# (p.373) Appendix B

# (p.373) Appendix B

**Real Spherical Harmonics**

(p.373) Appendix B

**Real Spherical Harmonics**

Wigner [350] has shown that if time is reversible in a quantum-mechanical system, then all wavefunctions can be made real. This theorem enables us to use real wavefunctions whenever possible, which are often more convenient than complex ones. Here we present a simplified proof of Wigner’s theorem, with some examples of its applications.

In the absence of a magnetic field, the Schrödinger equation of a particle moving in an external potential *U*(**r**) is

This equation is explicitly time-reversal invariant, because it only contains the square of the momentum **p** = −*iħ*∇. By taking the complex conjugate of Eq. B.1, we have

Obviously, if *ψ* is a solution of Eq. B.1, then *ψ* ^{*} is also a solution of Eq. B.1 *with the same energy eigenvalue E*. Consequently, the linear combinations of ψ and *ψ* ^{*} are also solutions of Eq. B.1 with the same energy eigenvalue. If ψ and *ψ* ^{*} are linearly independent (that is, they do not differ by a constant multiplier), then the following two real wavefunctions,

represent two states with the same energy eigenvalue. In other words, that energy level is degenerate.

We discuss a few examples in the following. A plane wave

*ψ* = *e* ^{ikx} (B.5)

represents a particle running in the positive *x* direction. In the absence of magnetic fields, its complex conjugate

(p.374)
*ψ** = *e* ^{−ikx}, (B.6)

which represents a particle running in the negative *x* direction, should be a solution of the Schrödinger equation with the same energy eigenvalue. The wavefunctions can be written in real form:

*ψ* _{1} = cos *kx*, (B.7)

*ψ* _{2} = sin *kx*. (B.8)

Those wavefunctions represent a pair of standing-wave states with a 90° phase difference in space.

In a central field of force, the wavefunctions can always be written in terms of spherical harmonics,

*ψ* _{nlm} = *u*(*r*)*P _{l} ^{m}* (cos

*θ*)

*e*, (B.9)

^{imø}where *P _{l} ^{m}*(

*x*) is an associated Legendre function [123], and

*u*(

*r*) is a radial function. A solution with

*m*> 0 represents a counterclockwise circular motion of the particle around the

*z*axis. A solution with

*m*< 0 represents a clockwise circular motion of the particle around the

*z*axis. In the presence of an external magnetic field, those states might have different energy levels. In the absence of an external magnetic field, that is, when the system exhibits time reversal symmetry, two states with equal and opposite quantum number

*m*are at the same energy level. Therefore, all those wavefunctions can be written in forms of sin

*mø*or cos

*mø*, which represent standing waves with respect to the azimuth

*ø*.

The spherical harmonics in real form have explicit nodal lines on the unit sphere. Morse and Feshbach [144] have given a detailed description of those real spherical harmonics, and gave them special names. Here we list those real spherical harmonics in normalized form. In other words, we require

The first of the real spherical harmonics is a constant, which does not have any nodal line:

The ones with *l* ≠ 0 and *m* = 0 have nodal lines dividing the sphere into horizontal zones, which are called *zonal harmonics*. The first two are

The ones with *m* = *l* divide the sphere into sections with vertical nodal lines, which are called *sectoral harmonics*. Those are:

The rest of the real spherical harmonics are called *tesseral harmonics*, which have both vertical and horizontal nodal lines on the unit sphere:

These real spherical harmonics are graphically shown in Figure B.1.