The Quantum Theory of Angular Momentum

I have primarily devised this essay by adapting information from the section on angular momentum in Paul Dirac’s 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’n 1)

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

(Eq’ns 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’n 3)

Thus we get Equation 1 in the form

(Eq’n 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

[L |
[L |
[L |
||

[L |
[L |
[L |
||

[L |
[L |
[L |

(Eq’ns 5)

and

[L |
[L |
[L |
||

[L |
[L |
[L |
||

[L |
[L |
[L |

(Eq’ns 6)

and

[L |
[L |
[L |
||

[L |
[L |
[L |
||

[L |
[L |
[L |

(Eq’ns 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’n 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’s 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’n 9)

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

(Eq’n 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’s statement that L^{2} represents a true scalar and,
therefore, it commutes with the angular momentum operator,

(Eq’n 11)

Pick one of the components of the angular momentum
operator (physicists typically use L_{z}). The fact that L^{2}
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’ns 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’n 13)

In light of that equation, we see that multiplying L_{y} by ±i gives
us

(Eq’n 14)

For convenience we can define two operators by writing

(Eq’n 15)

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

(Eq’n 16)

which gives us

(Eq’n 17)

Likewise we can apply L_{±} to the first of Equations 12 to get

(Eq’n 18)

in which we exploit the fact that L^{2} commutes with the components
of **L**. So now we know that if we apply L_{±} to a simultaneous
eigenfunction ψ
of L^{2} and L_{z}, we get a new simultaneous eigenfunction L_{±}ψ
that has eigenvalues A and B±S.

We have next

(Eq’n 19)

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

(Eq’n 20)

Also from Equations 12 we know that A≥B^{2}
and we know it because A represents the square of the system’s 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 ψ_{0},
we get a new state function ψ_{1}=L_{±}ψ_{0}.
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’s
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’s 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
ψ_{l}
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’n 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’n 22)

Subtracting the second of those equations from the first yields

(Eq’n 23)

Because B_{max}≥B_{min},
the first factor in that latter formula can never go to zero. Therefore, the
second factor must zero out and we find that

(Eq’n 24)

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

(Eq’n 25)

in which n≥0
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 B_{min} to transform it into the eigenfunction of B_{max}.
Combining Equations 24 and 25 then yields

(Eq’n 26)

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

(Eq’n 27)

with l=0, ½, 1, 3/2, 2,.... For the z-component of the angular momentum we have

(Eq’n 28)

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

Niels Bohr’s 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’n 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’s formula to calculate frequency instead of wavelength and generalized it beyond hydrogen;

(Eq’n 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’s hypothesis in the form

(Eq’n 31)

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

(Eq’n 32)

Further analysis enabled Bohr to calculate the Rydberg frequency,

(Eq’n 33)

For the case Z=1 (hydrogen) we have R_{z}=3.290x10^{15}
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 m_{0} 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’n 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’s indeterminacy principle as it pertains to angular momentum and angular displacement,

(Eq’n 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’n 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’n 37)

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

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