Heisenberg=s Indeterminacy Principle

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Born=s Theorem

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    In this essay I want to look at the most famous feature of the quantum theory, commonly (and improperly) called Heisenberg=s Uncertainty Principle. First, I want to rededuce it from fundamental axioms to ensure that we understand it properly. Then I want to see how far I can extend it as a description of Reality.

    Let me note here and emphasize that Heisenberg himself called it an indeterminacy principle and not an uncertainty principle. Uncertainty denotes the fact that we lack certain information; indeterminacy denotes the fact that the information does not actually exist. The difference between those two concepts still roils discussions among physicists on the foundations of the quantum theory.

The Least Observable Action

    I have already described elsewhere the finite-value theorem, which tells us that no property of Reality or of any of Reality= s contents can exist in an infinite quantity if that property conforms to a mathematical description that always adds up to a net zero, such as in a conservation law. We have the principle of least action,

(Eq=n 1)

which tells us that if we calculate the action that a particle accumulates as it moves between two fixed points and calculate the action that it would accumulate if it moved along a path between those same two points that differs from the true path by a minuscule amount, the difference between those two calculations equals zero. The calculation looks like the zeroing out of a derivative of some function in order to locate the point on the real-number line where the function achieves its minimum value and it lets us work out the mathematical description of the path of least action, the path that the particle actually follows. It also brings the finite-value theorem into play and thereby tells us that there must exist a minimum finite amount of action that any system can accumulate or enact.

    Now we know that in phase space we have a minimum sized granule of a fixed magnitude that represents the least action that can come true to Reality. But that minimum action, which we identify in numerical form with Planck=s constant, cannot represent any specific granule in phase space, one with determinate boundaries in space and in linear momentum. We must assert that any phenomenon that involves an action smaller than Planck=s constant must remain perfectly indeterminate.

    If we apply the principle of least action to a particle, we can plot the evolution of the particle=s location in phase space using Maupertuis= form of the principle,

(Eq=n 2)

Working that description out gives us

(Eq=n 3)

Neither pδx nor xδp gives a proper minimum, since we can change the values of p and x at will, so we must have as necessarily true to Reality

(Eq=n 4)

which we call Heisenberg=s Indeterminacy Principle. Of course, that expression does not properly zero out Equation 2, but we can fix that deficiency readily enough.

    We want to shift our point of view among inertial frames, so we must rewrite the action integral in Lorentz-invariant form. We have our displacement four-vector as x=(x,y,z,ct) and our momentum four-vector as p=(px,py,pz,E/c). We know that the dot-multiplication of two four-vectors yields a Lorentz-invariant product, so we have

(Eq=n 5)

In that equation the matrix represents the metric tensor, which we will use later to combine the quantum theory with General Relativity. If we confine the momentum to the x-direction only, we have px-Et as the action function that we must insert into Equation 2 as pdx-Edt. But the integral of a difference equals the difference between the integrals, so we have Equation 2 properly zeroed out if we have Heisenberg=s Indeterminacy Principle expressed in two statements B

(Eq=n 4)

and

(Eq=n 6)

de Broglie=s Theorem

    The theory of Relativity tells us that a particle that carries no linear momentum does, nonetheless, possess a non-zero amount of energy. For a particle carrying a rest mass of m0 we have the energy of existence as

(Eq=n 7)

Louis de Broglie noted in 1924 that Planck=s quantum theorem also necessitates that we associate a frequency v with that energy in accordance with

(Eq=n 8)

Equating those expressions for the energy lets us calculate the frequency that we associate with the bare existence of the particle,

(Eq=n 9)

Assigning a frequency to the particle necessarily implies an oscillation of some kind associated with the particle, a sinusoidal variation of something with the elapse of time. But what aspect of the particle could possibly oscillate?

    Because the particle has a definite rest mass, the only property left for us to assert as indeterminate denotes the particle=s existence per se. We know that the particle exists B the conservation law pertaining to energy guarantees that fact B but we cannot say from instant to instant when it exists. That proposition conforms to the version of Heisenberg=s indeterminacy principle expressed in Equation 6: indeed, if we recall the fact that the frequency of an oscillation equals the reciprocal of the oscillation period, we can see that Equation 8 expresses the content of the indeterminacy principle.

    It may seem strange to us that a particle that we conceive as fully existent would actually exist on a part-time basis. But, though we must feign at this stage not knowing about the fact that virtual particles emerge from the quantum vacuum by borrowing the energy that they need to come real through the indeterminacy principle, we can assert that the indeterminacy principle covers a temporary non-existence of the particle and of the energy that makes it real. In essence the particle lends its mass-energy to the vacuum and then gets it back within a time interval

(Eq=n 10)

    Does the particle, then, actually exist? To answer that question we apply the criterion presented in 1936 by Albert Einstein, Boris Podolsky, and Nathan Rosen;

AIf, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.@

The particle=s fundamental properties certainly meet that criterion: for an electron, for example, the particle=s electric charge, spin, and rest mass have definite, perfectly predictable values guaranteed by the quantum theorem by way of the appropriate conservation laws. Thus, if we put a particle in a box, we can predict with certainty the electric charge, spin, and rest mass that we would measure, but we cannot predict with any certainty the values of the particle=s potential energy, orientation of its spin axis, kinetic energy, linear momentum, or any other of the particle=s contingent properties. We conclude that the particle does, indeed, actually exist, albeit in an indefinite, blurry kind of way.

    If we have a particle flickering into and out of existence in accordance with the above description, then we can describe its behavior with a state function,

(Eq=n 11)

In that function we have yet to determine the amplitude A (presumably a function of the spatio-temporal coordinates that somehow encode the location of the particle. Heisenberg=s principle ensures that we can never know the actual value of the phase angle α; we know only that it lies within the range between zero and two pi radians.

    How can I assert that Equation 11 gives us a correct description of a phenomenon that we cannot possibly observe or measure? Physics, since the time of Galileo, has consisted of people devising mathematical theorems whose algebraic formulae mimic, as perfectly as possible, the formulae inferred from analysis of measurements made on the relevant phenomena. How can I claim, in light of that statement, that Equation 11 describes the behavior of a particle=s existence (indeed, what does it mean for a particle=s existence to have a behavior)?

    First, we take existence as a property of the particle, just like any other property and just as susceptible to mathematical description. That fact means that we must describe the particle=s existence as occurring at a specific place and a specific time within the space-time continuum. For greatest generality we assume that we would describe the particle=s existence by way of an existence density function, much like the electric-charge density function or the mass density function that we use to describe bulk matter.

    Second, whatever function describes the particle=s existence must possess the property of continuity; so we say, in other words, it must have a finite value at each and every point in the space-time continuum. It must represent the fundamental possibility of the particle existing anywhere at any time. Further, that function must possess the property of smoothness; so we say, in other words, that the derivatives of that function have finite values at each and every point in the space-time continuum. The function must have no discontinuities (infinite first derivatives), no kinks (infinite second derivatives), and so on.

    Third, the function describing the particle=s quantity of existence must take the real-number line as the domain of each of the variables that we put into it, because each of those variables represents a measurement that we can conceivably make and measurements only yield real numbers. But that function must take the complex plane (the set of all possible numbers) as its range, at least tentatively, because we don= t yet know what the function must look like.

    Fourth, by virtue of the particle=s rest energy the particle=s action of existence increases monotonically with the elapse of time, but its quantity of existence can=t change monotonically, so the state-of-existence function must map an infinite domain onto a finite range. In one possible model of that mapping we conceive the action as an endless thread wrapped around a cylinder. As a point, presumably occupied by the particle, moves along the thread it cycles through the two pi radians on the cylinder=s circumference. That model does, in fact, conform to Equation 11, but we have not yet properly deduced that fact: Reality might, as far as we must feign to know, conform to a different model.

    Fifth, the state-of-existence function must have such a form that the statement

(Eq=n 12)

stands true to Reality for any integer n. We know the truth of that statement because the mapping of an infinite domain onto a finite range obliges us to use a multi-argument function, a function for which many values of the argument yield the same value of the function. The indeterminacy principle necessitates that each and every value of the function correspond to equal intervals, equally spaced, on the domain: we know that statement stands true to Reality because the quantum principle tells us that the particle can accumulate definite actions only in whole units of Planck=s constant, so at the end of any interval Δt in which the particle accumulates one Planck unit of action, in accordance with

(Eq=n 13)

the value of the state-of-existence function must equal the value that the function had at the beginning of the interval. If that statement did not stand true to Reality, then we could conceive a way for the particle to enact a definite amount of action less than one Planck unit, which the quantum theorem forbids: our mathematical description, Equation 12, must conform to that fact.

    Sixth, the measure of the particle=s existence correlates with the particle=s accumulated action through the function ψ=ψ(S). As additional action accumulates to the particle, that function increments in proportion to the increment of action dS, some suitable proportionality constant K, and the quantity of the particle= s existence, so we have

(Eq=n 14)

That statement must stand true to Reality because;

    1. we have associated existence and action through the state-of-existence function, so through the definition of a differential we expect the increment in that function to stand in direct proportion to the increment in its argument, the accrued action,

    2. the more existence the particle has for a given action, the more that existence must change with a given change of the action (because the existence emanates from the action ab initio). The change in existence must stand in direct proportion to (proportional to the first power of) the measure of existence, because the amount of change in the particle= s existence must reflect the amount of existence available for change, just as the rate at which a radioactive material decays reflects the amount of material available for decay, and

    3. Aproportional to@ does not mean the same as Aequal to@, so we must introduce some number K, whose value we will determine later, to convert proportionality into equality.

    Dividing Equation 14 by the state-of-existence function and integrating the result gives us

(Eq=n 15)

which gives us as its anti-logarithm

(Eq=n 16)

We have in this case

(Eq=n 17)

so now we want to determine the numerical value of K.

    The requirement that ψ represent a non-monotonic function necessitates that K represent a purely imaginary number, the only argument that makes an exponential non-monotonic. That fact makes ψ a periodic function, as we also require. And the requirement that the function repeat itself over every accumulation of one Planck unit of action necessitates that

(Eq=n 18)

Thus we have Equation 16 as

(Eq=n 19)

which describes a sinusoidal pulsation.

    How can we describe that particle to an observer moving past us in the negative x-direction at some speed V? In that person=s frame the particle moves in the positive x-direction at that speed, so in addition to the action accumulated by its energy over the elapse of time it also accumulates action due to its linear momentum over the distance it crosses in that time. In this case we have the action as

(Eq=n 20)

so we have the state-of-existence function as

(Eq=n 21)

We could have obtained the same result by applying the fourth equation of the Lorentz Transformation directly to the time in Equation 11 and putting the prime on the amplitude to acknowledge the possibility that it differs between inertial frames. I have also taken the subscript off the mass to indicate that we must now use the particle=s relativistic mass.

    The algebraic expression in Equation 21 describes a wave that propagates from left to right on the standard Cartesian coordinate grid at a speed equal to c2/V, a speed greater than the speed of light. At this stage I don=t want to introduce into the Map of Physics anything that moves faster than light, even though we may have to revisit the notion in order to resolve the Einstein-Podolsky-Rosen paradox. For now I want to use an alternative formulation of the wave function.

    Instead of assuming that Equation 11 describes a function pulsating at a point, let=s assume that it represents a standing wave. This model fits the quantum theory better than the pulsating point does because it conforms better to Heisenberg=s indeterminacy principle. In stating that we have a stationary particle we have tacitly assumed that we have perfectly precise knowledge of the particle=s linear momentum, which necessitates that we have perfectly imprecise knowledge of the particle=s position. A standing wave spread uniformly throughout space gives us that imprecision.

    A standing wave consists of two identical waves propagating in opposite directions at the same speed. We require that the mathematical form of our wave function remain invariant under a Lorentz Transformation, so we must assert that the component waves of our standing wave propagate at the speed of light. We thus have Equation 11 as

(Eq=n 22)

in which the wave number k and the angular frequency ω bear the relation kc=ω. Bearing in mind the relation between frequency and angular frequency, we have Equation 9 as

(Eq=n 23)

which gives us the wave number as

(Eq=n 24)

Through those equations we rewrite the state function as

(Eq=n 25)

    How does that wave appear to our observer moving in the negative x-direction? In this case they would apply the Lorentz Transformation in the form of the relativistic Doppler shift, subjecting the wave moving to the right (the first term) to a Doppler upshift,

(Eq=n 26)

and subjecting the wave moving to the left (the second term) to a Doppler downshift,

(Eq=n 27)

Thus that observer represents the particle with the state function

(Eq=n 28)

That function describes any particle moving in a direction parallel to an observer= s x-axis. And it also gives mathematical form to Louis de Broglie=s theorem uniting the concepts of wave and particle.

    Before we move on we notice that the complex conjugate of the function in Equation 28 gives us a description of the particle identical to that given by the function itself. Reversing the algebraic sign on each and every imaginary element in that function has no effect on the description of the particle, so we infer that ψ, ψ*, or some linear combination of them represents the particle equally well. For convenience we simply take ψ to represent the particle and all of its properties.

The Correspondence Principle

    Beginning in 1913, Niels Bohr applied what he at first called the analogy principle and then called the correspondence principle to the quantum theory that had just recently emerged from Max Planck=s theory of blackbody radiation. In essence the principle states that in an event involving actions substantially greater than Planck=s constant the quantum dynamics describing the event must revert to the classical Newtonian-Maxwellian dynamics. More or less equivalently we can say that in the limit as Planck=s constant approaches zero quantum dynamics becomes classical dynamics. However, in that statement we have made a tacit assumption: we have assumed that the quantum theory and classical dynamics have the same mathematical formalism. Now we have to justify that assumption.

    So far in the quantum theory we have described our particle with a function that represents its existence smeared out over some volume in accordance with its dynamic properties, linear momentum and energy. On the other hand, in classical dynamics the equation x-vt=0 represents the set of points comprising the particle=s trajectory. But a particle does not ever occupy a single point: the Greeks defined a mathematical point, correctly, as having zero extent and a particle with zero extent simply does not exist. The smallest possible particle must occupy at least an infinitesimal volume around some point (which point we retain as a useful fiction for the sake of calculation). Instead of describing the particle=s trajectory, then, we need a mathematical theory that describes the particle on its trajectory; we thus say, in other words, that we must represent the particle with a function, f(x-vt), that has non-zero values only when the argument takes values equal to or infinitesimally different from zero: that function will thus track the particle as time elapses. Dirac=s integral, which equals one, gives us just what we need; the Dirac delta, the function that we integrate, thus represents the particle=s existence density. We thus have ρ=δ(x-vt), which moves as we require.

    We seem to have done something perfectly and completely trivial. It looks as if we have done nothing of any significance. In other words, we have gained exactly what we wanted. We have simply taken the algebraic description of the particle=s trajectory and made it the argument of a function that only has non-zero values when that argument equals zero or some number infinitesimally different from zero. In either case, whether we set the algebraic formula for the trajectory equal to zero directly or incorporate it into the Dirac delta as an argument, we get exactly the same description of the particle=s motion. However, in the latter case we assign to the particle an existence density, albeit confined to an infinitely narrow spike. Even though the particle effectively does not spread away from its trajectory in the classical case, the use of the existence density makes it possible for the description of the particle to smear out in accordance with Heisenberg=s indeterminacy principle when Planck=s constant becomes different from zero.

    Now we have classical dynamics in a form consistent with the mathematical form of the quantum theory. We can say now that in the limit as Planck=s constant approaches zero something must become the Dirac delta. Born=s theorem gives us that something.

Born=s Theorem

    Max Born (1882 Dec 11 B 1970 Jan 05) gave physicists the interpretation of the state function that they still use today, that the square of the state function equals the probability density of the particle or system manifesting the properties encoded in the state function. In 1926 he presented the idea in a paper that inspired Einstein to proclaim in a letter to Born that AThe Old One does not play dice.@

    Born also noted that applying certain mathematical operators, usually based on a differentiation, to the wave function yielded values for the relevant quantity, such as energy or linear momentum. Applying derivatives to the wave function gives the same result as multiplying the wave function by a number, so we have

(Eq=n 29)

Integrating that number over the volume occupied by the system yields the expectation value of the measurable quantity, P in this case, the value that we would expect to obtain from an actual measurement made on the system. In other words, we calculate a kind of average over the portion of space the particle could occupy.

    But Born obtained that result empirically, by looking at the patterns of numbers obtained from experiments. He drew inspiration from the matrices that Werner Heisenberg had devised to represent electronic energy levels in atoms and the transition probabilities among them. In Heisenberg=s matrix mechanics we draw the state functions of the states available to our quantum system across the top and again down the left side of the matrix and then fill the matrix with the products of the state functions, the one from the horizontal array and the other from the vertical array corresponding to the square that we want to fill. On that matrix the off-diagonal terms represent the probabilities associated with the transitions between the two states whose state functions appear in each term and the diagonal terms, each manifesting the square of a given state function, then represent the probabilities associated with the system occupying the state each term represents. Born understood that if he extended the matrix, the energy difference between neighboring states would diminish toward zero and the array of states would approach a continuum, so he asserted that the probability of finding a system in a given state on that continuum would stand in direct proportion to the square of the state function. Thus he obtained what we now call Born=s theorem.

    As usual in modern science, Born used the empirical-inductive method to obtain his hypothesis that the product of the state function with its own complex conjugate equals the probability density of finding the system in the state represented by the state function. Can we find a deductive path to the same result? Do we have a way in which we can add Born=s theorem to our axiomatic-deductive Map of Physics?

    We have claimed that the state-of-existence function encodes all of the properties, both necessary and contingent, belonging to the particle. With that statement standing before us, we can see that if we want to calculate the value of some property of the particle, we need only apply the appropriate mathematical operator to the state-of-existence function. Instead of giving us a single real-valued number, though, that process gives us a real-valued function of the coordinates (think of the potential energy of the particle in a gravitational field, for example) that tells us what value the property would have if the particle were to occupy a given point.

    In the quantum theory particles do not exist at single points. We need the state-of-existence function to express the amount of existence that a particle has at a given point in order to express that idea. Once we have calculated the value that a given property has at any point, we must then multiply that value by the amount of existence that the particle has at that point. Thus we have weighted the values of the property over the entire region that the particle could occupy. If we then integrate that product over the entire volume of space available for the particle to occupy, we get a kind of average value for the property in accordance with

(Eq=n 30)

If we have a situation in which we have the particle tightly localized, that calculation gives us the value of the property that we can expect to measure if we perform the appropriate experiment; thus, we call <p> in general the expectation value of the property.

    At this point we have not yet determined the value of the amplitude of the state-of-existence function. We also have not encountered any phenomenon that would oblige us to assign a specific value to it. Thus we remain free to assign an arbitrary number to that amplitude without changing the validity of the description that the state-of-existence function gives of the particle. To make this matter easy on ourselves we define that amplitude to have the value that makes the second integral in Equation 30 equal one. That choice gives us

(Eq=n 31)

    The number one can exist as a mathematical operator, so if we set P=1, then we have <p>=1. That operator extracts from the state-of-existence function the amount of existence that the particle has at a point, so the statement that <p>=1 in that case represents the fact that the particle exists somewhere in space. If we restrict the volume over which we carry out the integration, we expect to obtain a real number between zero and one. That fraction represents the probability of the particle existing in the region of integration. If that statement did not stand true to our mathematical description, then Equation 31 would not represent the calculation of a proper average. So, in the light from those statements, we see that ψ*ψ represents the probability density of the particle existing at a given point. That statement gives us the content of Born=s theorem.

    Now imagine that instead of a continuous function of the coordinates, we have the state-of-existence function as a polynomial function of certain quantum numbers. In that case the partial products that comprise the Born product (ψ*ψ) appear to us as the elements of a matrix: the diagonal elements represent the probabilities of finding the particle or system in the state identified by the quantum number indexing the state-of-existence function and the off-diagonal elements representing the probabilities of transitions between the states identified by the two quantum numbers in the product. In this derivation we have reversed Born= s inference, going from the general continuous probability density to a discontinuous array of probabilities.

    And, also as Born did, we note that if we examine the probability matrix farther to its lower right, where the energy differences between adjacent states approach zero the sum of probabilities shades into an integral and thence into Equation 31. Thus we can make the transition between the quantum description of a bound system to that of a system of free particles.

    Finally, we find, though I won=t work it out in this essay, that if we have an event involving an action with a value vastly greater than that of Planck=s constant, the state-of-existence function describing a particle evolves toward a Dirac delta as a limit. That calculation, when we work it out, proves and verifies the fact that the quantum theory, in the appropriate limit, corresponds to classical dynamics. We now face the task of working out that result in more detail.

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