The Dicke-Wittke Postulates

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    In the sixth chapter of their 1960 book "Introduction to Quantum Mechanics", Robert H. Dicke and James P. Wittke presented seven postulates as comprising the mathematical foundation of the quantum theory. We have those postulates as:

    I. "It is assumed for a system consisting of a particle moving in a conservative field of force (produced by an external potential), that there is an associated wave function, that this wave function determines everything that can be known about the system, and that it is a single-valued function of the coordinates of the particle and of the time. In general it is a complex function, and may be multiplied by an arbitrary complex number without changing its physical significance."

    II. "With every physical observable (the energy of the system, the x-position coordinate of the particle, etc.) there is associated an operator. Denote by Q the operator associated with the observable q. Then a measurement of q gives a result which is one of the eigenvalues of the eigenvalue equation:

(Eq’n 1)

This measurement constitutes an interaction between the system and the measuring apparatus."

    III. "Any operator associated with a physically measurable quantity is Hermitian."

    IV. "The set of functions Rj which are eigenfunctions of the eigenvalue equation

(Eq’n 2)

can be used to express an infinite number of possible functions."

    V. "If a system is described by a wave function R, the expectation value of any observable q with corresponding operator Q is given by

(Eq’n 3)

    VI. "The development in time of the wave function R, given its form at an initial time and assuming the system is left undisturbed, is determined by the Schrödinger Equation

(Eq’n 4)

where the Hamiltonian operator H is formed from the corresponding classical Hamiltonian function by substituting for the classical observables their corresponding operators."

    VII. "The operators of quantum theory are such that their commutators are proportional to the classical Poisson brackets according to the prescription

(Eq’n 5)

where {q,r} is the classical Poisson bracket for the observables q and r. The variables, if any, in the Poisson bracket are to be replaced by operators."

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    Let’s take a look at those postulates in light of what we have deduced for the Map of Physics. In historical terms, those postulates reflect the results of the 1923-1926 revolution in which the new quantum theory of de Broglie, Schrödinger, Born, et al. extended the old quantum theory of Planck, Einstein, and Bohr. Because we have deduced the content of those postulates in a different order from the one in which Dicke and Wittke put them, we will consider the Dicke-Wittke postulates out of order.

Postulate VII

    If we use generalized coordinates for linear momentum pi and location xi (and Dicke and Wittke tell us that we must use Cartesian coordinates for this), then for observable quantities represented by numbers q and r the classical Poisson bracket looks like this:

(Eq’n 6)

If we take the linear momentum and the location as our observables, so that q=xi and r=pi, then we have

(Eq’n 7)

the Dirac delta, which equals one when i=j and equals zero otherwise.

    Poisson brackets have no special meaning in classical physics; that is, they don’t encode any special knowledge of the nature of physical Reality. They merely provide a convenient notation for describing some aspects of classical physics, in particular the Hamiltonian formulation of classical dynamics. Because physicists used the Hamiltonian formulation in devising their description of the quantum theory, Poisson brackets come readily into the quantum theory.

    Dicke and Wittke posit that in the quantum realm the commutation relation between two operators is proportional to Planck’s constant and the Poisson bracket for the observable quantities that those operators represent. We define the commutation relation between two mathematical operators as the difference between the two ways of multiplying them together. This gives us a very important statement about the fundamental nature of the quantum theory by telling us that the order in which we measure observable quantities, represented by mathematical operators, may make a difference in what we actually measure. That fact conforms to the requirement of Heisenberg’s indeterminacy principle, which we can verify by deducing it from that principle.

    From the finite-value theorem we deduced the quantum principle, which tells us that between any two events we cannot measure an action less than some fundamental, finitely-small unit, which unit we represent with Planck’s constant. We claim that we can measure the location of a particle as precisely as we like and we claim that we can measure the linear momentum of a particle as precisely as we like, so the quantum theorem tells us that if we try to measure both of those quantities of the same particle participating in the same two events, then the measurements themselves create indeterminacies *p and *x such that

(Eq’n 8)

which expresses Heisenberg’s indeterminacy principle. To see how that analysis comes true to Reality, consider a single particle of mass m floating motionless before us.

    At times t0 and t1 we measure the location of the particle and at some instant before t1 or after t1 we measure the particle’s linear momentum. Those measurement acts constitute events. Between t0 and t1 the particle does not change its location ()x=0), so its linear momentum must also equal zero ()p=0) and we infer that the action accrued to the particle between those two times equals zero (s=)p)x=0; after all, the particle just sat there doing nothing). But we cannot have zero action between two events. To understand why that must be a fact of Reality we need only consider this scenario as viewed by another observer.

    Imagine that the alternate observer moves in the negative x-direction (right to left) at some very slow speed v relative to the inertial frame occupied and marked by our particle. In that frame the particle carries linear momentum p=mv and between the instants t0 and t1 it moves a distance x=v(t1-t0), so the alternate observer attributes to the particle a gain in action

(Eq’n 9)

But we can make v as small as we like, so that formula gives us the possibility of violating the quantum rule by creating an action less than the Planck unit. We cannot have that come true to Reality, so we must have instead

(Eq’n 10)

That mathematical description must remain true to Reality for any value of v, including zero.

    I have written the classical term in Equation 10 as a subtraction; indeed, all classical calculations of action come to us as subtractions, explicit or implicit. If we write Equation 10 as

(Eq’n 11)

then we must have as true to Reality

(Eq’n 12)

and

(Eq’n 13)

in which ti=0 represents the instant at which all of our clocks show zero, the instant from which we measure the actions that accrue to all of the bodies in our inertial frame. So now we see that the Planck unit of indeterminacy gives us another universal absolute like the speed of light: just as no addition of velocities can make a ray of light move past an observer at any speed more or less than 299,792.458 kilometers per second, so no subtraction of actions can remove the Planck unit of indeterminacy from the resulting difference.

    So at the instant t0 two observers see our particle crossing the point x=0. At t1 one of the observers measures the particle’s location and then measures its linear momentum, calculating the action as sA=px+h. Just before t1 the second observer measures the particle’s linear momentum and then measures the particle’s location at t1, calculating the action as sB=xp+h. Classically we would have sA-sB=0, but in the quantum realm we must have

(Eq’n 14)

If the particle moves in more than one dimension, we must have that equation in the form

(Eq’n 15)

in which *ij represents the Dirac delta. But we know that in Hamiltonian dynamics

(Eq’n 16)

so Equation 15 gives us the most significant part of Postulate VII to within multiplication by a constant (i/2B).

Postulate II

    We have grown accustomed to using numbers to represent physical quantities that we measure, either directly (such as distance and duration) or indirectly (such as potential energy or linear momentum). And when we combine those numbers to calculate other quantities we use the rules of basic arithmetic. Among those rules we find the commutative rule of multiplication, which rule tells us that the order in which we write the multiplier and the multiplicand has no effect upon their product.

    Postulate VII tells us that we cannot use that rule in our representation of the quantum realm. In particular, the factors that we multiply together to calculate an action cannot commute with each other. We must thus represent those factors by mathematical entities that do not commute under multiplication.

    Algebraic variables, representing simple numbers, and the algebraic formulae into which we combine them do not conform to that requirement. Tensors, basically arrays of numbers, of which vectors and matrices offer the most common examples, submit to several kinds of multiplication (think of the vector dot product and the vector cross product), of which at least one will not obey the commutation rule. If we use matrices or higher-rank tensors to represent our measurements, then we will devise Werner Heisenberg’s matrix mechanics version of the quantum theory (which I don’t want to do here). Mathematical operators, entities that transform algebraic formulae, also do not usually commute with each other under multiplication. Among the various possibilities, differential operators come readily to mind as a possibility.

    Certainly we should represent the things that we measure directly (e.g. distance, duration, orientation) with the appropriate algebraic variables (x, t, 2). If we then represent the action conjugates of those coordinates (linear momentum, energy, angular momentum) by multiplying Planck’s constant onto the partial differentiation operators respecting those coordinates (p=hM/Mx, E=hM/Mt, L=hM/M2), then we get a representation that conforms to the commutation rule of Equation 15. Those operators each represent an act of measurement.

    However, we don’t actually know the fully correct form of the operators, so we really should represent them in a more general form, such as

(Eq’n 17)

in which alpha and beta represent quantities that we must determine later.

Postulate I

    We cannot have a differentiation operator standing by itself: it must have something to differentiate. Thus with every quantum-mechanical system we must associate a function that represents the state of that system. If the operator acting on that function represents the act of measurement, then that state function must represent that upon which we perform the act of measurement. We now know three things about the state function:

    a. Because it represents a unique, real system, the state function must be a single-valued algebraic formula.

    b. Because it must be differentiable with respect to the coordinates, the state function must be a function of the position coordinates, of the time, and of the orientation coordinates.

    c. Because the measurement of a quantity represented by an operator Q must yield a real number q (the reading on a gauge), the state function must have such a form that we can equate the operator Q, whatever its form, with the operation of multiplying by q; that is, we must have

(Eq’n 18)

We know that we can satisfy that criterion because mathematics has such an extent as to make it possible and thereby give us the freedom to devise the function that we need. This completes Postulate II by giving us the eigenvalue equation.

    Now we can try applying the operator of Equation 17 to that equation and then solving it. For the linear momentum of a particle we get, then

(Eq’n 19)

We can solve that easily enough for $=0 and

(Eq’n 20)

in which A and " represent constants whose values we must determine later.

    We thus have a representation of the quantum realm that conforms to the commutation rule of Postulate VII; to wit,

(Eq’n 21)

    At this stage we cannot infer that the state function is a complex-valued function or that multiplying it by an arbitrary complex number does not alter its physical meaning, so we must leave our discussion of Postulate I incomplete. We will return to it a little further on.

Postulate VI

    Now I want to contemplate a quantum system that consists of a single particle trapped in a square potential well. In this imaginary experiment I have established force generators in such a way that the potential energy that the particle would have is U=0 between the points x=-1/2x0 and x=+1/2x0 and U=N everywhere else. I then say that the particle has kinetic energy T<N in the region between -x0/2 and +x0/2, which means that the particle remains in that region. The particle, pondering mass m, has linear momentum

When the particle reaches one of the discontinuities in the potential energy, it encounters an "infinite" force that reverses its motion and bounces it off the discontinuity like a tennis ball bouncing off a mountain.

    We have here a quantum system, so we must have an integer number of Planck units of action between any two abrupt events. Bouncing the particle off of one of the potential-energy walls is sufficiently abrupt for our purpose, so between any two successive bounces the particle must enact

(Eq’n 22)

If the width x0 of the potential-energy well has finite extent, then the particle can only possess one of a finite number of values for its linear momentum,

(Eq’n 23)

We give those values the name eigenvalue and we extract them from the eigenfunction Rn(x) by way of the eigenvalue equation

(Eq’n 24)

    The particle also carries kinetic energy that has its own eigenvalues

(Eq’n 25)

That value comes to us from the energy eigenfunction Rn(t) by way of the equation

(Eq’n 26)

in which U represents the particle’s potential energy. We have the eigenvalue equation

(Eq’n 27)

Equation 26 tells us that we can also write that equation as

(Eq’n 28)

But the kinetic energy is an eigenvalue of the system, so we must have an operator that will extract it from the state function. In accordance with Equation 25, we have

(Eq’n 29)

With that operator and a putative potential-energy operator U we rewrite Equation 28 as

(Eq’n 30)

or

(Eq’n 31)

in which I have removed the explicit representation of the time coordinate in the state function. I thus prepare the state function for an exploration of the implicit variables in it to see which ones emerge with the re-emergence of the time variable.

    We know that the energy operator extracts an eigenvalue from Rn(t) but not from Rn(x). And we know that the linear momentum operator extracts an eigenvalue from Rn(x) and not from Rn(t). Both operators appear in Equation 31, which physicists call Schrödinger’s Equation, so the complete state function R must involve both Rn(t) and Rn(x) in some simple combination, either a sum or a product. We already know that applying the natural logarithm operator to the complete state function extracts a description of the action accrued to the system between two events, so we must have as true to Reality that

(Eq’n 32)

That state function satisfies Schrödinger’s Equation. The Hamiltonian function (the function on the right side of Equation 31) doesn’t include only a function of the linear momentum: the Hamiltonian in any given situation will include kinetic energies based on generalized momenta (such as angular momentum for kinetic energy of rotation). A state function that satisfies Schrödinger’s Equation with a Hamiltonian function involving generalized momenta will also satisfy the eigenvalue equations for those generalized momenta, so that state function gives us a proper and correct description of quantum reality, as Dicke and Wittke postulated.

Postulate IV

    If I create a quantum system by putting a particle into an array of forcefields, I will describe that system with an appropriate state function. If at some other time I create precisely the same system by putting the same particle into the same array of forcefields, I will describe that system with the same state function. If I were to use a different state function in the second instance, I could only do so on the basis of a claim that the laws of physics possess no invariance with respect to the elapse of time. But that claim, in accordance with Nöther’s Theorem, entails a claim that Reality does not conserve energy and, except at the boundary of space, we disallow that claim.

    A particle trapped in a square potential-energy well, as described above, automatically creates a perfect duplicate of its initial state after an elapse of any integer multiple of the interval

(Eq’n 33)

To the initial state of that system we assign the state function

(Eq’n 34)

and to the subsequent state we assign the state function

(Eq’n 35)

But now we know that we must have R(1)=R(2), so we must have

(Eq’n 36)

If we extract the natural logarithm of that equation, we can solve the result for alpha. In that solution we must look beyond the common ln1=0 and acknowledge the result from complex analysis, which tells us that ln1=in2B for all integer values of n. That fact then gives us

(Eq’n 37)

Recalling to mind the fact that px0=nh, we deduce that

(Eq’n 38)

So now we have the state function as

(Eq’n 39)

in which we define the rationalized Planck constant as S=h/2B. The state function has become a wave function and, thus, we see that the quantum theory comes to us as a wave theory of matter and its motions.

    We can also rewrite the operators shown above as

(Eq’n 40)

(Eq’n 41)

and

(Eq’n 42)

We can also rewrite the eigenvalue equation in a special way, taking the eigenvalue equation for the linear momentum as an example. We begin by multiplying it by the complex conjugate of the state function:

(Eq’n 43)

If we divide that equation by A’2, we get

(Eq’n 44)

    I just bumped into the table on which I have built my experimental apparatus and in so doing I have jostled the apparatus and may have changed the particle’s linear momentum. I know that the particle must occupy one of the system’s eigenstates and possess the appropriate eigen-momentum, but without making an actual measurement I cannot know which state it has settled into. In that circumstance can I write out a state function that properly and correctly describes that jostled system?

    I cannot merely list the eigenfunctions of the system: that does not give me a single state function. Nonetheless, all of the system’s eigenfunctions must take part in the state function describing the system. We have two fundamental methods of combining eigenfunctions to create a single state function: addition and multiplication; that is, we can either add all of the eigenfunctions together or we can multiply them together to create the overall state function of the system.

    We know that the overall state function must differ depending upon which eigenstate the particle actually occupies. If we did not have that proposition true to Reality, then we would be unable to distinguish among the different eigenstates and we would have only one eigenstate. Thus we cannot use the product of the eigenfunctions to represent the overall state function: we must use the sum of the eigenfunctions, each multiplied by a suitable coefficient, as the overall state function. We use multiplication, as we did above, to assemble the single-coordinate eigenfunctions into multiple-coordinate eigenfunctions, but then we must use addition to assemble those eigenfunctions into the overall state function of the quantum system under consideration.

    At this point we have begun the transition from the deterministic physics of the classical realm to the fully stochastic physics of the quantum realm.

    I point out the fact that in accordance with Euler’s theorem we can write the imaginary exponential functions in our eigenfunctions as the sums of real cosines and imaginary sines. We now have our overall state function as a sum of sines and cosines. That fact brings the overall state function into the realm governed by Fourier’s theorem, which tells us that we can represent any function at all as a series of sines and/or cosines if those series have enough terms, even tending toward the infinite. So now we know that for any conceivable quantum system (and maybe some inconceivable ones as well) subject to jostling we can devise a proper and correct overall state function as a sum of the eigenfunctions of that system. If the number of quantum systems conceivable to us tends toward the infinite, then we get Dicke and Wittke’s Postulate IV.

Postulate V

    Look again at Equation 44 describing the direct calculation of an eigenvalue. Can we write that same equation using the overall state function; that is, can we have

(Eq’n 45)

as a kind of eigenvalue?

    First, we must acknowledge the fact that Equation 45 gives us a complex number, not the real number yielded by Equation 44. The imaginary part of that complex number comes from the cross terms (e.g. Rj*PRk for jk), terms that don’t exist in Equation 44. Further, the complex conjugate of the exponent in the first eigenfunction in each cross term does not fully cancel the exponent in the second eigenfunction in that term, so the cross term calculates values that differ at different locations in space. Those facts may tempt us to dismiss Equation 45 as false to Reality, but if we set all but one of the coefficients on the eigenfunctions to zero, the equation reverts to Equation 44, which we cannot dismiss. We must somehow modify Equation 45 in a way that eliminates the indeterminate value and the imaginary value without altering that reversion to Equation 44.

    To meet that criterion we must apply to the numerator and the denominator in Equation 45 a process that, when worked out in full, multiplies each direct term (j=k) by a real constant and eliminates the imaginary part of each cross term while converting the real part into a real constant. If we consider only the linear momentum of the system, then in part we have

(Eq’n 46)

We also have that equation as

(Eq’n 47)

If we multiply that equation by dx and integrate the product over the interval [-)x, )x], then we get

(Eq’n 48)

If we make

(Eq’n 49)

for any integer n, then Equation 48 becomes

(Eq’n 50)

which satisfies the criterion, with and without the operator P.

    But what have we calculated? We have Equation 45 now as

(Eq’n 51)

If all of the coefficients in the state function but one go to zero, then that equation becomes Equation 45 and we have calculated the eigenvalue of the linear momentum possessed by the particle occupying the eigenstate the nonzero coefficient represents. But if the state function has more than one nonzero coefficient, then we have a state function that seems to describe a state in which the particle occupies more than one eigenstate. We might imagine that the particle cycles rapidly among the eigenstates, occupying them in sequence, or that the particle somehow partially occupies each eigenstate, but in either case Equation 51 seems to give us a kind of average of the eigenvalues of the momentum that the particle possesses as it occupies each of the available eigenstates. Physicists call it an expectation value, but it doesn’t actually tell us what linear momentum we should expect the particle to have if we measure it: a measurement can only give us one of the eigenvalues.

    In Equation 49 we can make n represent any integer at all, however big we may want to make it. Thus we can make the integration of Equation 51 span the full width of space if we feel the need. Further, we can extend the state function into all three spatial dimensions, take dxdydz=dV as the geometric element of integration, and integrate over all space: we obtain

(Eq’n 52)

which is Postulate V.

    That integration was originally invisible and we found it through further analysis of our description of the quantum system vis-a-vis the features that the theory must possess. This is the heart of theoretical physics, using logic to reveal hidden features of our description of Reality.

    If we multiply the state function by a complex constant to obtain

(Eq’n 53)

then we get from Equation 52

(Eq’n 54)

which means that multiplying the state function by an arbitrary complex constant has no effect on the physical meaning of that state function. Thus we obtain the last piece of Postulate I.

Postulate III

    We have the operators of the quantum theory as function-like combinations of complex numbers and differentiation operators. If an operator applied to the state function yields the equivalent of multiplying the state function by a physically measurable quantity, then the operator must possess the mathematical property of being Hermitian.

    By definition we say that an operator P is Hermitian if

(Eq’n 55)

or, alternatively,

(Eq’n 56)

is true to Reality. That definition guarantees that the eigenvalues of an operator are real numbers and that the eigenstates that yield different eigenvalues under that operator are orthogonal to each other.

    To prove and verify the proposition that the eigenvalues of a Hermitian operator consist only of real numbers, we follow Dicke and Wittke in choosing an eigenstate

(Eq’n 57)

and writing Equation 56 as

(Eq’n 58)

which necessitates that

(Eq’n 59)

But only a real number can equal its own complex conjugate, so when the operator P acts on the eigenfunctions it can only yield real numbers. And that fact gives us what we need because the operators of quantum mechanics represent the measurements of physical quantities, which come to us as real numbers read off our measuring apparati.

    The fact that the operators of the quantum theory must be Hermitian leads immediately to another theorem: if two eigenvalues of the operator P do not equal each other, then their eigenfunctions must be orthogonal to each other (in the sense of vectors lying orthogonal to each other). Consider Equation 56 in the form

(Eq’n 60)

If

, then we must have

(Eq’n 61)

which satisfies the definition of orthogonal functions. Now we know that the state functions of physically realizable quantum systems comprise a set of vectors in an infinite-dimensional complex abstract space analogous to physical space. We can that abstract space a Hilbert space.

Epilogue

    Here we can re-present the Dicke-Wittke postulates as a series of propositions that, with some additional work, will eventually become proper theorems in the Map of Physics. We have then:

Proposition 1.

    We begin with the quantum theorem itself: between any two events no physical system can accrue less than one Planck unit (h=6.62x10-34 Joule-second) of action. We calculate the action accrued to a particle participating in two events by multiplying the particle’s generalized momentum by the particle’s generalized position (both measured relative to suitable origins on their respective coordinate axes) at both events and then subtracting one product from the other. Thus, no subtraction of one action from another can ever yield less than one Planck unit.

Proposition 2.

    We have a particle that participates in two events. One observer measures, at each event, the particle’s position and then its linear momentum and a second observer measures, at each event, the particle’s linear momentum and then its position. Both observers then calculate the action accrued to the particle between the two events. Employing the convention that the first variable measured goes on the right side of the product, we write the difference between the actions calculated by the two observers as

(Eq’n 62)

That statement must give us a description of the particle that is true to Reality. If it did not do so, then the first observer could measure the particle’s position with perfect precision and accuracy, the second observer could measure the particle’s linear momentum with perfect precision and accuracy, and both observers could multiply those results together to obtain a description of the particle’s action that violates Heisenberg’s Indeterminacy Principle and, ultimately, the quantum theorem.

    If we multiply a particle’s linear momentum by the distance that the particle moves between two events, we obtain the proper action accrued to the particle. If we multiply that particle’s linear momentum by a distance measured in a direction perpendicular to the direction in which the particle moves, we obtain an improper action, which is not subject to Equation 62. Thus, if we represent the vectors of the particle’s linear momentum and distance traveled by their Cartesian coordinates, Equation 62 becomes

(Eq’n 63)

in which *jk represents the Dirac delta. This equation, the basic commutation relation of quantum systems, remains valid if we assert that pj and xk represent generalized momentum and generalized position respectively.

Proposition 3.

    Equation 63 tells us that we cannot properly represent the calculation of an action by the simple multiplication of two numbers: we must have a non-commuting multiplication-like combination of mathematical entities representing the two variables under study. Because we measure distances directly and measure linear momenta indirectly, we represent distance with a pure real number and represent linear momentum with a suitable mathematical operator. Taking Equation 62 in the form

(Eq’n 64)

we see that one good candidate for a linear momentum operator is

(Eq’n 65)

The equivalent operators for energy and angular momentum are, then, respectively

(Eq’ns 66)

Proposition 4.

    A differential operator cannot stand by itself; it must operate upon some function R of the coordinate to which it refers. The operator represents an act of measurement, so that function, the state function of the quantum system under study, represents that upon which we perform the act of measurement. Because it must represent a real, unique system, the state function must be a single-valued algebraic function. Because the measurement of a quantity represented by the operator P must yield a real number p (as would be read off a gauge), the state function must have such a form that we can equate the operator P, whatever its form, with the operation of multiplying the state function by p; that is, we must have

(Eq’n 67)

the eigenvalue equation.

    We can rewrite that equation as

(Eq’n 68)

If we multiply that equation by dx and divide it by h and R, we get

(Eq’n 69)

which integrates to

(Eq’n 70)

Taking the anti-logarithm of that equation gives us

(Eq’n 71)

Proposition 5.

    We can combine operators in the same way that we combine variables in algebraic formulae, so we can create a kinetic energy operator from the linear momentum operator. We have

(Eq’n 72)

If we can devise a potential energy operator U, then we can add it to the kinetic energy operator to create a Hamiltonian operator H=T+U. As in classical dynamics, we can then equate the Hamiltonian to the total energy of the system and thereby obtain Schrödinger’s Equation,

(Eq’n 73)

The state function that solves that equation gives us a proper and correct description of quantum Reality. Such a function is

(Eq’n 74)

Proposition 6.

    The state function encodes a complete description of the quantum system that we have under consideration. It must also encode the laws of physics that guide the evolution of that description so that when operators extract numbers from it those numbers will match the numbers obtained from the measurements that the operators represent. Among those laws we must find Nöther’s theorem.

    To accord with Nöther’s theorem the state function must have a form that encodes the fact, derived from the conservation laws pertaining to linear momentum and energy, that the laws of physics do not change with any displacement of a physical system in space or in time. If we have two identical physical systems enacting identical pairs of events in both of which identical particles participate and if we separate those systems and their events in space by x0 and/or in time by t0, then the state functions that describe the two systems must equal each other. We must have, then,

(Eq’n 75)

which necessitates that

(Eq’n 76)

Taking the natural logarithm of that equation gives us

(Eq’n 77)

in which n represents any integer. We can only solve that equation by assuming the existence of a hidden factor on the left side, so that we have

(Eq’n 78)

Because px0/h and Et0/h must both equal integers, in accordance with the quantum theorem, we have "=i2B and

(Eq’n 79)

in which S=h/2B=1.05457x10-34 Joule-second. Thus, the state function is a complex-valued function of the dynamical properties of the physical system and of the coordinates in which we measure that system’s extent. We must now rewrite the physical operators as

(Eq’n 80)

(Eq’n 81)

and

(Eq’n 82)

Proposition 7.

    If we add a constant phase angle $ to the state function, we get

(Eq’n 83)

Applying any of the operators representing measurements of physical properties of the system to that state function yields the same eigenvalue that we would get if we didn’t add the phase angle to the original state function. Thus we infer that multiplying a state function by an arbitrary complex constant, exp(i$)=a+ib, does not change the physical meaning of the state function.

Proposition 8.

    In general every physical system has available to any particle in it more than one eigenstate, each with its own eigenfunction, Rn, and its own eigenvalues, pn, En, etc. We can contrive to put a particle in a given quantum system into a given eigenstate, but if something jostles that system, the particle may be knocked into a different eigenstate. With the exception of specially prepared systems, all quantum systems are subject to jostling by other quantum systems or by macroscopic systems, so we need a state function that we can use to describe a jostled quantum system.

    In order to constitute a complete description of the system, the overall state function must be a function of all of the system’s eigenfunctions. We know that the function cannot be a product of the eigenfunctions: if it were, then applying a measurement operator to it would yield the sum of all of the system’s eigenvalues, even if the system had been so established that it occupies only one of its eigenstates, which clearly gives us a wrong description. And we have no way in which we can so modify a product of eigenfunctions that it will yield a correct eigenvalue in such a case. That leaves us with a sum of the eigenfunctions, which gives us the same problem with correctly representing a measurement.

    In this case we can solve that problem by requiring that we multiply each eigenfunction by a weighting coefficient before we add it to the overall state function. If the system occupies only one of its eigenstates, the coefficient of that state’s eigenfunction equals one and the coefficients of all of the other eigenfunctions equal zero: applying a measurement operator to that state function will thus yield the correct result. If we jostle the system, the values of the coefficients change to reflect the possibilities of the system occupying other eigenstates. Applying a measurement operator to that jostled state function yields a sum of eigenvalues, each multiplied by its eigenfunction and a fraction indicative of the degree to which the system participates in that eigenstate; we get a kind of average.

    This overall state function resembles the partition function of statistical thermodynamics. Those two functions differ primarily in the fact that the exponentials of the quantum mechanical state function have imaginary arguments. Euler’s theorem tells us that our grand state function consists of a sum of a sequence of weighted cosines and weighted imaginary sines. By way of Fourier’s theorem, we now know that the state function can properly and correctly describe any conceivable quantum system.

Proposition 9.

    If we have put a quantum system into one of its eigenstates, then we can calculate the eigenvalue q of the measurable operator Q applied to that eigenstate as

(Eq’n 84)

If we jostle the system, then the weighting coefficients in Q change and that equation no longer gives us a proper measured property: the cross terms introduce imaginary numbers that cannot describe a measurable quantity. In that case we have direct terms of the form an2qn and cross terms of the form anamqmexp(i(qm-qn)x/S). We must apply to the series consisting of those terms an operation that multiplies each direct term by a constant and makes each cross term vanish. We gain that result by multiplying the series by dx and integrating it between the limits at +x0 and -x0, the points at which the particle rebounds from a potential-energy wall. That integration has the effect of multiplying each direct term by 2x0 and of zeroing out each cross term. With each cross term (XT) the integration gives us a bunch of constants (BoC) multiplying the exponential, so we have

(Eq’n 85)

in which we obtain the integer J=M-N from the integer relations qmx0=Mh and qnx0=Nh. And as before, we extend this to all three dimensions of space.

    The calculation

(Eq’n 86)

gives us an eigenvalue that does not represent an actual eigenvalue of the system, but rather comes to us as a kind of average. Physicists call that an expectation value, but we can only rightly regard it as such if we derive it for a large number of identical quantum systems. It gives us a purely statistical measurement; nonetheless, it is a legitimate part of the quantum theory.

Proposition 10.

    If we multiply the state function of a quantum system by an arbitrary complex number

(Eq’n 87)

then the expectation value calculated by Equation 86 remains unchanged. Thus, multiplying a state function by an arbitrary complex constant has no effect upon the physical meaning of that state function.

Proposition 11.

    Because we must represent measurable quantities with real numbers, the operators representing those measurements must be Hermitian.

Proposition 12.

    If the eigenvalues of two eigenstates are not equal to each other, then the two eigenstates are orthogonal to each other; that is,

(Eq’n 88)

habg

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