phenomenon is empirically observable. It is described to high
accuracy by the QM formalism, but not by a classical mechanical
formalism. Therefore, you might as well presume that the origin of the
interference effect is the QM formalism. Beyond that, I presume you
already know how to solve the Schroedinger equation.
Hope this helps.
Igor
Lester Welch
After a bit more thought about this problem, I realize that the y-
momentum is not quantized - otherwise there would be sharp spectrum-
like lines instead of the observed cos^2 shape. Sorry.
X-Phy
This issue has already be discussed at length, here and on
sci.physics.foundations. It would need a little search, but that led
more or less to what Igor answered. Moreover, there is also an
entanglement, for the recoil of the screen can't be known before the
electron is detected. To describe that, the global wave function
including the screen, rather than only the electron, must be used.
p.ki...@ic.ac.uk
> After a bit more thought about this problem, I realize that the y-
> momentum is not quantized - otherwise there would be sharp spectrum-
> like lines instead of the observed cos^2 shape. Sorry.
Perhaps try an infinite (and otherwise ideal) grating, rather than
a simple double slit?
--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (Photonics) (ph) +44-20-759-47734 (fax) 47714
Imperial College London, Dr.Paul...@physics.org
SW7 2AZ, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/
Lester Welch
>
> Is momentum conserved?
> Yes. The joint electron + slit + screen system, if can be considered
> as isolated, conserves momentum. Therefore, its overall state (if it
> has definite momentum) is composed of a superposition of only states
> of the same definite momentum. Therefore, a measurement, which
> projects (or decoheres) onto a state where the electron has made a
> definite mark on the screen, will conserve momentum and result in the
> slit + screen subsystem absorbing whatever momentum the electron
> appears to have lost in the course of the experiment.
Thanks, Igor. Your answer leads me to consider the following "thought
experiment."
_In principle_ then, one could wait until the electron is between the
slits and the screen and by measuring the y-momentum of the system
sans electron, determine the y-momentum of the electron since the sum
must be conserved. Thus we don't need to let the electron hit the
screen (or anything for that matter) at all to determine the
interference pattern. Agree?
Igor Khavkine
Yes. In fact, let me make the description of such an experiment more
concrete. Considre the electron gun and slit setup as usual. Now,
however, make sure that the barrier with the slit is attached to a
shock absorber (say only in the directions orthogonal to the electron
beam). The shock absorber is to the barrier as the detection screen is
to the electron. The absorber measures the impulse absorbed from the
barrier to bring it back to a stationary position.
If you send a large number of electrons through the slit and you
histogram the readings from the shock absorber, you will reproduce the
interference pattern of the electron on the detection screen. In fact,
the two interference patterns will be correlated. For each scattering
event, the impulse absorbed from the barrier will exactly balance the
momentum of the electron perpendicular to the barrier (as evidenced by
its deflection on the screen).
This setup is very similar to the usual EPR entanglement setup with
two electrons or photons. In the slit experiment, the screen +
electron system play the same role as the 2-particle system of an EPR
experiment. In both cases, you are dealing with entangled pair of
subsystems, hence the correlations.
Igor
Rich L.
...
> Yes. In fact, let me make the description of such an experiment more
> concrete. Considre the electron gun and slit setup as usual. Now,
> however, make sure that the barrier with the slit is attached to a
> shock absorber (say only in the directions orthogonal to the electron
> beam). The shock absorber is to the barrier as the detection screen is
> to the electron. The absorber measures the impulse absorbed from the
> barrier to bring it back to a stationary position.
>
> If you send a large number of electrons through the slit and you
> histogram the readings from the shock absorber, you will reproduce the
> interference pattern of the electron on the detection screen. In fact,
> the two interference patterns will be correlated. For each scattering
> event, the impulse absorbed from the barrier will exactly balance the
> momentum of the electron perpendicular to the barrier (as evidenced by
> its deflection on the screen).
>
> This setup is very similar to the usual EPR entanglement setup with
> two electrons or photons. In the slit experiment, the screen +
> electron system play the same role as the 2-particle system of an EPR
> experiment. In both cases, you are dealing with entangled pair of
> subsystems, hence the correlations.
>
> Igor
You have to be careful about how you do this so you stay within the
limits of the uncertainty principle. If the screen is mounted softly
enough to move significantly during the scattering of the electron,
the interference pattern, and hence the momentum measured at the
screen, will be blurred out. If that happens this will just look like
the electron scattering from a single slit and the behavior will be
essentially classical. If the slit is made heavy enough to NOT move
significantly during the scattering, then the experimenter will have
to wait a long time to measure any movement. This can still work,
however, if the electrons are emitted in brief time periods with long
gaps between them. This chopping of the electron wave function will
add uncertainty to the momentum of the electrons, however! Electrons
with different momenta will form interference patterns with different
spacings. Further more, the measurement of the momentum of the slit
will become uncertain by this time as well. When you measure the
offset of the slit some time later, you don't know if the electron
scattered at the beginning of the time window or the end, thus you
don't know the slit momentum with perfect precision.
An interesting question is whether one could measure the momentum of
the slit sufficiently accurately to determine which interference
fringe gets the electron. I think the answer is "yes", but I'm not
certain. If the spacing of the slits is sufficiently close then there
would be a small number of fringes. If there are many slits, the
fringes will be very narrow. It would then only be necessary to
determine the momentum of the slit sufficiently accurately to
determine about which direction the electron was headed.
An earlier post suggested that it might not be possible to measure the
slit recoil before the electron is detected. This isn't correct. The
screen with the detectors at each fringe position could be, say, a
light hour away from the slits. This would allow almost an hour to
make an accurate measurement of the slit momentum. In particular, the
slit recoil will start at the time the electron is deflected by the
slit, and should be measurable shortly thereafter. (The term
"shortly" may mean several minutes, based on the accuracy of
measurement required.)
-Rich L.
Lester Welch
> On Dec 4, 5:03=A0pm, Igor Khavkine <igor...@gmail.com> wrote:
> ...
>
>
>
>
> limits of the uncertainty principle. �If the screen is mounted softly
> enough to move significantly during the scattering of the electron,
> the interference pattern, and hence the momentum measured at the
> screen, will be blurred out. �If that happens this will just look like
> the electron scattering from a single slit and the behavior will be
> essentially classical.
Other considerations:
Presumeably one can construct the "electron gun" so that the emitted
electrons are very close to mono-energetic (This is what I meant when
I referred to them as "coherent" - to which Uncle Al correctly
objected.) . If needed, one _in principle_ should be able to get a
time reference by measuring the x momentum of the electron gun. When
the electron leaves, the gun must have equal and opposite momentum.
When the x-momentum of the gun changes we start the clock. We know
the energy of the electron so we can calculate when the electron is
between the slits and the screen (which could be, as you point out,
very large). At that time we measure the y-momentum of the apparatus
sans electron and determine the interference pattern. I agree that
ultimately the uncertainty principle will constrict our accuracy but
it's not clear to me that it would limit us in this method of
measurement. Also, the classical picture of the electron scattering
off of a slit becomes a bit murky when we realize the wave function
is, in some sense, scattering off of _both_ slits.
Lots of subtleties.
X-Phy
> Yes. In fact, let me make the description of such an experiment more
> concrete. Considre the electron gun and slit setup as usual. Now,
> however, make sure that the barrier with the slit is attached to a
> shock absorber (say only in the directions orthogonal to the electron
> beam). The shock absorber is to the barrier as the detection screen is
> to the electron. The absorber measures the impulse absorbed from the
> barrier to bring it back to a stationary position.
>
> If you send a large number of electrons through the slit and you
> histogram the readings from the shock absorber, you will reproduce the
> interference pattern of the electron on the detection screen. In fact,
> the two interference patterns will be correlated. For each scattering
> event, the impulse absorbed from the barrier will exactly balance the
> momentum of the electron perpendicular to the barrier (as evidenced by
> its deflection on the screen).
>
> This setup is very similar to the usual EPR entanglement setup with
> two electrons or photons. In the slit experiment, the screen +
> electron system play the same role as the 2-particle system of an EPR
> experiment. In both cases, you are dealing with entangled pair of
> subsystems, hence the correlations.
Actually, we assume that it'll happen so, but basically we don't know,
since that experiment has never been performed. The slit recoil is
way to small to be detectible. But today, with the nanotechnologies
and using big molecules instead of electrons in a super cold
environment, it's becoming near feasible. The exact EPR setup has
never been realized. With the Aspect's experiment we can show from QM
that there is no information transfered, but that is not so clear for
a position - momentum correlation.
[[Mod. note -- This paper involving "big molecules" also looks relevant:
"Decoherence of matter waves by thermal emission of radiation"
Authors: Lucia Hackermueller, Klaus Hornberger, Bjoern Brezger,
Anton Zeilinger, Markus Arndt
Comments: 5 pages, 4 figures
Journal-ref: Nature 427, 711-714 (2004)
DOI: 10.1038/nature02276
Emergent quantum technologies have led to increasing interest in
decoherence - the processes that limit the appearance of quantum
effects and turn them into classical phenomena. One important
cause of decoherence is the interaction of a quantum system with
its environment, which 'entangles' the two and distributes the
quantum coherence over so many degrees of freedom as to render it
unobservable. Decoherence theory has been complemented by experiments
using matter waves coupled to external photons or molecules, and
by investigations using coherent photon states, trapped ions and
electron interferometers. Large molecules are particularly suitable
for the investigation of the quantum-classical transition because
they can store much energy in numerous internal degrees of freedom;
the internal energy can be converted into thermal radiation and
thus induce decoherence. Here we report matter wave interferometer
experiments in which C70 molecules lose their quantum behaviour by
thermal emission of radiation. We find good quantitative agreement
between our experimental observations and microscopic decoherence
theory. Decoherence by emission of thermal radiation is a general
mechanism that should be relevant to all macroscopic bodies.
-- jt]]
Rich L.
...
> Other considerations:
>
> Presumeably one can construct the "electron gun" so that the emitted
> electrons are very close to mono-energetic (This is what I meant when
> I referred to them as "coherent" - to which Uncle Al correctly
> objected.) . If needed, one _in principle_ should be able to get a
> time reference by measuring the x momentum of the electron gun. When
> the electron leaves, the gun must have equal and opposite momentum.
> When the x-momentum of the gun changes we start the clock. We know
> the energy of the electron so we can calculate when the electron is
> between the slits and the screen (which could be, as you point out,
> very large). At that time we measure the y-momentum of the apparatus
> sans electron and determine the interference pattern. I agree that
> ultimately the uncertainty principle will constrict our accuracy but
> it's not clear to me that it would limit us in this method of
> measurement.
The measurement of the gun momentum and the time of emission are
fundamentally linked. To measure momentum requires two measurements
of position at different times. Thus the very small change in gun
momentum will result in a significant uncertainty in the time of
emission of the electron. Furthermore, if the gun is soft-mounted so
that it CAN recoil (and thus make the gun momentum measurable) then
there will be some uncertainty in the emitted momentum of the electron
simply for this reason. There really isn't any way to get around the
uncertainty principle.
Also, the classical picture of the electron scattering
> off of a slit becomes a bit murky when we realize the wave function
> is, in some sense, scattering off of _both_ slits.
>
> Lots of subtleties.
Rich L.
Igor Khavkine
> On Dec 4, 5:03 pm, Igor Khavkine <igor...@gmail.com> wrote:
> > Consider the electron gun and slit setup as usual. Now,
> > however, make sure that the barrier with the slit is attached to a
> > shock absorber (say only in the directions orthogonal to the electron
> > beam). The shock absorber is to the barrier as the detection screen is
> > to the electron. The absorber measures the impulse absorbed from the
> > barrier to bring it back to a stationary position.
> You have to be careful about how you do this so you stay within the
> limits of the uncertainty principle. If the screen is mounted softly
> enough to move significantly during the scattering of the electron,
> the interference pattern, and hence the momentum measured at the
> screen, will be blurred out. If that happens this will just look like
> the electron scattering from a single slit and the behavior will be
> essentially classical.
You are assuming somehow that the transverse momentum imparted to the
electron by scattering of the slitted barrier can tell you which slit
it supposedly went through. I see no reason to make that assumption,
especially if the distance of the screen to the barrier is much larger
than the slit spacing. In that case, the position of the electron
detection spot on the screen is mostly sensitive the the direction of
motion of the electron (essentially its transverse momentum) and not
the spatial offset of its trajectory (which could indicate which slit
it went through). So, the double slit interference pattern will not
disappear. Moreover, even with only a single slit, there are still
observable diffraction effects, which are manifestations of quantum
interference.
Igor
Rich L.
I am NOT suggesting that it is possible to determine which slit the
electron went through, only which interference fringe the electron
ended up in. Measuring the momentum of the slit tells you nothing
about where the electron is when it interacted with the slit. In fact
I think it is futile to even ask that question: the electron interacts
with the entire array of slits.
What I did say was that if the array of slits is sufficiently light
weight that it would move significantly during the electron scattering
event, then the interference pattern would be destroyed and what you
would see on the screen would more nearly resemble the single slit
interference pattern. I suppose I could be wrong about this, since it
still doesn't involve determining "which slit" the electron went
through, but I suspect if the slit was so flexible, that the wave
function from different slits would no longer add up coherently.
Rich L.
Rock Brentwood
> If the electrons are only classical and scatter off the slits, is
> there an interference pattern?
> No.
The momentum vector P = (P_1, P_2, P_3) and position r = (r_1, r_2,
r_3) have Poisson brackets
{r_i, r_j} = 0, {r_i, P_j} = delta_{ij}, {P_i, P_j} = 0.
If you use spherical coordinates for P, with
k = (sin T cos F, sin T sin F, cos T)
i = (cos T cos F, cos T sin F, -sin T)
j = (-sin F, cos F, 0)
and write
P = pk, r = xi + yj + zk
then, p = |P| is the magnitude of the momentum; z is the longitudinal
coordinate, while (x,y) are the transverse coordinates.
This is particularly useful in discussing what happens when the
DIRECTION of the momentum is kept constant or (more generally) what
"Heisenberg spread" corresponds to the directional unit vector k = P/
p.
In fact, in spherical form, the Poisson bracket relations become
{T, F} = 0, {p, T} = 0, {p, F} = 0
{z, T} = 0, {z, F} = 0, {p, z} = 1
{x, p} = 0, {y, p} = 0
{x, T} = 1/p, {x, F} = 0
{y, T} = 0, {y, F} = 1/(p sin T)
Classically, this shows that (px, T) and (py sin T, F) and (z, p) are
each conjugate sets.
So in a quantum setting, a dispersion comes by confining the angular
coordinates T and F; resulting in spreads of the transverse
coordinates x and y in accordance with the Heisenberg relations
[x, T] = i h-bar/p; [y, F] = i h-bar/(p sin T).
(This assumes the system is canonically quantized with respect to the
3 spherical coordinate pairs; the results are a bit different than
what comes about by canonically quantizing with respect to the 3
cartesian coordinate pairs; but everything is still valid up to the
first order in h-bar).
So, in the quantum setting, single electrons display dispersion in
transverse directions when you confine the direction of their
momentum. That comes about, for instance, by passing the electron
through a hole, since the direction of the momentum is then confined
to the line joining the hole to the source. So, coming out of the hole
it disperses in transverse directions.