A physical ket is antisymmetric Û Pauli's exclusion principle.
Þ: [Coh, p.1389, (C-7)].
By [Coh, p.1386, the symmetrization postulate], a physical ket can be represented by
G|1:φ;2:ψ>, where G =S or A [Coh, p.1380, (B-19 a,b)].
By Pauli's exclusion principle, G|1:φ;2:φ>=0,
G|1:ψ;2:ψ>=0, and G|1:φ+ψ;2:φ+ψ>=0.
Remark. For the equivalence in terms of wave functions, see [Pat, p.128, l.−12].
Since Maxwell’s equations are symmetric with respect to (E,
e ) and (H, - m ),
the theorems for TM waves can be derived from the corresponding theorems for TE
waves by substituting (E, e ) with (H,
- m ) [Born, p.52, l.19].
Symmetry (i.e. constant motion) vs. degeneracy.
H]=0 (algebraic symmetry) means that H is invariant under rotation (geometric
symmetry: [Coh, p.704, (94) &(95)]). In other words, H is symmetric with respect
to J. Similarly, [A, H]=0 (i.e. A is a
constant motion) Û H is symmetric with respect to A.
Remove degeneracy Û Add a perturbation to the
Hamiltonian to destroy its symmetry. Example. [Mer2, p.268, l.10-l.19; Lan3, p.127, Problem 3].
Descriptions of momentum in terms of translational symmetry.
(Classical mechanics) The Lagrangian of a closed system is invariant under
any infinitesimal translation [Lan1, p.15, l.6-l.9]. We define momentum as a
vector in [Lan1, p.15, (7.2)].
(Quantum mechanics) The Hamiltonion of a closed system is invariant under
any infinitesimal translation. In view of [Lan3, p.42, (15.1)], we define momentum
as an operator in [Lan3, p.42,
(Solid state) We view momentum as a quantum number characteristic of
the fuller translational symmetry of free space [Ashc, p.139, l.-8-l.-5].
How symmetry simplifies the calculation of an integral over an area.
Uniform infinite line charge [Wangs, p.60, l.-6-p.61, l.20].
Spherically symmetric spherical charge distribution [Wangs, p.63, l.-19-p.65,
Group representation theory is is the
most suitable language to describe the impact of the symmetry on various
concepts in quantum mechanics.
Selection rule [Hei, p.43, l.19-l.21].
Classify the wave functions using angular momentum quantum numbers [Hei, p.43, l.22-24].
Remark 1. [Hei, p.58, l.9] shows that it is impossible to reduce the state space further with the full rotation group.
In this sense, group representation theory helps
strengthen the structure of quantum mechanics. It is possible to
use other algebraic languages to deal with symmetry [Hei,
p.viii, l.16-l.17], but the use of group representation theory gives the
clearest image in the most organized manner by using the most economical words.
This is because the definitions in group representation theory are designed
mainly for such a purpose.
Remark 2. (Addition of angular momenta) [Hei, pp.67-71, §9 Reduction of
the Product Representations] and [Coh, chap. X, Addition of Angular Momenta]
basically discuss the same material. By comparison, Cohen-Tannoudji's exposition has the following drawbacks:
out that the key
idea of adding two angular momenta is the reduction of product representation.
It would be difficult to describe this key idea in one sentence by using an
language other than group representation theory.
Heine provides a blue print of
how to reduce the product representation [Hei, p.67, l.-7-p.68,
l.7], while Cohen-Tannoudji does not.
Unless we trace back to rotations (compare [Hei, p.49, (7.3); p.68, (9.1a)]
with [Coh, p.1014, (C-24)]), we would not recognize the role the axial rotation group plays in the early stage of the reduction of the product representation.
Unless we trace back to rotations [Hei, p.69, l.-14;
p.66, Problem 8.10], we would not understand the precise meaning of the formal
given in [Coh, p.1020, l.-1].
Degeneracy [Hei, p.44, Theorem 2].
A perturbation splits an energy level [Hei, p.43, Theorem 3].
Descriptions of axial symmetry.
In terms of geometry: It is symmetric with respect to an axis.
In terms of mathematical analysis: Associated Legendre functions reduce to Legendre polynomials (i.e., [Chou, p.145,
l.2 & (3.122) reduces to [Chou, p.146, (3.123)]).
In terms of quantum mechanics: m=0 [Chou, p.145, l.2 & l.-1].
solutions) We start with Coulomb's law to develop the theory of electrostatics and start with Ampère's law [Wangs p.218, l.8] to develop the
the theory of magnetostatics. However, if we proceed in separate ways like this, we would
never obtain the complete picture of electromagnetism because the above two
theories are somehow related to each other. Therefore, it is more proper to treat the macroscopic
Maxwell equations as axioms. For a given distribution of charges and
currents, it is difficult to find their solutions for the general case [Fan,
p.20, l.1]. However, we may find interesting solutions when the sources have
spherical or cylindrical symmetry. For example, Coulomb's law [Fan, p.20,
(1.19)] can be derived from [Fan, p.19, (1.17)] using spherical symmetry; [Fan, p.22, (1.21); Wangs,
p.243, (15-19) & (15-20)] can be derived as the solution of [Fan, p.16, (1.13)]
using cylindrical symmetry. Similarly using this correct set of tools
(axioms), we may
derive both Biot-Savart's law and Ampère's
law from the Maxwell equations [Sad, §7.8; Fan, p.124, (4.70)]. Thus, the above formulas are derived from solid
rather than regarded as empirical laws (trial-and-error bases). To have the correct big picture is also the reason why [Fan, p.11, l.-7] says that axioms should be discussed
after the presentation of a theory.
If the distribution of sources has a certain kind of symmetry, then the
corresponding field produced by the sources will have the same kind of symmetry [Grif, p.178, l.-11-l.-10].
I. Spherical symmetry: The charge density r(r) depends only on the distance between
r and the origin. D(r) = D(r)rˆ.
II. Cylindrical symmetry: The charge density r(r) depends only on the perpendicular distance
between r and a cylindrical axis. This requires a charge distribution that is infinitely
long. D(r) = D(r)rˆ.
III. Plane symmetry: The charge density r(r) depends only on the perpendicular distance
between r and a plane. This requires a charge distribution that extends infinitely in two
directions. D(r) = sgn(z)D(|z|)zˆ [let xy-plane be the plane of symmetry].