In an atom, the motion of an electron is not restricted to a plane (ring). Since its total energy depends only on its distance from the positive nucleus, it might be concluded that it can roam anywhere on the surface of a sphere of radius r . However, because of quantum conditions its roaming is limited to particular places. This leads to a complex mathematical description which is why the case of planar motion on a ring was considered first. Now, before the full complexity is considered, an intermediate case will be discussed in order to minimize the complications in going from two to three dimensions. The spherical case that will be initially discussed differs somewhat from an atom in that it will be assumed that there is no interaction (potential energy) between the particle on the surface of the sphere and the center of the sphere. The particle is arbitrarily confined to motion on the spherical surface.
To describe motion on a sphere, a new angleθ is needed (Figure 6.3). In the interme-diate case, the particle then lies at a fixed distance r from the center of the sphere, and at longitudeφ, as well as latitude θ. In these coordinates:
x= r sin θ cos φ y= r sin θ sin φ z= r cos φ
Figure 6.3 Definition of the spherical coordinates: r radius,θ latitude, and φ longitude.
and since r = constant, the r derivatives drop out of the Laplacian operator. It becomes
2= (1/r2)2, where2= f (θ, φ) is the Lagrangian. Thus the Schr¨odinger equation becomes:
2(θ, φ) = −
82E I h2
(θ, φ) (6.16)
The solution of this is separable into two angular parts with(θ, φ) = (θ) (φ) where the equation for is:
d2
dφ2 = −m2 (6.17)
which has solutions like those for the particle on a ring (for motion around the polar axis):
= eim (6.18)
Notice that the symbol,α, that was previously used for the angular momentum is now m.
The m stands for “magnetic angular momentum” for historical reasons. The “mechanical angular momentum” quantum number will be called l (“el”), shortly.
The additional coordinate,θ, leads to further quantization consisting of a series of ring-like states at discrete values of the latitudeθ (Figure 6.4). The equation for (θ) is more complicated than the one for (φ), and will not be discussed in detail here. Interested readers are referred to other books such as Pauling and Wilson (1983).
Equation (6.16) has the same form as the standard differential equation for functions known as spherical harmonics, Ylm(θ, φ). The equation that these functions satisfy is:
2Ylm(θ, φ) = −l(l + 1)Ylm(θ, φ) (6.19) and Ylm is separable: Ylm(θ, φ) = lm(θ)m(φ) with the m(φ) being given by Equa-tion (6.17). Thelm(θ) are called associated Legendre functions. For l = 1, 2, 3 and m = 0,
Figure 6.4 Permitted orientations of the angular momentum vector, J(l), relative to a reference field vector, z. The values of l, and therefore J, are fixed. In this case, l= 2, so m = 0, ±1, ±2 is the magnetic quantum number. The component along z interacts with the field.
the first few are given by:
00 = (1/2)1/2 circular shape
10 = (3/4)1/2cosθ figure of eight shape
20 = (5/16)1/2(3cos2θ − 1) cloverleaf shape
From the Schr¨odinger equation then, various shapes emerge for the spacial distributions of the probability amplitudes without further assumptions. Also, the energy and the angular momentum emerge. Comparison of Equations (6.16) and (6.19) shows that the energy is:
E = h2
82I [l(l+ 1)] (6.20)
where l= 0, 1, 2, . . . , (n − 1), and the angular momentum is J = I ω, with ω the rotational frequency,ω = 2ν. Therefore, the rotational energy is Erot= J2/2I . Substituting into Equation (6.20) gives an expression for the angular momentum:
J = h
2 [l(l+ 1)]1/2 (6.21)
So l is the angular momentum quantum number, and it can have the values:
l= 0, 1, 2, . . . , n
For a total amount of energy determined by the value of n, if l= 0, there is no angular momentum, solm(θ) = 00 = constant and the symmetry of the probability amplitude is spherical. However, for example, if n= 2, so l = 2 is permitted, then the angular mo-mentum, according to Equation (6.21) is J = (√
6)--h. This is a vectorial quantity as in classical mechanics, but it is unusual in that it can have only particular directions relative to a reference vector in space. The reference might be a magnetic field, H, or an electric field, E. Historically, the magnetic case was discovered first, so it is called the magnetic
Figure 6.5 Cones on which the angular momentum vector may lie.
quantum number, and is designated m. The particular values ofθ it can have are m = 0,
±1, ±2, . . . , ±l. This is illustrated by Figure 6.4 which shows how the orientation varies with m, at constant l. As the orientation of the angular momentum vector varies, its component along the reference direction varies. This means, of course, that the interaction with the reference field vector varies, and hence the interaction energy. For each value of m, although the total angular momentum l is constant, the permitted components in a particular direction are determined by m.
If there is no magnetic, or electric, field present to affect the angular momentum, the spacial quantum number m is zero.
For a given set of n, l, and m in the presence of a reference field, the value ofθ is fixed, butφ can have any value, so the angular momentum vectors lie on cones centered on the reference vector (Figure 6.5).
The second derivatives (with respect to x, y, z, or r ,θ, φ) of the probability amplitudes represent curvatures. Thus the kinetic energy is proportional to the curvature of the prob-ability amplitude. This is consistent with the highest frequency Fourier component of the amplitude (shortest wavelength), and follows from Planck’s Principle, E= hν which gives high energies for short wavelengths.
The total energy of a state depends only on the value of the principal quantum number n, not on l or m.