One of the most prominent American physicists of the twentieth century, Julian Schwinger (1918–94) taught at Harvard, MIT, and UCLA, among other institutions. In addition to his many other awards, Schwinger, jointly with Richard Feynman and Shinichiro Tomonaga, received the Nobel Prize in Physics in 1965 for his work in quantum electrodynamics.
One of the most prominent American physicists of the twentieth century, Julian Schwinger (1918–94) taught at Harvard, MIT, and UCLA, among other institutions. In addition to his many other awards, Schwinger, jointly with Richard Feynman and Shinichiro Tomonaga, received the Nobel Prize in Physics in 1965 for his work in quantum electrodynamics.
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One of the most prominent American physicists of the twentieth century, Julian Schwinger (1918–94) taught at Harvard, MIT, and UCLA, among other institutions. In addition to his many other awards, Schwinger, jointly with Richard Feynman and Shinichiro Tomonaga, received the Nobel Prize in Physics in 1965 for his work in quantum electrodynamics.
Product Details
| ISBN-13: | 9780486801889 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 02/09/2015 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 64 |
| File size: | 9 MB |
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On Angular Momentum
By Julian Schwinger
Dover Publications, Inc.
Copyright © 2015 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-80188-9
CHAPTER 1
Introduction
One of the methods of treating a general angular momentum in quantum mechanics is to regard it as the superposition of a number of elementary "spins," or angular momenta with j = 1/2. Such a spin assembly, considered as a Bose-Einstein system, can be usefully discussed by the method of second quantization. We shall see that this procedure unites the compact symbolism of the group theoretical approach with the explicit operator techniques of quantum mechanics.
We introduce spin creation and annihilation operators associated with a given spatial reference system, a+ζ = (a++, a+- and aζ = (a+, a-), which satisfy
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
The number of spins and the resultant angular momentum are then given by
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
With the conventional matrix representation for σ, the components of J appear as
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
Of course, this realization of the angular momentum commutation properties in terms of those of harmonic oscillators can be introduced without explicit reference to the composition of spins.
To evaluate the square of the total angular momentum
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
we employ the matrix elements of the spin permutation operator
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
Thus
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
and
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
According to the commutation relations (1.1),
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
whence
J2 = 1/2n (1/2n + 1); (1.9)
a given number of spins, n=0, 1, 2, ..., possesses a definite angular momentum quantum number,
j = 1/2n = 0, 1/2, 1, .... (1.10)
We further note that, according to (1.3), a state with a fixed number of positive and negative spins also has a definite magnetic quantum number,
m = 1/2(n+ - n-), j = 1/2(n+ + n-). (1.11)
Therefore, from the eigenvector of a state with prescribed occupation numbers,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)
we obtain the angular momentum eigenvector
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)
Familiar as a symbolic expression of the transformation properties of angular momentum eigenvectors, this form is here a precise operator construction of the eigenvector.
On multiplying (1.13) with an analogous monomial constructed from the components of the arbitrary spinor xζ = (x+, x-)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)
we obtain, after summation with respect to m, and then with respect to j,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)
in which we have written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)
To illustrate the utility of (1.16), conceived of as an eigenvector generating function, we shall verify the orthogonality and normalization of the eigenvectors (1.13). Consider, then,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)
According to the commutation relations (1.1), and aζΨ0 we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19)
whence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)
We have thus proved that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)
As a second elementary example, we shall obtain the matrix elements of powers of J± by considering the effect of the operators eλJ ± on (1.16). We have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)
and therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)
which, on expansion, yields the nonvanishing matrix element
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)
Similarly
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.25)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.26)
A particular consequence of (1.24) and (1.26) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.27)
which details the construction of an arbitrary eigenvector from those possessing the maximum values of \m\ compatible with a given j.
It is also possible to exhibit an operator which permits the construction of an arbitrary eigenvector from that possessing the minimum value of j compatible with a given m. Indeed, (1.13), written in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28)
states that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.29)
where K+ and two associated operators are defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)
It is easily seen that
[J3, K±] = [J3, K3] =0, (1.31)
and that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.32)
The latter are analogous to the commutation properties of J, save for the algebraic sign of the commutator [K+, K_]. In keeping with this qualified analogy we also have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.33)
as compared with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.34)
Noting that the eigenvalue of K3j + 1/2, is we see that the roles of j and m are essentially interchanged in K. The hyperbolic nature of the space in which the latter operates is thus related to the restriction |m| ≤ j.
If (1.29) is multiplied by a similar numerical quantity, and then summed with respect to j, one obtains
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.35)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.36)
and Ir is the cylinder function of imaginary argument. A simpler generating function is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.37)
CHAPTER 2Rotations
A significant interpretation is obtained for (1.15) by introducing the operators
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)
where
[xy] = x+ y- - x- y+. (2.2)
With the restriction
(x*x) = 1, (2.3)
these operators also obey the commutation relations (1.1), and must therefore constitute spin creation and annihilation operators associated with an altered spatial reference system. Accordingly, (1.15) can be viewed as the expression of the state m = j, in a rotated coordinate system, as a linear combination of the eigenvectors in a fixed coordinate system,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)
The unitary nature of this transformation is here easily verified,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)
In general
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)
where the coefficients are to be inferred from
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)
It is useful to introduce the unitary operator that generates Ψ'(jm') from Ψ(jm)
Ψ'(jm') = UΨ(jm), (2.8)
which permits an alternative construction of the coefficients in (2.6),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)
In terms of the successive rotations characterized by Eulerian angles, φ, θ, ψ, U is given explicitly by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)
are the operators appropriate to the coordinate systems produced by the previous rotations. The resulting expression for U(φθψ) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)
The angular momentum operators associated with the new coordinate system,
J' = UJU-1, (2.13)
can be constructed from the transformed creation and annihilation operators,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)
In evaluating (2.14), we have made use of the relations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)
of which the former follows immediately from the significance of a+± as a positive (negative) spin creation operator, while the latter may be verified by differentiation with respect to θ, in conjunction with the commutation relations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)
The form of (2.14) is in agreement with (2.1) and (2.3), where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)
To construct the matrix of U, we consider
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18)
in which the a+ are the operators (2.14). On writing
(ya'+) = (a+ uy) (2.19)
where u is the matrix
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)
we immediately obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)
Since (2.12) implies that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.22)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23)
we may simplify (2.21) by placing = φ = ψ = 0, thereby obtaining
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)
The matrix u is unitary and unimodular, that is, possesses a unit determinant. Its representation in terms of spin matrices has, as it must, the form of (2.12),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.25)
Any such unitary matrix can be presented as
u = e-iH (2.26)
where H is a Hermitian matrix. Since
det u = e-i trH, (2.27)
H must be a traceless Hermitian matrix and, accordingly, is a linear combination of the spin matrices, with real coefficients. Hence u can be written as
u = e-(i/2)λn x σ (2.28)
where n is a unit vector, specified by two angles, α and ß. The fact that (2.28) is the matrix describing a rotation through the angle γ about the axis n affirms the well-known equivalence between an arbitrary rotation and a simple rotation about a suitably chosen axis. The rotation angle γ is easily obtained by comparing the trace of u, in its two versions,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.29)
More generally, the trace of U for a given j depends only upon the rotation angle γ. We define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.30)
in which Pj is the projection operator for the states with quantum number j. If we remark that U must also have the form of (2.28),
U = e-iγn x J (2.31)
we immediately obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.32)
However, we can also derive this directly from the generating function (2.21).
For simplicity we shall assume the reference system to be so chosen that u is a diagonal matrix, with eigenvalues e±(i/2)γ We replace x*ζ with t([partial derivative]/[partial derivative]yζ) and evaluate the derivatives at yζ = 0. According to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.33)
we then have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.34)
in which the notation reflects the necessity of placing the derivatives to the left of the powers of yζ. Now
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.35)
and therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.36)
which is a generating function for the χ(j). On writing
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.37)
and expanding in powers of t, one obtains
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.38)
Symmetry properties of U(j)mm' (φθψ) are easily inferred from (2.21). According to the in variance of (x*uy) under the substitutions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.39)
Among the additional equivalent forms produced by successive application of these transformations are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.40)
We also note that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.41)
On removing the angles φ and ψ with the aid of (2.22), we find that the content of (2.39) and (2.40) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.42)
In view of these relations, it is sufficient to exhibit U(j)mm'(θ) for non-negative values of m and m'.
On expanding the generating function (2.24) in terms of φjm(x*) or of φjm(y) we obtain the equivalent expressions
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.43a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.43b)
of which the latter is the counterpart of (2.7). As a convenient means of constructing U(j)mm'(θ), we place
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
so that (2.43b) reads
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.44)
Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.45)
The structure of the right side will be recognized as that of the Jacobi polynomial,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.46)
whence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.47)
Other forms can be obtained from (2.43), corresponding to the variety of transformations permissible to hypergeometric functions. Thus the known relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.48)
applied to (2.47), gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.49)
Another aspect of reference system transformation is best discussed in terms of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.50)
This quantity is the transformation function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.51)
in which we have used ω to designate collectively the angles φθψ relating the new reference system to the fixed one. We shall be interested in the differential characterization of this transformation function, in its dependence upon the Eulerian angles. Now
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.52)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.53)
And, therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.54)
This is a differential operator representation of an arbitrary angular momentum vector. The familiar differential operators associated with an orbital angular momentum emerge if the transformation function is independent of ψ. Since this corresponds to m = 0, the quantum number j must then be an integer.
The differential operators (2.54) are well-known in connection with angular momentum of a rigid body, and, accordingly, the eigenvalue equation for J2 in this representation will be identical with the symmetrical top wave equation. To construct this equation directly, we remark that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.55)
Since
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.56)
On referring to (2.52), we immediately obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.57)
and the analogous differential equation for (ω|), including the eigenvalue equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.58)
An integral theorem concerning the angular dependence of U, or U–1, is stated by
∫ U dω = P0, (2.59)
where P0 is the projection operator for the state j = 0, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.60)
The integration domain is here understood to be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.61)
To prove this theorem we subject (2.57) to the angular integrations contained in dω. In virtue of the periodicity possessed by U-1 over Aφ intervals of φ and ψ, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.62)
This result asserts the vanishing of U-1 dω, and the Hermitian conjugate ∫ U dω, except for the state with j = 0. The fact that the rotation operator U reduces to unity for this spherically symmetrical state completes the proof of (2.59). We shall defer application of this theorem to the next section.
(Continues...)
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Table of Contents
Contents
1. Introduction,2. Rotations,
3. Addition Of Two Angular Momenta,
4. Three And Four Angular Momenta,
5. Tensok Opeeators,
Appendix A,
Appendix B,
Appendix C,