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1 Deriving the Born Rule from a model of the quantum measurement process Alan Schaum Abstract
The quantum mechanics postulate called the Born Rule attributes a probabilistic meaning to a
wave function. This paper derives the Born Rule from other quantum principles along with a
model of the measurement process.
The nondeterministic nature of quantum measurements is hypothesized to arise from an
ignorance of the quantum states of a measuring device’s microscopic components. Their
interactions with a system to be measured are modeled heuristically with any member of a
particular class of stochastic processes, each of which generate the Born Rule. One member
of the class appears particularly compatible with properties expected of quantum interactions.
Background
Measurement in the Copenhagen interpretation remains an unexplained
process, since there is nothing in the mathematics of quantum mechanics
that specifies how or why the wave function collapses. - ([1] p. 318)
In the early twentieth century new principles were being integrated with classical physics ideas
in order to explain quantum phenomena. Orbital angular momentum was quantized in the Bohr
model of the atom, then magnetic quantum numbers and electron spin were introduced,
ultimately to be replaced by the self-consistent quantum mechanics of Heisenberg,
Schrödinger, and Dirac. Electronic interactions with photons and the physical basis of the
heuristic Pauli exclusion principle had to await the more general framework of quantum field
theory for deeper explanations.
But one axiom appended to early quantum principles, the Born Rule, continues to lack a
consensus explanation after 100 years. This paper offers a less exotic explanation than most,
describing how the effect of a quantum measurement can be modeled as a series of simple
interactions of a quantum system with the components of the measuring device.
Introduction
First a general model is introduced to describe the “collapse” of a quantum mechanical wave
function that a measurement induces. Then a more detailed class of associated mathematical
processes is introduced that explains how a mixed quantum state S could be transformed by a
measuring device D into an eigenstate of the observable O that the device is designed to
measure.
In classical physics no general definition of measurement seems necessary. One knows it when
one sees it. But more is required in quantum mechanics, given the mystery of what happens to
a wave function during a measurement. Quantum mechanics predicts that some systems can
exist as superpositions of discrete eigenstates of O, and because a conventional measurement
can produce only one of the associated eigenvalues, it is generally agreed that the act of
measuring transforms perform a mixed quantum state into an eigenstate. It also seems
incontrovertible that:
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I.
A measurement of an observable O is always made by a macroscopic device D.
II.
Associated with D is a preferred basis in Hilbert space, the eigenvectors of O.
III. D reports any pure state faithfully and never reports a state absent from S.
The evolution of S into an eigenstate is governed by probabilities and so should be
interpretable as a stochastic process. In the model proposed here, D’s microscopic
components perturb S serially according to any one of a well-defined class of stochastic
processes. The perturbations produce a series of complex rotations (unitary transformations) of
the Hilbert-space vector associated with S. The models all predict that every such process
ends in some eigenstate of O, and with probabilities defined by the Born Rule. Those
eigenstates also define the basis in which the stochastic models are most easily formulated.
The effect of a measurement is described better as a random realignment of the quantum state
vector, which remains normalized, rather than a collapse (as in a projection). Thus the models
preserve vector length at every step. Also, like the Born Rule itself, the models do not relate
directly to any particular physical mechanism of measurement. The hope is that, for any
particular type of device, ultimately some member of the proposed class of models will be
recognized as consistent with a more focused quantum mechanical description of the
associated measurement process. However, one member of the class of model processes
(defined by equation (12)) seems particularly appropriate as a general candidate.
The quantum state S initially consists of a superposition of some
eigenstates of O with
amplitudes whose squared magnitudes (here called intensities) are perturbed sequentially upon
each encounter with a component of D. Each encounter is in turn described by a sequence of
more elementary binary processes, in which a small intensity amount
is removed from a
randomly chosen eigenstate, and then is recaptured by a random eigenstate. The many
microscopic components of the macroscopic D accommodate a potentially large number of
such encounters (~1023 if necessary). The idea of modeling these interactions with S as a
sequence of small changes is motivated by considering a measuring device that has been
reduced in complexity down to a single molecule, which can be expected only to perturb a
quantum state S, not necessarily transform it into an eigenstate.
Elementary stochastic process
The initial
intensity values are modeled as integers
times , with
, where
is a
large integer and
. (1)
The state S is represented here as S
, not the usual Hilbert space vector,
because the complex phases of the wave function do not figure in the models. The condition of
equation (1) is maintained throughout the progression of S to its final pure eigenstate.
The evolution of intensity
proceeds as follows. Some model transition probability distribution
is defined that obeys the constraints:
. (2)
M
ϵ
M
ai
ϵ
ϵ = 1/N
N
M
∑
i=1
ai = N
= (a1, a2, … , aM)ϵ
ai
pM(ai)
M
∑
i=1
pM(ai) = 1, pM(0) = 0, pM(ai) ≠0 if ai ≠0
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3 A transaction between two randomly selected intensities is initiated in which an amount can be exchanged, according to the following process. With probability , intensity
donates an amount
to a repository. Or with probability
, some
other intensity donates . Regardless of the source of the donation,
re-acquires
with
probability
, and some other intensity acquires it with total probability
.
The second equality in conditions (2) is required by the general model, because once an
intensity reaches zero, it can no longer contribute to the repository. That equality then also
insures that once a component exits the evolution process, it will not return. This effects a
“stopping rule” inherent to the process. Once
reaches zero, it stays there. Also, during the
evolution of S, with probability one some
will reach zero (at least in the limit of an infinite
number of interactions). Thereafter the process continues until only one nonzero intensity
survives. The inequality in condition (2) insures that a nonzero component of the evolving S
cannot defect from the evolution process. Thus (2) guarantees requirement III above.
Note that the evolution of intensity
is governed by only
and is not dependent on the
individual probabilities
. Therefore, in the stochastic process of
evolving into
either
or 0, the labels on the intensities where the other
-intensities reside have no
effect. Similarly, binary exchanges occurring amongst those eigenstates do not effect the
probabilistic fate of intensity
. Consequently, the probability
that
will evolve asymptotically to the value 1 has the same value as it would in a competition with a pooled version of all other intensities. Thus, the result can be used to predict the more general result, and the probability that the th eigenstate is reached can be expressed as . (3)
To find the probabilities for the case , define , the probability that the initial state evolves into the pure eigenstate . The stochastic evolution of the first intensity is governed at each step according to various transition possibilities: (4)
The first line corresponds to where can land if is the donor, the second if the other intensity donates to the repository. Dividing equation (4) by yields the surprising result: , (5) ϵ pM(ai) ϵai ϵ M ∑ j=1,j≠i pM(aj) = 1 −pM(ai) ϵ ϵai ϵ pM(ai) 1 −pM(ai) ai ai ϵai pM(ai) pM(aj), j ≠i ai N N −ai ϵ ai Pi(a1, … , ai, … , aM) ai M = 2 i Pi(a1, … , ai, … , aM) = P1(ai, N −ai) M = 2 M = 2 P(a) ≡P1(a, N −a) S = (a, N −a)ϵ (1, 0) ϵa P(a) = pM(a)((pM(a))P(a) + (1 −pM(a))P(a −1)) +(1 −pM(a))(pM(a)P(a + 1) + (1 −pM(a))P(a)) ϵ ϵa pM(a)(1 −pM(a)) P(a + 1) = 2P(a) −P(a −1)
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which is independent of
! The division is always allowed because, according to (2), either
divisor is zero only when the evolution process for terminates, at either
or
.
Because equation (5) requires that second-order differences of
be zero, its general
solution has the form
. The boundary conditions
and
then
yield
. From equation (3) it then follows that
. (6)
This is the Born Rule: The probability that the state evolves into its th eigenstate equals that
eigenstate’s initial intensity. Thus, the Rule is satisfied by the above measurement model in
combination with a family of associated stochastic processes, constrained only by equation (2).
Suggested experimentation
“The great tragedy of science–the slaying of a beautiful hypothesis by an ugly fact.”
-Thomas Henry Huxley
The main premise of this paper, that the measurement process can be modeled as a series of
perturbative interactions, is falsifiable, in principle. As a test of the premise, components of a
measurement device could be pared down to minimalist essentials to reduce the number of
hypothesized interactions, producing a new kind of partial measurement. The surviving
modified version of S produced by such a device could be examined for evidence of the
interruption of an evolutionary process such as modeled above. It could produce a final state
that is neither the original nor an eigenstate.
One can envision, for example, Stern-Gerlach-type experiments [2] in which horizontally
polarized spin-1/2 silver atoms (corresponding to the
case) impinge on a modified
device (see figure). A second half-sized detection screen with a hole larger than the upper
pM
a
a = 0
a = N
P(a)
P(a) = α + βa
P(0) = 0
P(N ) = 1
P(a) = a /N = aϵ = P1(a, N −a)
Pi(a1, … , ai, … , aM) = aiϵ
S
i
M = 2
Polarized silver atom source
Magnet
Screen 1
S
Thin Screen 2
Notional Modified Stern Gerlach Apparatus
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beam width is first inserted between the magnet and the original screen, an arrangement that
should not disrupt the original experiment, with half spin up and half spin down detections on
screen 1. If the hole is then filled, the beam detected by screen 1 is polarized with spin down.
But if instead a thin enough film of the detection material can be inserted into the hole, the
above model predicts that the original quantum state S of the silver atoms could be converted
into other mixed states, and screen 1 could be used to collect statistics to test that hypothesis.
The feasibility of the test relies on certain contingencies. For example, there must be a large
enough number of interactions with small ’s to enable conversion of S into a measurable
mixed state. It should be noted also that in general the number of elementary binary exchanges
corresponding to the interaction of S with a single component of D might vary with device, as
well as during the entire measurement process. The mean values computed below refer to the
total number of binary interactions over all detector components required for conversion from a
superposition of eigenstates into a pure eigenstate.
Number of steps to complete the process
As in the above argument generalizing the
probabilities to
, the mean number of
interactions
for a state whose th component is
to evolve into completion, that is, into a
value of either 0 or
, can be computed by considering the
case. Then, as in equation
(4), four transitions can occur, starting from the value
, but now each adds one
interaction to the process. Therefore,
, (7)
where it has been assumed that the probabilities in equation (2) are independent of
, which
ultimately must change in the evolution process, ending in the value one. Equation (7) can be
rewritten:
, with
. (8)
Next let
. Equation (8) becomes
, whose solution is
, because
. Because also
, this means that
.
On reversing the summation order, this simplifies to:
, [
].
To evaluate
, use the boundary condition
The final closed form for the mean time to
completion from a state with initial intensity
is then
ϵ
M = 2
M ≥2
vai
i
ai
N
M = 2
ai ( ≡a)
va = p(a)((p(a))va + (1 −p(a))va−1) + (1 −p(a))(p(a)va+1 + (1 −p(a))va) + 1
M
va+1 = 2va −va−1 −qa
qa = [p(a)(1 −p(a))]
−1
(0 < a < N )
da = va −va−1
da+1 = da −qa
da = d1 −
a−1
∑
i=1
qi
v0 = 0
d1 = v1
va −v1 =
a
∑
n=2
dn =
a
∑
n=2 [v1 −
n−1
∑
i=1
qi] = (a −1)v1 −
a
∑
n=2
n−1
∑
i=1
qi
va = av1 −
a−1
∑
i=1
qi(a −i)
a ≥2
v1
vN = 0.
aϵ
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, [
] ;
;
. (9)
Notice that
does not appear, and therefore neither does
, so that according to equation
(2), is well-defined in equation (9).
Each step in the process involves two intensities, so that the total mean time for completion of
the stochastic evolution of S
is
. (10)
It should be noted that the number of steps counted here includes those in which no change
occurs in any of the intensities. According to equation (4), these “null” interactions occur a
fraction
of the time for intensity
, which can vary with each iteration if depends on
.
The above analysis also describes probability distributions
for which
, because
changes only at the last step of the process. For the particular “uniform” probability model
(with
the Kronecker delta), it follows that
, and the sums in equation (9) are computable, with the result
.
Again, because both eigenstates participate in each interaction, the mean number of binary
interactions until a pure eigenstate arises is
. With equation (1) this simplifies
to
. Notice that null interactions occur in the
model during an average of
of the transitions, and so the mean number of nontrivial transitions is
only
.
Note finally that once a final eigenstate has been reached, the stochastic process can be
thought of as continuing indefinitely with repeated null interactions, as the final state continues
to be “measured” and remains pure.
Number of nontrivial steps in the process
If the above model is ever tested experimentally, it could be useful to know the mean number
of steps involved in the evolution of S that involve actual changes in intensities. According to
equation (7), given that the value of
changes, it has equal probabilities of increasing or
decreasing by one. Therefore, the mean number
of non-null steps involved in the evolution
of intensity satisfies
va = a N N−1 ∑ i=1 (N −i)qi − a−1 ∑ i=1 qi(a −i) a ≥2 v1 = 1 N N−1 ∑ i=1 (N −i)qi v0 = 0 q0 p(0) qi = (a1, … ai, … , aM)ϵ 1 2 M ∑ i=1 vai p2(ai) ai p ai pM(ai) M = 2 M p2(ai) = ( 1 2)(1 −δ0 ai)(1 −δN ai) + δN ai δ q1 = q2 = 4 vai = 2ai(N −ai) 1 2 2 ∑ i=1 2ai(N −ai) 2a1a2 M = 2 p2(a1) + p2(a2) = 1 2 a1a2 a wa a wa = 1 2 wa−1 + 1 2 wa+1 + 1, (0 < a < N ),
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which can be written
.
This was considered earlier (equation (8)) and is solved in equation (9), resulting (for
) in
. Then, as in equation (10), the mean number of non-trivial steps in the total
process of evolution to the final state is
, (11)
which is maximized at the value
when
. So, O
steps with changes
in intensity might be necessary for evolution to an eigenstate.
Not so spooky?
The proposed experiment and model resurrect the knotty question of “spooky action at a
distance.” Interactions with the atom’s wave function at the top of one screen affect the
amplitude of the wave function at the bottom of another. However, the mathematical
representation of the spin-1/2 atom’s quantum state includes two separate wave functions, one
for the spin up and one for the down component. That is, its mathematical representation is as
an outer product of a continuous spatial representation with a two-component spin
representation. The change in one component caused by the measuring screen is effected by a
change in the other component across the spin direction, not the spatial. The transformation is
achieved mathematically by a 2x2 unitary matrix operating in spin space. Thus, the distance
across which the action occurs may seem spooky, but it is transverse to the spatial
representation, and (as is commonly acknowledged) no problem arises with relativistic
causation.
A historical precedent for sleight of hand “God made the integers. All else is the work of man.” - Leopold Kronecker Max Planck regarded his seminal paper [3] in the year 1900 as an “act of desperation” [4]. He was motivated by his belief that “a theoretical interpretation (of his formula for the blackbody spectrum) had to be found at any cost” [ibid]. For many years after, he suspected that his method of quantizing the interactions between a blackbody cavity’s wall and its radiation field had “no more than a formal significance” [5]. “He did not believe (the energy of radiation) was really chopped up into quanta” ([1] p. 26). He and his contemporaries “all believed that it was nothing more than the usual theorist’s sleight of hand, a neat mathematical trick on the path to getting the right answer” ([1] p. 27). His fictional contrivance, the quantization constant , could be adjusted to account for the experimental data, but he did not understand the truth behind his heuristic model of the physics. By contrast, in this paper the limit can be taken and still give the desired result, the Born Rule. However, if the underlying premise of this model is necessary, then is finite, wa+1 = 2wa −wa−1 −2 qa = 2 wa = a(N −a) 1 2 M ∑ i=1 wai = 1 2 M ∑ i=1 ai(N −ai) = M ∑ i<j aiaj (M −1)N 2 2M ai = N/M (N 2) h ϵ →0 N = 1/ϵ
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because the number of components of a measuring device is finite. This would mean that
cannot be zero.
Planck found the fundamental quantum constant, which led to quantized physical entities,
such as angular momentum and energy. The various charges and masses of elementary
particles are also quantized. Theories of quantum gravity even raise the possibility of space
and time being quantized. In the above model, represents a small fraction of a probability,
which is a pure number. A pure number can be derived from the fundamental physical
constants, if one includes a mass estimate of the lightest known massive elementary particles.
The neutrino/Planck mass ratio is thought to be nonzero but <
, perhaps an upper limit to
the value of . Might reflect the quantization of number itself, when applied to the physical
world? Could underly all other quantizations?
Conclusion and Future Prospects
“Plurality should not be posited without necessity.” - William of Occam
A simple explication of the quantum measurement process as a series of elementary
perturbative interactions appears supportable. Exotic semi-metaphysical explanations such as
the “many worlds interpretation” [6] of quantum mechanics seem unnecessary. Nor is there a
need to give up on notions of objective reality. The moon is there, even if nobody is looking at
it, because it is macroscopic and constantly measures itself, or “decoheres.”
It appears that the probabilistic nature of a quantum measurement can be understood to arise
merely from an incomplete knowledge of the detailed microscopic interactions between the
system under study and the components of the measuring device. Analogous to the situation
with a classical gas. Quantum measurements give differing results despite “identical”
conditions, because no experiment has ever actually been repeated. Only the macroscopic, not
the quantum condition of the measurement device is replicated.
The models of this paper share with the Born Rule their generality. None explores the detailed
quantum processes of any specific type of measurement device. A synopsis of the main
assumption and reasoning:
1.
Anything qualifying as a measurement device has many microscopic components that
individually perturb the quantum system being measured.
2.
Associated with every quantum measurement is a probabilistic mechanism that transforms
any superposition of eigenstates into some pure eigenstate.
3.
Therefore, the course of a measurement must be describable as a stochastic process.
This paper has shown that some of the simplest, pairwise exchange models of such
processes, consistent with general quantum principles, are also governed by the Born Rule.
Coincidence perhaps. But this property could inform future attempts at more detailed physics
explanations for any particular measurement modalities.
Also, the demonstration that an entire class of stochastic models obeys the Born Rule raises
the prospect that the Rule’s universality might be associated not with a single transition
probability model
, but rather with a subset of those belonging to some class, such as
defined by equation (2).
ϵ
ϵ
10−29
ϵ
ϵ
ϵ
pM(a)
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9 However, the choice
(12)
seems intuitively attractive from a general physics perspective. It says that the stronger the
presence of an eigenvector in a quantum system, the stronger will be the degree of its
interaction with a measurement device.
Finally, it can be argued that these ideas at most replace the question of why a measurement
should obey the Born Rule (equation 6) with why transition probabilities should (perhaps) obey
the analogous equation (12). But at least the Rule’s demonstrated consistency with the above
microscopic measurement model might serve as the touchstone to a deeper truth.
And it has not escaped notice that the specific underlying model postulated here immediately
suggests possible implications for nascent quantum technologies, such as in computing and
cybersecurity.
References:
1.
Kumar, Manjit, Quantum - Einstein, Bohr and the great debate about the nature of reality,
W.W. Norton & Company, 2008.
2.
Gerlach, W.; Stern, O. (1922). “Der experimentelle Nachweis der Richtungsquantelung im
Magnetfeld”. Zeitschrift für Physik. 9 (1): 349–352.
3.
Planck, Max (1900), “On the theory of the energy distribution law of the normal spectrum,”
reprinted in Haar, Dirk ter (1967), The Old Quantum Theory, (Oxford: Pergamon), as cited in
[1].
4.
Hermann, Armin, (1971) The Genesis of Quantum Theory, quoted p. 23, letter from Planck
to Robert Williams Wood, 7 October 1931, (Cambridge, MA: MIT Press), as cited in [1].
5.
Planck, Max (1949), Scientific Autobiography and Other Papers, [New York: Philosophical
Library], p. 41, as cited in [1].
6.
Everett lIl, Hugh,“Relative State Formulation of Quantum Mechanics,” REVIEWS
OF MODERN PHYSICS, VOLUME 29, NUMBER 3. JULY, 1957.
pM(ai) = ϵai
Canonical Hub: CANONICAL_INDEX