Pauli’s Exclusion Principle

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    Let ψ represent an eigenfunction of Schrödinger’s Equation. For the x-component of the linear momentum of the particle covered by ψ we have the eigenvalue px given by the eigenvalue equation

(Eq’n 1)

But we also have, through the definition of the partial derivative, the relation

(Eq’n 2)

In that equation Päx represents the transformation operator that shifts the contours of ψ(x) by the minuscule distance δx relative to a set of fixed coordinate axes. Comparing that equation with Equation 1 allows to write

(Eq’n 3)

in which δx has the nature of a differential (that is, it represents a pseudo-infinitesimal number).

    If H represents the Hamiltonian function describing a system and we have

(Eq’n 4)

in which θz represents an angle by which a system rotates about a fixed z-axis in the counterclockwise sense, then we have conservation of angular momentum about the z-axis and the eigenvalue equation

(Eq’n 5)

That gives us the differential transformation

(Eq’n 6)

or

(Eq’n 7)

in which Pδθz represents the operator that rotates the contours of ψ(θ) about the z-axis by an amount equal to δθz in the sense of a right-hand screw. In this case jz represents the eigenvalue of the system’s angular momentum about the z-axis.

    We can represent any rotation about the same axis as a sum of smaller rotations, so for some finite rotation about the z-axis let θz=nδθz and let the number n approach infinity as δθz approaches zero, doing so in such a way that θz remains constant. The rotation operator Pθz corresponding to that rotation comes from applying n=θz/δθz iterations of the differential rotation operator Pδθz to the state function encoding the system. Thus we have

(Eq’n 8)

In going from the second to the third line in that equation I exploited the mathematical fact that

(Eq’n 9)

    To see what that can mean, consider a system that consists of two identical particles. We feign having the ability to label them A and B and we describe each particle’s state with ψ1(A) and ψ2(B). A description of the state function of the whole system comes from multiplying the particles’ individual state functions together. But we can have an identical state of the system that corresponds to an interchange of the particles, so we must add the state function encoding that state to the system’s total state function. Thus we have

(Eq’n 10)

If both of the particles exist each in the same component state, then ψ1=ψ2. In that case interchanging the particles has the same result as does rotating the system by 180 degrees (pi radians).

    We must necessarily carry out that rotation about an axis perpendicular to the straight line passing from one particle to the other. We define that axis as the z-axis in our system of fixed axes and note that, by the rules of quantum mechanics, the z-component of each particle’s angular momentum is quantized (that is, it becomes an eigenvalue of the state function). So now we have the second component on the right side of Equation 10 as

(Eq’n 11)

in which the z-component of each particle’s total angular momentum equals the sum

(Eq’n 12)

in which sum jz represents the z-component of the particle’s orbital angular momentum and sz represents the z-component of the particle’s spin. We can thus rewrite Equation 10 as

(Eq’n 13)

In devising the second line of that equation I made use of the fact that the orbital angular momentum (and, thus its z-component) always comes out as an integer multiple of Dirac’s constant (aitch-bar), so that for identical states the argument of the first exponential in the first line of the equation comes out as an even multiple of pi, which makes the exponential itself equal +1.

    As for spin, the inherent angular momentum residing in each particle, each particle can possess an amount equal to an integer multiple of Dirac’s constant or an integer-plus-half multiple of Dirac’s constant. In the case of an integer multiple our identical states give us szA+szB as an even multiple of Dirac’s constant and Equation 13 becomes

(Eq’n 14)

The particles thus obey Bose-Einstein statistics, in which any number of particles can occupy any given angular momentum state. In the case of integer-plus-half multiples (½, 3/2, 5/2, etc.) of Dirac’s constant our identical particles state gives us szA+szB as an odd multiple of Dirac’s constant, so the exponential, consequently, equals -1 and we get Equation 13 as

(Eq’n 15)

The particles carrying half-integer spin obey Fermi-Dirac statistics, which physicists base on the fact that no two identical fermions can occupy the same angular momentum state.

    Thus we have Pauli’s exclusion principle, which Wolfgang Ernst Pauli (1900 Apr 25 – 1958 Dec 15) devised in 1925 to explain certain features in the spectroscopy of the chemical elements. Although Pauli inferred the principle through empirically-based induction (the usual scientific method), the axiomatic-deductive approach used above has become the standard method of inferring the principle, albeit usually expressed in the matrix-mechanics form. I have used the Schrödinger wave-mechanics form to set the principle more clearly in my version of the Map of Physics.

    With that principle we have the means to understand the electronic structure of atoms and, thus, of the fundamental facts of chemistry, including the layout of Dmitri Mendeleev’s table of the elements. It also gives us the means to work out the inner structure of neutron stars, the strange remnants of supernovae.

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