A hallmark of Albert Einstein's career was
his use of visualized thought experiments
(German: Gedankenexperiment) as a fundamental
tool for understanding physical issues and
for elucidating his concepts to others. Einstein's
thought experiments took diverse forms. In
his youth, he mentally chased beams of light.
For special relativity, he employed moving
trains and flashes of lightning to explain
his most penetrating insights. For general
relativity, he considered a person falling
off a roof, accelerating elevators, blind
beetles crawling on curved surfaces and the
like. In his debates with Niels Bohr on the
nature of reality, he proposed imaginary devices
intended to show, at least in concept, how
the Heisenberg uncertainty principle might
be evaded. In a profound contribution to the
literature on quantum mechanics, Einstein
considered two particles briefly interacting
and then flying apart so that their states
are correlated, anticipating the phenomenon
known as quantum entanglement.
== Introduction ==
A thought experiment is a logical argument
or mental model cast within the context of
an imaginary (hypothetical or even counterfactual)
scenario. A scientific thought experiment,
in particular, may examine the implications
of a theory, law, or set of principles with
the aid of fictive and/or natural particulars
(demons sorting molecules, cats whose lives
hinge upon a radioactive disintegration, men
in enclosed elevators) in an idealized environment
(massless trapdoors, absence of friction).
They describe experiments that, except for
some specific and necessary idealizations,
could conceivably be performed in the real
world.As opposed to physical experiments,
thought experiments do not report new empirical
data. They can only provide conclusions based
on deductive or inductive reasoning from their
starting assumptions. Thought experiments
invoke particulars that are irrelevant to
the generality of their conclusions. It is
the invocation of these particulars that give
thought experiments their experiment-like
appearance. A thought experiment can always
be reconstructed as a straightforward argument,
without the irrelevant particulars. John D.
Norton, a well-known philosopher of science,
has noted that "a good thought experiment
is a good argument; a bad thought experiment
is a bad argument."When effectively used,
the irrelevant particulars that convert a
straightforward argument into a thought experiment
can act as "intuition pumps" that stimulate
readers' ability to apply their intuitions
to their understanding of a scenario. Thought
experiments have a long history. Perhaps the
best known in the history of modern science
is Galileo's demonstration that falling objects
must fall at the same rate regardless of their
masses. This has sometimes been taken to be
an actual physical demonstration, involving
his climbing up the Leaning Tower of Pisa
and dropping two heavy weights off it. In
fact, it was a logical demonstration described
by Galileo in Discorsi e dimostrazioni matematiche
(1638).Einstein had a highly visual understanding
of physics. His work in the patent office
"stimulated [him] to see the physical ramifications
of theoretical concepts." These aspects of
his thinking style inspired him to fill his
papers with vivid practical detail making
them quite different from, say, the papers
of Lorentz or Maxwell. This included his use
of thought experiments.
== Special relativity ==
=== 
Pursuing a beam of light ===
Late in life, Einstein recalled
...a paradox upon which I had already hit
at the age of sixteen: If I pursue a beam
of light with the velocity c (velocity of
light in a vacuum), I should observe such
a beam of light as an electromagnetic field
at rest though spatially oscillating. There
seems to be no such thing, however, neither
on the basis of experience nor according to
Maxwell's equations. From the very beginning
it appeared to me intuitively clear that,
judged from the standpoint of such an observer,
everything would have to happen according
to the same laws as for an observer who, relative
to the earth, was at rest. For how should
the first observer know or be able to determine,
that he is in a state of fast uniform motion?
One sees in this paradox the germ of the special
relativity theory is already contained.
Einstein's recollections of his youthful musings
are widely cited because of the hints they
provide of his later great discovery. However,
Norton has noted that Einstein's reminiscences
were probably colored by a half-century of
hindsight. Norton lists several problems with
Einstein's recounting, both historical and
scientific:
1. At 16 years old and a student at the Gymnasium
in Aarau, Einstein would have had the thought
experiment in late 1895 to early 1896. But
various sources note that Einstein did not
learn Maxwell's theory until 1898, in university.
2. The second issue is that a 19th century
aether theorist would have had no difficulties
with the thought experiment. Einstein's statement,
"...there seems to be no such thing...on the
basis of experience," would not have counted
as an objection, but would have represented
a mere statement of fact, since no one had
ever traveled at such speeds.
3. An aether theorist would have regarded
"...nor according to Maxwell's equations"
as simply representing a misunderstanding
on Einstein's part. Unfettered by any notion
that the speed of light represents a cosmic
limit, the aether theorist would simply have
set velocity equal to c, noted that yes indeed,
the light would appear to be frozen, and then
thought no more of it.Rather than the thought
experiment being at all incompatible with
aether theories (which it is not), the youthful
Einstein appears to have reacted to the scenario
out of an intuitive sense of wrongness. He
felt that the laws of optics should obey the
principle of relativity. As he grew older,
his early thought experiment acquired deeper
levels of significance: Einstein felt that
Maxwell's equations should be the same for
all observers in inertial motion. From Maxwell's
equations, one can deduce a single speed of
light, and there is nothing in this computation
that depends on an observer's speed. Einstein
sensed a conflict between Newtonian mechanics
and the constant speed of light determined
by Maxwell's equations.Regardless of the historical
and scientific issues described above, Einstein's
early thought experiment was part of the repertoire
of test cases that he used to check on the
viability of physical theories. Norton suggests
that the real importance of the thought experiment
was that it provided a powerful objection
to emission theories of light, which Einstein
had worked on for several years prior to 1905.
=== Magnet and conductor ===
In the very first paragraph of Einstein's
seminal 1905 work introducing special relativity,
he writes: It is well known that Maxwell's
electrodynamics—as usually understood at
present—when applied to moving bodies, leads
to asymmetries that do not seem to attach
to the phenomena. Let us recall, for example,
the electrodynamic interaction between a magnet
and a conductor. The observable phenomenon
depends here only on the relative motion of
conductor and magnet, while according to the
customary conception the two cases, in which,
respectively, either the one or the other
of the two bodies is the one in motion, are
to be strictly differentiated from each other.
For if the magnet is in motion and the conductor
is at rest, there arises in the surroundings
of the magnet an electric field endowed with
a certain energy value that produces a current
in the places where parts of the conductor
are located. But if the magnet is at rest
and the conductor is in motion, no electric
field arises in the surroundings of the magnet,
while in the conductor an electromotive force
will arise, to which in itself there does
not correspond any energy, but which, provided
that the relative motion in the two cases
considered is the same, gives rise to electrical
currents that have the same magnitude and
the same course as those produced by the electric
forces in the first-mentioned case.
This opening paragraph recounts well-known
experimental results obtained by Michael Faraday
in 1831. The experiments describe what appeared
to be two different phenomena: the motional
EMF generated when a wire moves through a
magnetic field (see Lorentz force), and the
transformer EMF generated by a changing magnetic
field (due to the Maxwell–Faraday equation).
James Clerk Maxwell himself drew attention
to this fact in his 1861 paper On Physical
Lines of Force. In the latter half of Part
II of that paper, Maxwell gave a separate
physical explanation for each of the two phenomena.Although
Einstein calls the asymmetry "well-known",
there is no evidence that any of Einstein's
contemporaries considered the distinction
between motional EMF and transformer EMF to
be in any way odd or pointing to a lack of
understanding of the underlying physics. Maxwell,
for instance, had repeatedly discussed Faraday's
laws of induction, stressing that the magnitude
and direction of the induced current was a
function only of the relative motion of the
magnet and the conductor, without being bothered
by the clear distinction between conductor-in-motion
and magnet-in-motion in the underlying theoretical
treatment.Yet Einstein's reflection on this
experiment represented the decisive moment
in his long and tortuous path to special relativity.
Although the equations describing the two
scenarios are entirely different, there is
no measurement that can distinguish whether
the magnet is moving, the conductor is moving,
or both.In a 1920 review on the Fundamental
Ideas and Methods of the Theory of Relativity
(unpublished), Einstein related how disturbing
he found this asymmetry:
The idea that these two cases should essentially
be different was unbearable to me. According
to my conviction, the difference between the
two could only lie in the choice of the point
of view, but not in a real difference .
Einstein needed to extend the relativity of
motion that he perceived between magnet and
conductor in the above thought experiment
to a full theory. For years, however, he did
not know how this might be done. The exact
path that Einstein took to resolve this issue
is unknown. We do know, however, that Einstein
spent several years pursuing an emission theory
of light, encountering difficulties that eventually
led him to give up the attempt.
Gradually I despaired of the possibility of
discovering the true laws by means of constructive
efforts based on known facts. The longer and
more desperately I tried, the more I came
to the conviction that only the discovery
of a universal formal principle could lead
us to assured results.
That decision ultimately led to his development
of special relativity as a theory founded
on two postulates of which he could be sure.
Expressed in contemporary physics vocabulary,
his postulates were as follows:
1. The laws of physics take the same form
in all inertial frames.
2. In any given inertial frame, the velocity
of light c is the same whether the light be
emitted by a body at rest or by a body in
uniform motion. [Emphasis added by editor]Einstein's
wording of the second postulate was one with
which nearly all theorists of his day could
agree. His wording is a far more intuitive
form of the second postulate than the stronger
version frequently encountered in popular
writings and college textbooks.
=== Trains, embankments, and lightning flashes
===
The topic of how Einstein arrived at special
relativity has been a fascinating one to many
scholars: A lowly, twenty-six year old patent
officer (third class), largely self-taught
in physics and completely divorced from mainstream
research, nevertheless in the year 1905 produced
four extraordinary works (Annus Mirabilis
papers), only one of which (his paper on Brownian
motion) appeared related to anything that
he had ever published before.Einstein's paper,
On the Electrodynamics of Moving Bodies, is
a polished work that bears few traces of its
gestation. Documentary evidence concerning
the development of the ideas that went into
it consist of, quite literally, only two sentences
in a handful of preserved early letters, and
various later historical remarks by Einstein
himself, some of them known only second-hand
and at times contradictory.
In regards to the relativity of simultaneity,
Einstein's 1905 paper develops the concept
vividly by carefully considering the basics
of how time may be disseminated through the
exchange of signals between clocks. In his
popular work, Relativity: The Special and
General Theory, Einstein translates the formal
presentation of his paper into a thought experiment
using a train, a railway embankment, and lightning
flashes. The essence of the thought experiment
is as follows:
Observer M stands on an embankment, while
observer M' rides on a rapidly traveling train.
At the precise moment that M and M' coincide
in their positions, lightning strikes points
A and B equidistant from M and M'.
Light from these two flashes reach M at the
same time, from which M concludes that the
bolts were synchronous.
The combination of Einstein's first and second
postulates implies that, despite the rapid
motion of the train relative to the embankment,
M' measures exactly the same speed of light
as does M. Since M' was equidistant from A
and B when lightning struck, the fact that
M' receives light from B before light from
A means that to M', the bolts were not synchronous.
Instead, the bolt at B struck first. A routine
supposition among historians of science is
that, in accordance with the analysis given
in his 1905 special relativity paper and in
his popular writings, Einstein discovered
the relativity of simultaneity by thinking
about how clocks could be synchronized by
light signals. The Einstein synchronization
convention was originally developed by telegraphers
in the middle 19th century. The dissemination
of precise time was an increasingly important
topic during this period. Trains needed accurate
time to schedule use of track, cartographers
needed accurate time to determine longitude,
while astronomers and surveyors dared to consider
the worldwide dissemination of time to accuracies
of thousandths of a second. Following this
line of argument, Einstein's position in the
patent office, where he specialized in evaluating
electromagnetic and electromechanical patents,
would have exposed him to the latest developments
in time technology, which would have guided
him in his thoughts towards understanding
the relativity of simultaneity.However, all
of the above is supposition. In later recollections,
when Einstein was asked about what inspired
him to develop special relativity, he would
mention his riding a light beam and his magnet
and conductor thought experiments. He would
also mention the importance of the Fizeau
experiment and the observation of stellar
aberration. "They were enough", he said. He
never mentioned thought experiments about
clocks and their synchronization.The routine
analyses of the Fizeau experiment and of stellar
aberration, that treat light as Newtonian
corpuscles, do not require relativity. But
problems arise if one considers light as waves
traveling through an aether, which are resolved
by applying the relativity of simultaneity.
It is entirely possible, therefore, that Einstein
arrived at special relativity through a different
path than that commonly assumed, through Einstein's
examination of Fizeau's experiment and stellar
aberration.We therefore do not know just how
important clock synchronization and the train
and embankment thought experiment were to
Einstein's development of the concept of the
relativity of simultaneity. We do know, however,
that the train and embankment thought experiment
was the preferred means whereby he chose to
teach this concept to the general public.
== General relativity ==
=== 
Falling painters and accelerating elevators
===
In his unpublished 1920 review, Einstein related
the genesis of his thoughts on the equivalence
principle: When I was busy (in 1907) writing
a summary of my work on the theory of special
relativity for the Jahrbuch für Radioaktivität
und Elektronik [Yearbook for Radioactivity
and Electronics], I also had to try to modify
the Newtonian theory of gravitation such as
to fit its laws into the theory. While attempts
in this direction showed the practicability
of this enterprise, they did not satisfy me
because they would have had to be based upon
unfounded physical hypotheses. At that moment
I got the happiest thought of my life in the
following form: In an example worth considering,
the gravitational field has a relative existence
only in a manner similar to the electric field
generated by magneto-electric induction. Because
for an observer in free-fall from the roof
of a house there is during the fall—at least
in his immediate vicinity—no gravitational
field. Namely, if the observer lets go of
any bodies, they remain relative to him, in
a state of rest or uniform motion, independent
of their special chemical or physical nature.
The observer, therefore, is justified in interpreting
his state as being "at rest."
The realization "startled" Einstein, and inspired
him to begin an eight-year quest that led
to what is considered to be his greatest work,
the theory of general relativity. Over the
years, the story of the falling man has become
an iconic one, much embellished by other writers.
In most retellings of Einstein's story, the
falling man is identified as a painter. In
some accounts, Einstein was inspired after
he witnessed a painter falling from the roof
of a building adjacent to the patent office
where he worked. This version of the story
leaves unanswered the question of why Einstein
might consider his observation of such an
unfortunate accident to represent the happiest
thought in his life.
Einstein later refined his thought experiment
to consider a man inside a large enclosed
chest or elevator falling freely in space.
While in free fall, the man would consider
himself weightless, and any loose objects
that he emptied from his pockets would float
alongside him. Then Einstein imagined a rope
attached to the roof of the chamber. A powerful
"being" of some sort begins pulling on the
rope with constant force. The chamber begins
to move "upwards" with a uniformly accelerated
motion. Within the chamber, all of the man's
perceptions are consistent with his being
in a uniform gravitational field. Einstein
asked, "Ought we to smile at the man and say
that he errs in his conclusion?" Einstein
answered no. Rather, the thought experiment
provided "good grounds for extending the principle
of relativity to include bodies of reference
which are accelerated with respect to each
other, and as a result we have gained a powerful
argument for a generalised postulate of relativity."
Through this thought experiment, Einstein
addressed an issue that was so well known,
scientists rarely worried about it or considered
it puzzling: Objects have "gravitational mass,"
which determines the force with which they
are attracted to other objects. Objects also
have "inertial mass," which determines the
relationship between the force applied to
an object and how much it accelerates. Newton
had pointed out that, even though they are
defined differently, gravitational mass and
inertial mass always seem to be equal. But
until Einstein, no one had conceived a good
explanation as to why this should be so. From
the correspondence revealed by his thought
experiment, Einstein concluded that "it is
impossible to discover by experiment whether
a given system of coordinates is accelerated,
or whether...the observed effects are due
to a gravitational field." This correspondence
between gravitational mass and inertial mass
is the equivalence principle.An extension
to his accelerating observer thought experiment
allowed Einstein to deduce that "rays of light
are propagated curvilinearly in gravitational
fields."
== 
Quantum mechanics ==
=== Background: Einstein and the quantum ===
Many myths have grown up about Einstein's
relationship with quantum mechanics. Freshman
physics students are aware that Einstein explained
the photoelectric effect and introduced the
concept of the photon. But students who have
grown up with the photon may not be aware
of how revolutionary the concept was for his
time. The best-known factoids about Einstein's
relationship with quantum mechanics are his
statement, "God does not play dice" and the
indisputable fact that he just didn't like
the theory in its final form. This has led
to the general impression that, despite his
initial contributions, Einstein was out of
touch with quantum research and played at
best a secondary role in its development.
Concerning Einstein's estrangement from the
general direction of physics research after
1925, his well-known scientific biographer,
Abraham Pais, wrote:
Einstein is the only scientist to be justly
held equal to Newton. That comparison is based
exclusively on what he did before 1925. In
the remaining 30 years of his life he remained
active in research but his fame would be undiminished,
if not enhanced, had he gone fishing instead.
In hindsight, we know that Pais was incorrect
in his assessment.
Einstein was arguably the greatest single
contributor to the "old" quantum theory.
In his 1905 paper on light quanta, Einstein
created the quantum theory of light. His proposal
that light exists as tiny packets (photons)
was so revolutionary, that even such major
pioneers of quantum theory as Planck and Bohr
refused to believe that it could be true.
Bohr, in particular, was a passionate disbeliever
in light quanta, and repeatedly argued against
them until 1925, when he yielded in the face
of overwhelming evidence for their existence.
In his 1906 theory of specific heats, Einstein
was the first to realize that quantized energy
levels explained the specific heat of solids.
In this manner, he found a rational justification
for the third law of thermodynamics (i.e.
the entropy of any system approaches zero
as the temperature approaches absolute zero):
at very cold temperatures, atoms in a solid
don't have enough thermal energy to reach
even the first excited quantum level, and
so cannot vibrate.
Einstein proposed the wave-particle duality
of light. In 1909, using a rigorous fluctuation
argument based on a thought experiment and
drawing on his previous work on Brownian motion,
he predicted the emergence of a "fusion theory"
that would combine the two views. Basically,
he demonstrated that the Brownian motion experienced
by a mirror in thermal equilibrium with black
body radiation would be the sum of two terms,
one due to the wave properties of radiation,
the other due to its particulate properties.
Although Planck is justly hailed as the father
of quantum mechanics, his derivation of the
law of black-body radiation rested on fragile
ground, since it required ad hoc assumptions
of an unreasonable character. Furthermore,
Planck's derivation represented an analysis
of classical harmonic oscillators merged with
quantum assumptions in an improvised fashion.
In his 1916 theory of radiation, Einstein
was the first to create a purely quantum explanation.
This paper, well known for broaching the possibility
of stimulated emission (the basis of the laser),
changed the nature of the evolving quantum
theory by introducing the fundamental role
of random chance.
In 1924, Einstein received a short manuscript
by an unknown Indian professor, Satyendra
Nath Bose, outlining a new method of deriving
the law of blackbody radiation. Einstein was
intrigued by Bose's peculiar method of counting
the number of distinct ways of putting photons
into the available states, a method of counting
that Bose apparently did not realize was unusual.
Einstein, however, understood that Bose's
counting method implied that photons are,
in a deep sense, indistinguishable. He translated
the paper into German and had it published.
Einstein then followed Bose's paper with an
extension to Bose's work which predicted Bose-Einstein
condensation, one of the fundamental research
topics of condensed matter physics.
While trying to develop a mathematical theory
of light which would fully encompass its wavelike
and particle-like aspects, Einstein developed
the concept of "ghost fields". A guiding wave
obeying Maxwell's classical laws would propagate
following the normal laws of optics, but would
not transmit any energy. This guiding wave,
however, would govern the appearance of quanta
of energy
h
ν
{\displaystyle h\nu }
on a statistical basis, so that the appearance
of these quanta would be proportional to the
intensity of the interference radiation. These
ideas became widely known in the physics community,
and through Born's work in 1926, later became
a key concept in the modern quantum theory
of radiation and matter. Therefore, Einstein
before 1925 originated most of the key concepts
of quantum theory: light quanta, wave-particle
duality, the fundamental randomness of physical
processes, the concept of indistinguishability,
and the probability density interpretation
of the wave equation. In addition, Einstein
can arguably be considered the father of solid
state physics and condensed matter physics.
He provided a correct derivation of the blackbody
radiation law and sparked the notion of the
laser.
What of after 1925? In 1935, working with
two younger colleagues, Einstein issued a
final challenge to quantum mechanics, attempting
to show that it could not represent a final
solution. Despite the questions raised by
this paper, it made little or no difference
to how physicists employed quantum mechanics
in their work. Of this paper, Pais was to
write:
The only part of this article that will ultimately
survive, I believe, is this last phrase [i.e.
"No reasonable definition of reality could
be expect to permit this" where "this" refers
to the instantaneous transmission of information
over a distance], which so poignantly summarizes
Einstein's views on quantum mechanics in his
later years....This conclusion has not affected
subsequent developments in physics, and it
is doubtful that it ever will.
In contrast to Pais' negative assessment,
this paper, outlining the EPR paradox, is
currently among the top ten papers published
in Physical Review, and is the centerpiece
of the development of quantum information
theory, which has been termed the "third quantum
revolution."
=== 
Einstein's light box ===
Einstein did not like the direction in which
quantum mechanics had turned after 1925. Although
excited by Heisenberg's matrix mechanics,
Schroedinger's wave mechanics, and Born's
clarification of the meaning of the Schroedinger
wave equation (i.e. that the absolute square
of the wave function is to be interpreted
as a probability density), his instincts told
him that something was missing. In a letter
to Born, he wrote:
Quantum mechanics is very impressive. But
an inner voice tells me that it is not yet
the real thing. The theory produces a good
deal but hardly brings us closer to the secret
of the Old One.
The Solvay Debates between Bohr and Einstein
began in dining-room discussions at the Fifth
Solvay International Conference on Electrons
and Photons in 1927. Einstein's issue with
the new quantum mechanics was not just that,
with the probability interpretation, it rendered
invalid the notion of rigorous causality.
After all, as noted above, Einstein himself
had introduced random processes in his 1916
theory of radiation. Rather, by defining and
delimiting the maximum amount of information
obtainable in a given experimental arrangement,
the Heisenberg uncertainty principle denied
the existence of any knowable reality in terms
of a complete specification of the momenta
and description of individual particles, an
objective reality that would exist whether
or not we could ever observe it. Over dinner,
during after-dinner discussions, and at breakfast,
Einstein debated with Bohr and his followers
on the question whether quantum mechanics
in its present form could be called complete.
Einstein illustrated his points with increasingly
clever thought experiments intended to prove
that position and momentum could in principle
be simultaneously known to arbitrary precision.
For example, one of his thought experiments
involved sending a beam of electrons through
a shuttered screen, recording the positions
of the electrons as they struck a photographic
screen. Bohr and his allies would always be
able to counter Einstein's proposal, usually
by the end of the same day.On the final day
of the conference, Einstein revealed that
the uncertainty principle was not the only
aspect of the new quantum mechanics that bothered
him. Quantum mechanics, at least in the Copenhagen
interpretation, appeared to allow action at
a distance, the ability for two separated
objects to communicate at speeds greater than
light. By 1928, the consensus was that Einstein
had lost the debate, and even his closest
allies during the Fifth Solvay Conference,
for example Louis de Broglie, conceded that
quantum mechanics appeared to be complete.
At the Sixth Solvay International Conference
on Magnetism (1930), Einstein came armed with
a new thought experiment. This involved a
box with a shutter that operated so quickly,
it would allow only one photon to escape at
a time. The box would first be weighed exactly.
Then, at a precise moment, the shutter would
open, allowing a photon to escape. The box
would then be re-weighed. The well-known relationship
between mass and energy
E
=
m
c
2
{\displaystyle E=mc^{2}}
would allow the energy of the particle to
be precisely determined. With this gadget,
Einstein believed that he had demonstrated
a means to obtain, simultaneously, a precise
determination of the energy of the photon
as well as its exact time of departure from
the system. Bohr was shaken by this thought
experiment. Unable to think of a refutation,
he went from one conference participant to
another, trying to convince them that Einstein's
thought experiment couldn't be true, that
if it were true, it would literally mean the
end of physics. After a sleepless night, he
finally worked out a response which, ironically,
depended on Einstein's general relativity.
Consider the illustration of Einstein's light
box:
1. After emitting a photon, the loss of weight
causes the box to rise in the gravitational
field.
2. The observer returns the box to its original
height by adding weights until the pointer
points to its initial position. It takes a
certain amount of time
t
{\displaystyle t}
for the observer to perform this procedure.
How long it takes depends on the strength
of the spring and on how well-damped the system
is. If undamped, the box will bounce up and
down forever. If over-damped, the box will
return to its original position sluggishly
(See Damped spring-mass system).
3. The longer that the observer allows the
damped spring-mass system to settle, the closer
the pointer will reach its equilibrium position.
At some point, the observer will conclude
that his setting of the pointer to its initial
position is within an allowable tolerance.
There will be some residual error
Δ
q
{\displaystyle \Delta q}
in returning the pointer to its initial position.
Correspondingly, there will be some residual
error
Δ
m
{\displaystyle \Delta m}
in the weight measurement.
4. Adding the weights imparts a momentum
p
{\displaystyle p}
to the box which can be measured with an accuracy
Δ
p
{\displaystyle \Delta p}
delimited by
Δ
p
Δ
q
≈
h
.
{\displaystyle \Delta p\Delta q\approx h.}
It is clear that
Δ
p
<
g
t
Δ
m
,
{\displaystyle \Delta p<gt\Delta m,}
where
g
{\displaystyle g}
is the gravitational constant. Plugging in
yields
g
t
Δ
m
Δ
q
>
h
.
{\displaystyle gt\Delta m\Delta q>h.}
5. General relativity informs us that while
the box has been at a height different than
its original height, it has been ticking at
a rate different than its original rate. The
red shift formula informs us that there will
be an uncertainty
Δ
t
=
c
−
2
g
t
Δ
q
{\displaystyle \Delta t=c^{-2}gt\Delta q}
in the determination of
t
0
,
{\displaystyle t_{0},}
the emission time of 
the photon.
6. Hence,
c
2
Δ
m
Δ
t
=
Δ
E
Δ
t
>
h
.
{\displaystyle c^{2}\Delta m\Delta t=\Delta
E\Delta t>h.}
The accuracy with which the energy of the
photon is measured restricts the precision
with which its moment of emission can be measured,
following the Heisenberg uncertainty principle.After
finding his last attempt at finding a loophole
around the uncertainty principle refuted,
Einstein quit trying to search for inconsistencies
in quantum mechanics. Instead, he shifted
his focus to the other aspects of quantum
mechanics with which he was uncomfortable,
focusing on his critique of action at a distance.
His next paper on quantum mechanics foreshadowed
his later paper on the EPR paradox.Einstein
was gracious in his defeat. The following
September, Einstein nominated Heisenberg and
Schroedinger for the Nobel Prize, stating,
"I am convinced that this theory undoubtedly
contains a part of the ultimate truth."
=== EPR Paradox ===
Both Bohr and Einstein were subtle men. Einstein
tried very hard to show that quantum mechanics
was inconsistent; Bohr, however, was always
able to counter his arguments. But in his
final attack Einstein pointed to something
so deep, so counterintuitive, so troubling,
and yet so exciting, that at the beginning
of the twenty-first century it has returned
to fascinate theoretical physicists. Bohr’s
only answer to Einstein’s last great discovery—the
discovery of entanglement—was to ignore
it.
Einstein's fundamental dispute with quantum
mechanics wasn't about whether God rolled
dice, whether the uncertainty principle allowed
simultaneous measurement of position and momentum,
or even whether quantum mechanics was complete.
It was about reality. Does a physical reality
exist independent of our ability to observe
it? To Bohr and his followers, such questions
were meaningless. All that we can know are
the results of measurements and observations.
It makes no sense to speculate about an ultimate
reality that exists beyond our perceptions.Einstein's
beliefs had evolved over the years from those
that he had held when he was young, when,
as a logical positivist heavily influenced
by his reading of David Hume and Ernst Mach,
he had rejected such unobservable concepts
as absolute time and space. Einstein believed:
1. A reality exists independent of our ability
to observe it.
2. Objects are located at distinct points
in spacetime and have their own independent,
real existence. In other words, he believed
in separability and locality.
3. Although at a superficial level, quantum
events may appear random, at some ultimate
level, strict causality underlies all processes
in nature.
Einstein considered that realism and localism
were fundamental underpinnings of physics.
After leaving Nazi Germany and settling in
Princeton at the Institute for Advanced Studies,
Einstein began writing up a thought experiment
that he had been mulling over since attending
a lecture by Léon Rosenfeld in 1933. Since
the paper was to be in English, Einstein enlisted
the help of the 46-year-old Boris Podolsky,
a fellow who had moved to the Institute from
Caltech; he also enlisted the help of the
26-year-old Nathan Rosen, also at the Institute,
who did much of the math. The result of their
collaboration was the four page EPR paper,
which in its title asked the question Can
Quantum-Mechanical Description of Physical
Reality be Considered Complete? After seeing
the paper in print, Einstein found himself
unhappy with the result. His clear conceptual
visualization had been buried under layers
of mathematical formalism.Einstein's thought
experiment involved two particles that have
collided or which have been created in such
a way that they have properties which are
correlated. The total wave function for the
pair links the positions of the particles
as well as their linear momenta. The figure
depicts the spreading of the wave function
from the collision point. However, observation
of the position of the first particle allows
us to determine precisely the position of
the second particle no matter how far the
pair have separated. Likewise, measuring the
momentum of the first particle allows us to
determine precisely the momentum of the second
particle. "In accordance with our criterion
for reality, in the first case we must consider
the quantity P as being an element of reality,
in the second case the quantity Q is an element
of reality."Einstein concluded that the second
particle, which we have never directly observed,
must have at any moment a position that is
real and a momentum that is real. Quantum
mechanics does not account for these features
of reality. Therefore, quantum mechanics is
not complete. It is known, from the uncertainty
principle, that position and momentum cannot
be measured at the same time. But even though
their values can only be determined in distinct
contexts of measurement, can they both be
definite at the same time? Einstein concluded
that the answer must be yes.The only alternative,
claimed Einstein, would be to assert that
measuring the first particle instantaneously
affected the reality of the position and momentum
of the second particle. "No reasonable definition
of reality could be expected to permit this."Bohr
was stunned when he read Einstein's paper
and spent more than six weeks framing his
response, which he gave exactly the same title
as the EPR paper. The EPR paper forced Bohr
to make a major revision in his understanding
of complementarity in the Copenhagen interpretation
of quantum mechanics.Prior to EPR, Bohr had
maintained that disturbance caused by the
act of observation was the physical explanation
for quantum uncertainty. In the EPR thought
experiment, however, Bohr had to admit that
"there is no question of a mechanical disturbance
of the system under investigation." On the
other hand, he noted that the two particles
were one system described by one quantum function.
Furthermore, the EPR paper did nothing to
dispel the uncertainty principle. Later commentators
have questioned the strength and coherence
of Bohr's response. As a practical matter,
however, physicists for the most part did
not pay much attention to the debate between
Bohr and Einstein, since the opposing views
did not affect one's ability to apply quantum
mechanics to practical problems, but only
affected one's interpretation of the quantum
formalism. If they thought about the problem
at all, most working physicists tended to
follow Bohr's leadership.So stood the situation
for nearly 30 years. Then, in 1964, John Stewart
Bell made the groundbreaking discovery that
Einstein's local realist world view made experimentally
verifiable predictions that would be in conflict
with those of quantum mechanics. Bell's discovery
shifted the Einstein–Bohr debate from philosophy
to the realm of experimental physics. Bell's
theorem showed that, for any local realist
formalism, there exist limits on the predicted
correlations between pairs of particles in
an experimental realization of the EPR thought
experiment. In 1972, the first experimental
tests were carried out. Successive experiments
improved the accuracy of observation and closed
loopholes. To date, it is virtually certain
that local realist theories have been falsified.So
Einstein was wrong. But after decades of relative
neglect, the EPR paper has been recognized
as prescient, since it identified the phenomenon
of quantum entanglement. It has several times
been the case that Einstein's "mistakes" have
foreshadowed and provoked major shifts in
scientific research. Such, for instance, has
been the case with his proposal of the cosmological
constant, which Einstein considered his greatest
blunder, but which currently is being actively
investigated for its possible role in the
accelerating expansion of the universe. In
his Princeton years, Einstein was virtually
shunned as he pursued the unified field theory.
Nowadays, innumerable physicists pursue Einstein's
dream for a "theory of everything."The EPR
paper did not prove quantum mechanics to be
incorrect. What it did prove was that quantum
mechanics, with its "spooky action at a distance,"
is completely incompatible with commonsense
understanding. Furthermore, the effect predicted
by the EPR paper, quantum entanglement, has
inspired approaches to quantum mechanics different
from the Copenhagen interpretation, and has
been at the forefront of major technological
advances in quantum computing, quantum encryption,
and quantum information theory.
== Notes ==
== Primary sources
