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The Einstein-Podolsky-Rosen Paradox Is Not So Paradoxical
By Jupiter Scientific staff scientist Dr. Stuart Samuel
Some Features of Quantum Mechanics
Quantum mechanics, which governs Nature
at the smallest scales such as those
inside an atom, can seem strange especially to those who have only
experienced the world at large scales. Since few of us have seen an
atom or a nucleus up close, it is not surprising that Nature behaves
different at such tiny distances. Why should the way an apple drops
from a tree have anything to do with the way an electron orbits
around a nucleus? The macroscopic world is governed by classical
mechanics and classical mechanics is very different from quantum
mechanics. Whereas an apple is a solid object, an electron is a wave.
Indeed, according to quantum mechanics, everything is a wave. Only
when the momentum is sizable and the wave is relatively well
localized or involves dense oscillations does the entity behave like
a particle or an object; and, in such a case, quantum mechanics
produces the results of classical mechanics, a property known as the
Momentum is sizable if mass or speed or both are big. Since everyday objects are
so much heavier than atoms, waves of everyday things behave like
particles or objects, and they are well described by classical
mechanics. High-speed electrons have high momentum and they too
behave like particles in this case.
Because quantum mechanics and the world of the small are so different from
our everyday experiences, surprising behavior occurs. For example, an
electron in an atom does not have a definite location. Instead, its
position can only be described probabilistically. Such electrons have
higher probabilities of being in certain places and lower
probabilities in being in other places. An electron bound to a
nucleus has a high probability of being in the vicinity of the
nucleus and a low probability of being far away from it. At any given
moment, one cannot say exactly where the electron is. This is related
to Heisenberg's uncertainty principle
Heisenberg's uncertainty principle.
From the point of view of the “wave” of the electron, this is
understandable; when you look at a water wave, for example, is it
possible to stay that it is located at one particular point?
A second example of the unusual nature of quantum mechanics is that the
energy of an electron in an atom (and in many other situations) is
only allowed to possess certain specific values; a property that
physicists call energy quantization. The quantization of these energy
values leads to the specific spectral lines associated with atoms.
Third, because waves are involved in quantum mechanics, there can
be constructive and destructive interference (meaning that wave
components can combine to create larger or smaller amplitudes) and
this does not happen in classical mechanics for objects; It does
happen classically for waves such as water, sound and light waves. A
fourth unusual feature of quantum mechanics is tunneling: although an
electron may not have enough energy to pass through a region (a kind
of “energy hill or barrier”), there is a certain,
typically very low probability of doing so. If this phenomenon were
to be blown up to macroscopic distances, it would be like seeing a
marble in the bottom of one cup suddenly disappear and appear
spontaneously in a neighboring cup. At tiny scales, these seemingly
magical things are part of quantum mechanics.
Not only is energy sometimes quantized but other entities also are.
Momentum and angular momentum are two examples. Angular momentum is
the amount of rotation a system undergoes. Electrons rotate and this
rotation is call spin. They are like little tops. The spin of electrons
and other elementary particles contributes to the total angular
momentum of a system. However, the amount of spin that an electron can
undergo is not arbitrary. If one picks an axis, then an electron can
spin about this axis by a fundamental amount, equal to ħ/2,
in one direction or the opposite direction. Hence, the spin of the
electron is quantized: it can only take on two values. Here, ħ
(“h bar”) is a fundamental parameter of quantum mechanics
known as Planck's constant; it determines the quantization spacing of
energy levels and other quantized entities and the degree of
uncertainty in Heisenberg's uncertainty principle. The waves
associated with situations in which entities such as energy,
momentum, angular momentum, et cetera are quantized are called
A state in quantum mechanics is the situation corresponding to a particular
The miniscule value of ħ of 1.055 x 10-34
Joule-seconds means that quantum mechanical effects are minute. The
spacing of energy levels is very “tight” and seems almost
continuous (the situation in classical mechanics) if not viewed in
detail. The other contribution to total angular moment, namely
orbital angular momentum, is also quantized. The only permitted
values are integral multiples of ħ, i.e., nħ where n is an
integer. When n=0, the orbital angular momentum is zero and there is
only one wave configuration associated with this state corresponding
to a wave with perfect spherical symmetry: the wave has the same
amplitude in all directions about a point. Orbital angular momentum
is always measured about a point, and the word “orbital”
is appropriate because it corresponds to the angular momentum of
thing going around (or orbiting) the point. For example, take the
point to be the center of the Sun. Then the planets contribute to the
orbital angular momentum of the solar system because they revolve
around the Sun.
Photons, the quantized versions of electromagnetic waves, also have quantized
spin, which in this case is known as helicity. Unlike the case of
electrons, the spin is quantized in the direction of the motion of
the photon. Like electrons, however, there are only two possibilities:
clockwise or counterclockwise rotation. Classically, this corresponds
to the polarization of
electromagnetic waves. The amount of spinning for a photon happens to
be twice that of the electron, that is, ħ.
Fundamental to quantum mechanics is the notion a
that depends on position and time and often on other variables.
It represents the “wave” in the discussion above. Among things, it encodes the
quantum-mechanical uncertainty of an entity: Since waves exist over
an extended region, there is no notion of the wave being located at
any particular point. The wavefunction is a function of the positions
(the x, y and z coordinates) of each entity (electron, atom, nucleus,
molecule, etc.) in the Universe. Positions where the wavefunction is
zero are regions where the entity cannot be found. The entity is more
likely to be found in regions where |Ψ|2
(absolute value of the wavefunction squared) is large. Indeed, the
probability of finding an entity at x
at time t is proportional to |Ψ(x,t)|2.
Quantum mechanics provides a fundamental wave equation describing how the
at time t changes during a small time interval Δt:
Ψ(t + Δt) = Ψ(t) − iHΨ(t)Δt/ħ
which can be written as
ħdΨ(t)/dt = -iHΨ(t)
where dt = Δt and ΔΨ(t) = Ψ(t + Δt) − Ψ(t).
In this equation, H is called the Hamiltonian, ħ
is the above-mentioned Planck's constant and
i is a special constant.
For those that are not mathematically
inclined, Equation (1) says that the wavefunction changes smoothly
with time in a very definite way determined by the Hamiltonian acting
on it. Because quantum mechanical probabilities are determined by the
square of the wavefunction, Equation (1) also means that probable
locations for an entity evolve smoothly and in a way specified by
quantum mechanics. In the non-relativisitic situation, Equation (1) is known as the
for relativistic quarks (the constituents of protons and neutrons)
and electrons, it is the
and for photons it is
the quantized version of
Maxwell's equations (the third column of this table).
When HΨ(t) = EΨ(t), where E is a constant, Ψ is called an eigenstate of
energy and E is the quantized value of energy. In other words, an
eigenstate of energy is just a particular wavefunction involving a
definite value of energy. As mentioned above, eigenstates exist for
operators other than the Hamiltonian such as the ones for momentum
and angular momentum; these operators are the quantum mechanical
analogs of the classical notions of momentum and angular momentum. An
eigenstate for momentum is just a wavefunction with a specific value
for momentum; an eigenstate for angular momentum is a wavefunction
with a specific value of angular moment.
This concludes the introduction to quantum mechanics. The reader is now in
a position to understand the
The Setup for an Einstein-Podolsky-Rosen Experiment
experiment is considered to be a
because it is so difficult to execute in practice. In a gedanken experiment
a scientist thinks of an experiment that could be performed
in practice given sufficient care and resources. Albert Einstein was
the master of the gedanken experiment. He used pure thought to imagine
how certain experiments might progress, produce interesting
consequences and provide insight into theories.
The Einstein-Podolsky-Rosen (EPR) “paradox” arises in a
number of different situations. A simple and commonly considered case
is the following. Two detectors capable of measuring spin are placed
on opposite sites of and some distance away from a source that produces
what is known as entangled spin pairs. The pairs may be atoms,
nuclei or particles. In what follows, we take take the pair to be a
positron (an electron's anti-particle) and an electron because it will be
easy to distinguish the two in the discussion below. One possible source for this situation
is a motionless neutral spin-zero particle that is unstable and capable of
decaying into an electron-positron pair.
Figure 1: The Experimental Setup for an Einstein-Podolsky-Rosen Experiment
When the spinless entity decays, the electron and positron take off in
opposite directions with the same speed because linear moment is
conserved. If one is lucky, the two particles head toward the two
detectors. Otherwise, many particles repeatedly decay in central
region until one of them produces an electron-positron pair moving in
the “correct” direction. Alternatively, a multitude of
detectors can be employed so that the full spherical surface
surrounding the central region is covered.
When the source particle decays, physicists often argue that the spin of
the electron heading to one detector is the opposite of the spin of
the positron heading to the opposite detector. This is due to the
conservation of angular momentum. Because the motionless particle is
spinless, the total angular momentum before the decay takes place is
zero. After the decay, the spins of the electron and positron should
combine to give zero angular momentum. This requires them to spin in
opposite directions. Choosing the z-axis (up direction) to quantize
the spin, there is a 50% chance that the electron heading to the left
in Figure 1 spins clockwise (or down) and a 50% chance that the
electron spins counterclockwise (or up).
It is very important for the region between the neutral particle and the
detectors to be free of interfering effects. If the electron or
positron interact with an atom, a magnetic field, or some other force
or entity, it can change its spin.
Suppose you, an experimentalist, are located at the detector on the left.
Then if you measure the electron's spin to be up, then you instantly
know that the spin of the positron heading to the right must be down.
If you measure the electron's spin to be down, then the spin of the
positron must be up. In principle, this can be verified
experimentally: After detecting the spin of the electron at the left
detector, you can proceed to the other detector to see whether the
spin at the right detector is the opposite. If such an experiment
were to be done carefully, then physicists argure that the spins
would be found to be oppositely aligned. If D is the distance between
the two detectors, then the verification process must last longer
than D/c where c is the speed of light because nothing can travel
faster than the speed of light. However, the measurement of the spin
of the electron at the left instantly provides knowledge of the
spin of the positron on the right.
Actually, given the way the experiment has been described above, the reader
probably does not find anything paradoxical about the
Einstein-Podolsky-Rosen experiment. However, one aspect of quantum
mechanics has been left out. If performed in a pristine environment
such as a perfect vacuum, then just before the left detector measures the
electron, there is still a 50% chance that its spin is up and a 50%
chance that its spin is down. It is not the case that the electron
heading to the left is emitted with its spin up in 50% of the decays
and is emitted with it spin down in 50% of the time. In a single
decay, the 50-50 probability situation persists right up until the
time of the measurement. Theorists understand quantum mechanics so
well that they know this to be true. Hence, just before the
measurement, the two spins exist in this uncertain probabilistic
situation: simultaneously being 50% up-down and 50% down-up, where
up-down means the left-moving electron has its spin up and the
right-moving positron has its spin down, and down-up means the
opposite. Some physicists say that the measurement of the electron's
spin causes the positron to assume a particular spin state. This is
like “action at a distant.” Indeed, Einstein called it
“spooky action at a distance.” Something done at the left
detector instantaneously causes something to happen to the positron
on the right. This bothered Einstein because it seemed to violate
special relativity where effects cannot happen instantly. Instead, an
action at one point can only affect something a distance D away at a
time greater than D/c since effects cannot propogate faster than the
speed of light. This bothers a lot of scientists and is the origin
for the word “paradox” in the Einstein-Podolsky-Rosen paradox.
In brief, there are two aspects of the Einstein-Podolsky-Rosen paradox.
The first is the instantaneous transfer of knowledge about a distance
object (which in the above discussion is the spin status of the
positron). The second aspect is that one act (the process of
measuring the spin of the electron on the left) causes something to occur
instantaneously to a far away object (the spin of the positron on the
right). In both cases, faster-than-the-speed-of-light propagation
seems to be occurring.
In the example of the Einstein-Podolsky-Rosen paradox discussed above,
it is possible to replace the electron-positron pair by any two
entities that possess non-zero spin as long as the spins get
“entangled.” In general, entanglement is the property of
one entity being correlated with the property of another entity. In
the case of spin 1/2 objects, entanglement means that the final spin
state is given by
means that entity moving to the left has spin up while the entity
moving to the right has spin down, and |↓>L|↑>R
means that the entity moving to the left has spin down while the
entity moving to the right has spin up. This spin state is an
admixture of these two possibilities with a 50% chance of either one
of the spins for the electron-positron case might be accomplished
using neutral pions, which sometime decay into electron-positron
pairs. To actually conduct an Einstein-Podolsky-Rosen experiment
using neutral pions, however, is very difficult for a number of
reasons that I will not discuss here.
Two Other Gedanken Experiments
Insight into the Einstein-Podolsky-Rosen paradox can be obtained by
considering two other similar gedanken experiments.
Gedanken Experiment 1: A Classical Einstein-Podolsky-Rosen Experiment
The setup for this experiment is similar to the above except that instead
of having the decay of a spinless particle into an electron and a
positron, sister-brother twins at the site of the source flip a coin.
If the coin comes up heads, the sister agrees to
hold her thump up and while the brother agrees to
hold his thumb down. If the coin comes up tails, the twins hold their
thumbs in the opposite directions. The the sister proceeds to the left while the brother
proceeds to the right. When the observer at the left sees
the sister approach him, he instantly knows the direction of the
thumb of the twin brother moving to right.
Gedanken Experiment 2: A Unknown Einstein-Podolsky-Rosen Experiment
An experimentalist determines the spin of an electron with a detector.
Unbeknownst to the experimentalist, this electron is one that
originated from the decay of a neutral pion and had its spin entangled with a positron. The positron necessarily
is moving in the opposite direction with its spin oppositely
oriented. However, the experimentalist does not know the source of
the detected electron and therefore is unaware of positron.
Discussion of the Two Other Gedanken Experiments
Gedanken Experiment 1 shows that transfer of knowledge at a rate faster than
the speed of light can occur even in classical mechanics. Gedanken
Experiment 2 shows that if the experimental situation has not been
arranged in advance, then transfer of knowledge at a rate faster than
the speed of light does not occur in quantum mechanics.
The key point is the following: To have instantaneous
faster-than-the-speed-of-light knowledge transfer, a “setup
agreement” must occur earlier and the events of the experiment
(the decay of the spinless particle and subsequent detection of the
electron and its properties, or flipping a coin to decide the
orientation of thumbs of the twins and the subsequent observation of
one twin's thumb) must be casually connected to the setup-agreement
event. An event B is casually connected to A if it is possible for A
to communicate with B without having the communication travel faster
than the speed of light. Scientists then say that B is inside the future
light cone of A.
Here is a diagram.
Figure 2: Light Cone Diagram for EPR Paradox
Hence, there is no violation of special relativity in the transfer of
knowledge in the Einstein-Podolsky-Rosen experiment. Indeed, one can
use the idea of a “setup event” to possible prevent
faraway intelligent life from invading Earth. See
“How to Defend Earth from Distant Aliens”.
As mentioned above, there is
a difference between the classical and quantum mechanical versions of
the Einstein-Podolsky-Rosen experiment. In the classical case, the
decision as to which thumb is up is made when the two twins are
together. In the quantum mechanical case, the uncertainty in the
direction of the spins of the electron and positron persists up until
the spins are measured. More precisely, just before the spin of the
left electron is measured, there is still an equal probability of its
spin being up or down.
Another difference between the classical and quantum mechanical versions is
that in the quantum case the selection of an axis direction to
determine spin-up or spin-down is arbitrary. In Figure 1, the spin is
measured in the z direction. However, the x or y (or any) direction
can be used and the electron spins would still point in opposite
directions. For the x-direction, this follows mathematically because
where the z and x subscripts indicate the axis being used to measure
the spins. (For the mathematically inclined, substitute
and |↓>z = (|↑>x
into the left-hand side of the equation to get the right-hand side.)
In other words, the entanglement of the spins of the electron and
positron do not depend on the spin-quantization axis.
knowledge of the spin state of the positron is instantaneously gained,
there is no way to use the Einstein-Podolsky-Rosen experiment to send
a message faster than the speed of light.
Proofs of this result exist.
In this sense, faster-than-the-speed-of-light communication is not violated.
One Perhaps Disturbing Aspect of the Einstein-Podolsky-Rosen Experiment
The previous section makes it clear that, although
faster-than-the-speed-of-light knowledge takes place in the
Einstein-Podolsky-Rosen experiment, there is nothing paradoxical
about this. However, there is another aspect of the
Einstein-Podolsky-Rosen experiment that bothers some scientists.
the spin of the electron at the left is measured
and determined to be up, one knows that the spin of the positron at
the right is down. When the spin of the electron at the left is measured
to be down, one knows that the spin of the positron is up. According
to several interpretations of quantum mechanics,
the measuring of
the electron at the left causes
the spin of the positron to be selected. So measuring something at
one location (at the left detector) is causing something to happen
at a location (at the right detector) that seems to violate
causality. Here, violating causality means causing something to
happen at a distant location instantaneously. If one measures the
spin of the electron on the left to be up and immediately sends a
light signal with a “message” in it to tell the positron
on the right that its spin should be down, then by the time that
message arrives at the positron, the spin of that positron is already
down. So it appears that one effect (the measuring of the spin of the
electron at the left) is causing
another effect (the quantization direction and value of the spin of
the positron at the right) with a “communication speed”
that exceeds the speed of light.
A Possible Resolution of the Einstein-Podolsky-Rosen Paradox
The content of this section is based on ideas
of the author (Dr. Stuart Samuel)
and has not been evaluated by the general physics community.
Among things, the author assumes that quantum mechanics as determined by
the “Schrödinger” equation (Equation (1)) governs
everything including the devices needed to measure spins in the
Einstein-Podolsky-Rosen experiment; some physicists believe this
while other physicists believe that “measurement” involves
something “in addition” to Equation (1). To get a
feeling of the diversity of opinions about aspects of quantum
mechanics, see the New York Times article “
Quantum Trickery: Testing Einstein's Strangest Theory.”
This section is rather technical requiring knowledge of quantum mechanics
at least at the level of an undergraduate physics major to understand
the arguments in detail. I shall try to provide simple descriptions
of what is going on with the hope that a more general audience will be able to follow.
With this in mind, let me just summarize the result.
The last step of the Einstein-Podolsky-Rosen paradox involves a measurement
causing a transition of the spin part of the wavefunction to an untangled state.
While one can image theoretically
doing this, I claim that there is no way to do this in a real
experiment. In short, the EPR experiment is not a valid gedanken
experiment and hence there is no paradox.
There is an assumption among many physicists that measurement in quantum
mechanics causes the
“collapse” of the wavefunction.
An extreme example of this is the following: If a wavefunction for an
electron spreads over a wide region and a measurement determines the
electron to be in a smaller region then the wavefunction must
collapse to a wavefunction that covers only the smaller region. In
the situation with spins in the Einstein-Podolsky-Rosen experiment,
the analogous measurement effect is to cause the spin state to
collapse from the entangled state (|↑>L|↓>R
to an untangled one: either |↑>L|↓>R
each with 50% probability.
This is the last step in the Einstein-Podolsky-Rosen experiment: If
then the measurement of the left spin has determined it to be up and
the right spin to be down. On the other hand, if (|↑>L|↓>R
then the measurement of the left spin has determined it to be down
and has caused
the right spin to be up. In either case, one has “spooky”
action at a distance.
In order for a gedanken experiment to be valid, it must in principle be
able to happen physically in a real experiment. If the gedanken
experiment involves several steps, then each step must be possible in
practice. It is my belief that the last step, the measurement of spin
causing the collapse of the spin state to one of the above two untangled spin
states cannot experimentally happen.
One standard way of measuring the spin of an entity (an atom, a nucleus,
a particle, etc.) uses a gradient magnetic field. When a spin 1/2
object passes in the region of a magnetic field, it is deflected in
the direction of or in the opposite direction of the magnetic field
depending on whether the spin is oriented with or opposite to the
The direction of the magnetic
field establishes the quantization axis for the spin. This method was
first used by
Otto Stern and Walther Gerlach in an important
experiment that established the quantization of spin values.
The Stern-Gerlach measuring method, however, does not cause a collapse of
the spin state. Suppose the magnetic field and the spin-quantization
axis are along the z axis (the up-down direction). If the spin state
of the spin 1/2 entity when it enters the magnetic field region of
the Stern-Gerlach experiment is (a|↑> + b|↓>)/N
(where N2 = a2 + b2),
meaning that the entity has a probability of a2/N2
of having its spin up and a probability of b2/N2
of having its spin down, then the spatial part of its wavefunction
gets split into two pieces: one piece Ψ+
moving up and one piece moving Ψ− down.
In other words, if Ψ(x,y,z,t)(a|↑> + b|↓>)/N
is the wavefunction before ending the gradient magnetic field region,
then (aΨ+(x,y,z,t)|↑> + bΨ−(x,y,z,t)|↓>)/N
is the wavefunction exiting the magnetic region. The Stern-Gerlach
measuring method does not cause the spin of the entity to collapse to
a particular spin state. We know this from quantum mechanics. Quantum
mechanics evolves “independent” states such as a
/N and bΨ(x,y,z,t)|↓>/N
independently. If the spin is purely up, then it will move up and the
Stern-Gerlach method allows us to observe
that the electron's spin is up; Similarly, if the electron spin is
purely down. In short, Stern-Gerlach allows up to observe spin but
not to measure it in the sense of causing a wavefunction collapse.
The question then arises: If the Stern-Gerlach method does not measure
spin 1/2 but merely observes it, then is there another Spin 1/2 Measurement that produces
the last step of the Einstein-Podolsky-Rosen paradox, namely, causing
(a|↑> + b|↓>)/N
to transition to either |↑>
with a probability of a2/N2
or to |↓>
with a probability b2/N2?
I now argue that such a Spin ½ Measurement would violate two
fundamental principles: conservation of angular moment and
what-is-know-as unitarity in quantum mechanics.
Conservation of angular moment is a fundamental property of nature that has never
been observed to be violated. In classical mechanics all three
components (corresponding to rotations about the x, y or z axes) of
angular momentum are measurable, but in quantum mechanics only
total angular momentum and a component about one axis
can be measured.
This is due to Heisenberg's uncertainty principle. Most people think
of Heisenberg's uncertainty principle as saying that momentum and
position cannot be simultaneously measured. However, the uncertainty
principle affects a number of other observables including any two
components of angular momentum.
Angular momentum has two contributions: orbital angular momentum due the
movements of objects around a selected point and spin, which is the
fundamental rotation of elementary particles. Total angular momentum
is the sum of these components.
I now show that if a Spin 1/2 Measurement is possible, then
conservation of angular momentum can violated in a gedanken
experiment: As a specific example, assume
that the initial situation before the creation of the
electron-positron pair is an eigenstate of total angular momentum J
of 0 and z-component Jz of 0 as
measured about the central location of the neutral pion. We assume
that after the creation of the electron and positron that the spins
are entangled. If the entangled spins cannot be created then the
Einstein-Podolsky-Rosen paradox cannot arise in the first place. Spin
entanglement means that just after the electron-positron pair is
created, the wavefunction for the system is Ψ×
where Ψ is the wavefunction component for the electron, the positron and
everything else except the spins of the electron and positron. The
angular momentum of Ψ
must be 0 and its z-component must be 0 since (|↑>−|↓>+
has spin 0 and does not contribute to total angular momentum. If a
Spin 1/2 Measurement exists, its use causes a transition to either
The problem with either of these final states is that the spin part
is a combination of spin 0 and spin 1 and it is not possible to
construct a state with the original values of J and Jz.
This follows from
addition rules for
combining angular momentum in quantum mechanics.
To create a state with the original J and Jz
for the spin 1 part, either |↑>−|↑>+
or both must be present. For example, in the first case above,
For the first term to have J and Jz
of 0, Ψ'
must be an eigenstate of angular momentum with values of J of 0 and
of 0 because (|↑>−|↓>+
is spin 0. Because (|↑>−|↓>+
is the Jz=0
component of a spin 1 objects, the
quantum mechanical rules for constructing an eigenstate of total angular momentum J of 0
require Ψ' to be spin 1 (already a contradiction) and the presence of terms of
the form Ψ'−|↑>−|↑>+
have total angular momentum 1 and z-components of -1 and +1
respectively. In short, a Spin 1/2 Measurement must involve spin
flipping for an initial state of angular momentum 0. Spin
flipping means |↑>
Hence, a transition from (|↑>−|↓>+
to either only Ψ'|↑>−|↓>+
or only Ψ''|↓>−|↑>+
is impossible for an initial state with angular momentum 0. A Spin
1/2 Measurement will also violate conservation of angular moment if
the initial state is an eigenstate with total angular moment J and
The existence of a Spin 1/2 Measurement also violates
in quantum mechanics.
Equation (1) allows one to determine Ψ(t + Δt)
in terms of Ψ(t). The equation can then be used again to determine Ψ(t + 2Δt)
in terms of Ψ(t + Δt),
and repeated use of the equation determines the wavefunction for all
future times. The solution can be expressed as
Ψ(tf) = U(tf,t0)Ψ(t0)
The value of the wavefunction at a latter time tf
is determined from the wavefunction at an earlier time t0
using the operator U(tf,t0).
It is known that U is a
have the property of preserving the normalization (roughly corresponds to the size) of states.
If the Spin 1/2 Measurement caused a collapse of
(a|↑> + b|↓>)/N
at time t0 just before the measurement to |↑>
at time tf just after the measurement then the implication is that
a|↑>/N → |↑> and |↓> → 0.
However, this transformation
would not perserve the normalization of either of these two states.
It is well know that a
collapse of a wavefunction violates unitarity.
Similarity if the Spin 1/2 Measurement caused (a|↑> + b|↓>)/N
to transition to |↓>
then preservation of normalization of states under U(tf,t0)
would be violated. The assumption that everything is governed by
quantum mechanics including measurement then precludes the
possibility of a Spin ½ Measurement of the type described above.
In brief, I believe the Einstein-Podolsky-Rosen paradox is not a paradox
because it is not a valid gedanken experiment. It requires a
measurement process that cannot be achieved in real world.
The basic argument of this section is that wavefunction collapse during a measurement does
not happen. If (a|↑> + b|↓>)/N cannot collapse to ↑> or to ↓>
cannot collapse to |↑>−|↓>+ or to
|↓>−|↑>+ under a Spin 1/2 Measurement
and therefore the observation of the spin of the electron at the left does
not cause the spin of the positron on the right to achieve a particular value.
While the issue of
measurement in quantum mechanics is not well understood,
I believe that
the “Einstein-Podolsky-Rosen paradox” simply observe
the quantum correlations of entangled spins.2
Difficulties in Setting Up the Entangled Spin State
for the e+ e- Case
A requirement of the entangled state is that the
the electron and positron head in opposite directions
to regions where their spins can be observed.
Now in classical mechanics
there are lots of states with zero orbital angular momentum as
measured about the location of the decaying spin-zero particle.
Indeed, any straight trajectory starting from this point has zero
orbital angular momentum. In quantum mechanics, however, there is essential
only one wavefunction with zero orbital angular momentum: the
completely spherically symmetric one. Hence, if both the electron and
the positron in the final state separately have zero orbital angular
momentum then each of their wavefunctions must consist of an outward
moving spherical shell. In such a case, the electron's and positron's
spatial wavefunctions would be equally distributed in all directions and be
overlapping everywhere. However, for the EPR paradox to work, the two
particles must be heading in particular (fairly narrow) angular
cones. Such a state is not inconsistent with the decay of a spin-zero
particle but it means that all quantized values of angular momentum
are involved. In addition, the angular momentum values of the
electron must match and be opposite to those of the positron in order
to create an overall zero orbital angular momentum state. The same is
true for linear momentum: The momentum of the electron and positron
must be equal in magnitude and opposite in direction. Thus, for the
EPR experimental setup to be realized, not only are the spins
entangled but the orbital angular momentum and linear momentum states
must also be.
If all values of orbital angular momentum are present, it would seem to
be a miracle that only one spin state is present. Two 1/2 spins
can combine to give spin zero or spin one. If the spin
is one, then the orbital angular momenta of the electron and positron could
combine to give angular momentum one. Then the spin and orbital
angular momentum can combine to give total angular momentum of zero,
which is needed because the initial state has total angular momentum
zero. Once there are spin one components in the final state there is
no reason why the final spin state should be of the form
Indeed, as discussed above, if a
(|↑>−|↓>+ + |↓>−|↑>+)/√2 component is present,
would necessarily have to be present also. Although there is no
violation of a fundamental principle, it would seem extremely
difficult to generate the pure spin-zero state in practice.
By the way, the state
has the property that the first term |↑>−|↓>+
is orthogonal to the second term |↓>−|↑>+,
and by unitarity of the evolution operator U(tf,t0) this will be true
throughout the future evolution of the Universe.
Although there are unitary transformation that can eliminate the entanglement,
an example being
(− |↑>−|↓>+ +
which leads to
− |↓>−|↑>+)/√2. →
the position and electron would need to return to the same region for these transformations
to physically occur.
Hence, the spin entanglement is likely to persist forever.
The EPR experimental would also have to unfold in a perfectly pristine
environment. Any magnetic fields might cause spatial separations of
the two components Ψ1
or the flipping of a spin. There are only isolated points in the
universe where no magnetic field is present. Hence, there is no place
where the perfectly pristine environment exists.
Suppose the electron moving to the left passes by a spin 1/2 object with its
spin up. Then the component with the spin down can undergo spin
exchange with it : (|↑>−|↓>+
Here the subscript “e” stands for “environment” because the
spin 1/2 object is located in a region surrounding the left-moving
electron. After this spin exchange, the spin of the left-moving
electron is no longer entangled with the positron moving to the
right. Instead, the right-moving positron's spin is entangled with
the environmental spin 1/2 object. Such spin-exchange effects can
ruin the entanglement of the spins of the electron and positron.
Indeed, if the electron or positron interacts with anything in the environment, the EPR
experiment would be in jeopardy. In short, the EPR paradox in the
case of charged massive objects of spin-1/2 would seem to exist only as a gedanken
experiment in terms of setting up entangled spins at two distant locations.
1For objects with charge such as electrons,
there is also a a curved motion in a plane perpendicular to the
direction of the magnetic field with the “curving direction”
depending on the charge of the object.
2In the case of the
some physicists might say that until the electron hits the screen that its spin is not measured.
This involves the
which which is unsolved.
Nevertheless, I believe I can guess what takes place for the Stern–Gerlach case.
If the wavefunction for the electron is of the form Ψ(a|↑> + b|↓>)/N
when entering the gradient magnetic field
(with spin-quantization and the magnetic field both in the positive z direction)
and a pristine environment exists all the way to the screen,
then by the time the electron arrives just in front of the screen
the wavefunction will have the form
(aΨ+|↑> + bΨ−|↓>)/N
where Ψ+ is mostly non-zero above the z-plane
and Ψ− is mostly non-zero below the z-plane.
When the electron reaches the screen, the environment is no longer pristine,
the electron feels the forces produced by molecules in the screen,
and the (whole) wavefunction reacts to them evolving according to Equation (1).
In addition, the electron interacts with the molecules, which in turn, produce an abundance of photons.
There is no reason why the spin of the electron should be purely |↑> or |↓>
after the electron hits the screen.
Hence, there is no point at which the spin collapses to |↑> or |↓>.
What an experimentalist observes are electromagnetic waves coming from a rather focused
location on the screen and not the spin of the electron.
Now because the forces associated with the screen tend to be localized,
the probability that the electron interacts with the molecules at (or near) a particular location x
on the screen to produce a plethora of photons
is proportional to the probability that the electron is located there and hence proportional to
|Ψ(x)|2 = |aΨ+(x)|2/N2 +
So the chances of producing a flash above the z-plane is approximately equal to |a|2/N2
and the chances of producing a flash below the z-plane is approximately equal to |b|2/N2,
but not exactly so.
Indeed, because of the nature of solutions of Equation (1),
the non-zero region of Ψ+ extends below the z-plane and
the non-zero region of Ψ− extends above the z-plane,
so that there is contribution, albeit quite small, of have a flash above the z-plane
due to the bΨ−|↓> component of the wavefunction.
Also, the spin of the electron actually varies continuous from above the z-plane to below the z-plane.
This follows because (a|↑> + b|↓>)/N for any a and b can be written
as |↑>n for some spin-axis quantization direction n.
Defining n(x) by |↑>n(x) =
(aΨ+(x)|↑> + bΨ−(x)|↓>)/N,
one realizes that n(x) rotates from approximately +e3
to approximately -e3 continuously as one
moves from values of x well above the z-plane to values of x well below the z-plane
(Here, +e3 is a unit vector in the plus z-direction).
This means that flashes are also occurring on the screen for electrons whose spin does not point
in either the plus or minus z-direction!
Two questions arise: (i) Why does the flash occur at a particular localized region on the screen
for a single electron and (ii) Why are there not several flashes at several places on the screen?
Both these questions are related to the measurement problem and how a microscopic quantum state
interacts with its environment to create to a macroscopic effect. This is currently a
poorly understood problem.
My guess is that the screen creates a “sensitivity to conditions.”
So, each time the experiment is run, the conditions are not precisely the same and this causes the
electron to interact with one region of the screen rather than another.
It is like placing a pencil vertically on a desk:
a tiny difference causes the pencil to topple over in one particular direction.
As to question (ii), when the electron interacts with molecules in the screen either it loses
enough energy so as not to be able to interact with molecules elsewhere in the screen to produce a flash
or the wavefunction evolves (via Equation (1)) sufficiently so that it is not possible to produce a second flash.
The second possibility
allows the wavefunction to quickly become localized in the region of the flash (a weak version
of wavefunction collapse in which Ψ+(x) and Ψ−(x)
must evolve to a "common region", and such a thing
must happen when an electron with an extended wavefunction is captured
in a bound state of an atom or molecule).
Given the above, it is conceivable to me that there could be a very tiny probability for a single electron
to produce two “weak” flashes in two different screen locations.
The observation of photons emanating from a particular spot on the screen
neither determines that the wavefunction is concentrated in a particular
localized region nor that the electron's spin is a particular value
if the spin is of the form (a|↑> + b|↓>)/N with a ≠ 0 and b ≠ 0.
Multiple observations for the same initial experimental setup
allows the experimentalist to determine the value of the absolute square of the wavefunction
as a function of screen position and the values of |a| and |b| in the spin part of the total wavefunction.
This is why I say that Stern–Gerlach experiments allow one to observe spin if the spin
is up or down in the z-direction but not to measure spin,
where by “measure” I mean in the sense of forcing
a spin to assume a particular value.
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