The Quantum Theory of Angular Momentum

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    I have primarily devised this essay by adapting information from the section on angular momentum in Paul Dirac痴 book "The Principles of Quantum Mechanics".

    The fourth postulate of the quantum theory states that there exists a mathematical operator that, when applied to the state function describing a quantum system, will extract a description of the angular momentum held by that system. In order to determine the form of that operator we must look at the classical description of the angular momentum,

(Eq地 1)

which gets expressed in terms of the vector components in a Cartesian space as

(Eq地s 2)

We want to use the standard representation of the state function, in which representation the distance operator consists of a simple multiplication by distance and the linear momentum operator involves a differentiation with respect to distance. To create the angular momentum operator we substitute those operators into Equations 2. For the last of those equations, for example, we get

(Eq地 3)

Thus we get Equation 1 in the form

(Eq地 4)

    With the operator forms of Equations 2 we can work out the commutation relations between the angular momentum operator and its factors and between the angular momentum operator and its own components. We get

[Lx,x]=0

 

[Lx,y]=iSz

 

[Lx,z]=-iSy

[Ly,x]=-iSz

 

[Ly,y]=0

 

[Ly,z]=iSx

[Lz,x]=iSy

 

[Lz,y=-iSx

 

[Lz,z]=0

(Eq地s 5)

and

[Lx,px]=0

 

[Lx,py]=iSpz

 

[Lx,pz]=-iSpy

[Ly,px]=-iSpz

 

[Ly,py]=0

 

[Ly,pz]=iSpx

[Lz,px]=iSpy

 

[Lz,py]=-iSpx

 

[Lz,pz]=0

(Eq地s 6)

and

[Lx,Lx]=0

 

[Lx,Ly]=iSLz

 

[Lx,Lz]=-iSLy

[Ly,Lx]=-iSLz

 

[Ly,Ly]=0

 

[Ly,Lz]=iSLx

[Lz,Lx]=iSLy

 

[Lz,Ly]=-iSLx

 

[Lz,Lz]=0

(Eq地s 7)

In that last matrix the commutators in the upper center, middle right, and lower left positions give us the components of the angular momentum operator cross multiplied with itself. Contrary to the classical expectation that the cross product of a vector with itself zeroes out, we get

(Eq地 8)

That equation means that the vector components do not commute with each other and that fact, in turn, means that we cannot measure two vector components of a body痴 angular momentum to arbitrary precision. Unlike action conjugate pairs, such as location and linear momentum or time and energy, angular momentum manifests the indeterminacy principle within itself.

    Given that the vector cross product of the angular momentum with itself gives us new information, we may reasonably expect to learn something from a consideration of the vector dot product of the angular momentum operator with itself. We have

(Eq地 9)

Will that function commute with its components? We calculate, using Equations 7,

(Eq地 10)

We can carry out that same calculation with the other components of the angular momentum operator and we will get the same result. These calculations confirm Paul Dirac痴 statement that L2 represents a true scalar and, therefore, it commutes with the angular momentum operator,

(Eq地 11)

    Pick one of the components of the angular momentum operator (physicists typically use Lz). The fact that L2 commutes with that component means that the two operators can only legitimately act on a state function from which each operator extracts a mathematical entity that the other operator acts on just as a differential operator acts on a scalar constant. That fact necessitates the existence of a state function ψ that conforms to two eigenvalue equations,

(Eq地s 12)

Next we want to see how the other components act on the state function. The most direct way to get to that destination goes through the commutators, which sum up as

(Eq地 13)

In light of that equation, we see that multiplying Ly by アi gives us

(Eq地 14)

For convenience we can define two operators by writing

(Eq地 15)

    If we combine Equation 14 with the second of Equations 12, we get

(Eq地 16)

which gives us

(Eq地 17)

Likewise we can apply L to the first of Equations 12 to get

(Eq地 18)

in which we exploit the fact that L2 commutes with the components of L. So now we know that if we apply L to a simultaneous eigenfunction ψ of L2 and Lz, we get a new simultaneous eigenfunction Lψ that has eigenvalues A and BアS.

    We have next

(Eq地 19)

Applying that operator to the state function and referring to the eigenvalue Equations 12 gives us

(Eq地 20)

Also from Equations 12 we know that AB2 and we know it because A represents the square of the system痴 total angular momentum and B represents the magnitude of the z-component of that angular momentum.

    Look at that calculation from a different angle. Equation 17 tells us that if we apply either L+ or L- to an initial state function ψ0, we get a new state function ψ1=Lψ0. Applying one or the other operators, in accordance with Equation 15, is equivalent to measuring a combination of the x- and y-components of the system痴 angular momentum. That measurement, in accordance with the indeterminacy principle, changes the state of the system. If we then measure the z-component of the angular momentum, we find that the previous measurement shifted that component closer to the maximum possible value, positive or negative, by one Dirac unit (represented by the ubiquitous aitch-bar). When the z-component takes up the entire magnitude of the system痴 angular momentum it cannot grow any more, so we know that any subsequent measurement of the x- and/or y-components must necessarily yield a zero. We represent that state with a state function ψl and note that if we apply to it the operator L that would increase the magnitude of the z-component, the result must equal zero. The state function still exists, but now it represents a state that the system cannot occupy. If we apply the reverse operator L to that state function, the eigenvalue extracted must still equal zero, so we get Equation 20.

    Taking the eigenvalue part of Equation 20, dividing out the state function, and carrying out a little algebraic rearrangement yields

(Eq地 21)

But in this case B represents the maximum and the minimum eigenvalues of the z-component of the angular momentum, so we can rewrite that equation in two parts,

(Eq地 22)

Subtracting the second of those equations from the first yields

(Eq地 23)

Because BmaxBmin, the first factor in that latter formula can never go to zero. Therefore, the second factor must zero out and we find that

(Eq地 24)

    Finally we note that we must have as true to Reality the statement that

(Eq地 25)

in which n0 represents an integer. We know that n represents an integer because, as the coefficient of the difference between the extreme eigenvalues, it represents the number of times that we must apply the operator L+ to the eigenfunction of Bmin to transform it into the eigenfunction of Bmax. Combining Equations 24 and 25 then yields

(Eq地 26)

in which the lower-case ell represents the quantum number corresponding to the system痴 angular momentum. And, of course, that result transforms the first of Equations 12, by way of the first of Equations 22, into

(Eq地 27)

with l=0, ス, 1, 3/2, 2,.... For the z-component of the angular momentum we have

(Eq地 28)

in which m=l, l-1, ..., 1-l, -l.

    Niels Bohr痴 theory of the basic electronic structure of the atom stands as the most fundamental application of that theory of quantized angular momentum. In 1911 Ernest Rutherford conducted his famous experiment, which demonstrated the existence of the atomic nucleus and confirmed that the atom resembled something like the model proposed by Hantaro Nagaoka (1865 Aug 15 1950 Dec 11) in 1904. In his Saturnian model of the atom Nagaoka proposed that each atom consists of a dense nucleus with the electrons swirling around it like the rings of Saturn. Bohr understood that the electrons would act more like the moons of Saturn rather than the rings and, further, that the atomic "moons" would follow orbits that differ from each other by integer multiples of the Dirac unit of angular momentum. That latter assumption, Bohr understood, would explain a weird formula familiar to spectroscopists, a formula that had been devised in 1885 by Johann Jakob Balmer (1825 May 01 1898 Mar 12) and improved in 1888 by Johannes Rydberg (1854 Nov 08 1919 Dec 28).

    Using the wavelengths measured for four bright lines in the spectrum of hydrogen, Balmer inferred the formula

(Eq地 29)

in which equation p>q (with p and q taking only integer values) and K=364.56 nanometers. Later Balmer found that his formula correctly calculated the wavelengths of another twelve lines of the hydrogen spectrum, which wavelengths astronomers had derived from spectroscopic studies of white stars, those stars whose photospheres have high enough temperatures to drive the more energetic transitions that underlie the generation of those additional bright lines in the spectra. Rydberg then inverted Balmer痴 formula to calculate frequency instead of wavelength and generalized it beyond hydrogen;

(Eq地 30)

in which the integers obey the relation a>b.

    In 1913 Niels Bohr published his explanation for the Balmer-Rydberg formula. He had simply applied Planck痴 hypothesis in the form

(Eq地 31)

calculating the difference between the energies that an electron would have in orbit-a and orbit-b, so that Rydberg痴 formula became

(Eq地 32)

Further analysis enabled Bohr to calculate the Rydberg frequency,

(Eq地 33)

For the case Z=1 (hydrogen) we have Rz=3.290x1015 hertz, which corresponds to an electron energy of 13.605 electron-volts. In that formula Z represents the atomic number of the atom under consideration and the numbers corresponding to m0 and e represent the rest mass and the electric charge carried by a single electron.

    A simple interpretation of Equation 32 tells us that an electron occupying the orbit to which we have assigned the quantum number n has energy

(Eq地 34)

Of course we cannot have the case n=0, but clearly the smaller the value of n, the closer the electron comes to the nucleus of the atom; thus, n provides an index for numbering the atomic orbits from the smallest orbit outward to the largest.

    Next we invoke Heisenberg痴 indeterminacy principle as it pertains to angular momentum and angular displacement,

(Eq地 35)

The minimum change in angular momentum corresponds to the maximum possible change in angular displacement, Δθ=2π radians, so we get ΔL=h/2π=S. We infer that each orbit differs from any other by an integer multiple of the Dirac constant (S), so the angular momentum of an electron occupying the n-th orbit equals

(Eq地 36)

In that equation the integer n corresponds exactly to the quantum number n in Equation 34, so we can rewrite Equation 34 as

(Eq地 37)

Thus we contrive a relation between an electron痴 energy and its angular momentum.

habg

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