The One-Electron Atom
107
should be appreciated that the results of the Bohr model follow from ad-hoc,
though interesting, assumptions, while those from the Schrödinger equation are
based on fundamental physical principles.
The solutions of the nonrelativistic Schrödinger equation for the one-electron
atoms provide a satisfactory basis for the understanding of the general features
of the energy spectra of these atoms. However, these spectra have a fine
structure, to understand which small relativistic corrections should be included
and additional properties for the electron proposed.
4.2 ELECTRON SPIN
The l-degeneracy of the energy levels of the one-electron atom is removed by
any additional interaction which is noncoulombic in form. Such an interaction
may be provided by the relativistic corrections to the Schrödinger equation, or
by the non-point structure of the nucleus. As a consequence, the energy levels
for a given n split into several closely-spaced levels corresponding to different
l values. This leads to multiplets of spectral lines which are described as the
fine structure of the spectrum.
There are some fine-structure lines which cannot be described in terms of
the l-multiplets. Examples of these are the alkali metal spectra which show
doublets of closely spaced lines. Prominent among these is the sodium yellow
line for the (n = 3, l = 1) transition to (n = 3, l = 0), which actually consists of two
closely-spaced lines of wavelengths 5890 Å and 5896 Å. To explain these splitting,
Goudsmit and Uhlenbeck (1925) proposed that the electron has an intrinsic
angular momentum called the spin (the origin of spin is not the spatial motion of
the electron), and an associated magnetic moment. If S describes the spin of the
electron, the associated magnetic moment is
µ = –b S
(4.33)
where, since the electron has negative charge, it is assumed that b is positive
constant of proportionality. It is the interaction of this magnetic moment with the
magnetic field seen by the electron due to the motion of the nucleus around it,
that contributes to the fine structure of the atomic energy levels.
A definitive support to the hypothesis of spin is provided by the experiment
of Stern and Gerlach (1922). In this experiment, a beam of neutral silver atoms
was passed through an inhomogeneous magnetic field in the z direction
(Fig. 4.1). If the atom has a magnetic moment µ, it has a potential energy
U = –µ ⋅
µ ⋅
µ ⋅
µ ⋅
µ ⋅ B
(4.34)
in this field, and is subjected to a force in the z-direction,
F z = x
B
z
∂
µ ∂
(4.35)
107
should be appreciated that the results of the Bohr model follow from ad-hoc,
though interesting, assumptions, while those from the Schrödinger equation are
based on fundamental physical principles.
The solutions of the nonrelativistic Schrödinger equation for the one-electron
atoms provide a satisfactory basis for the understanding of the general features
of the energy spectra of these atoms. However, these spectra have a fine
structure, to understand which small relativistic corrections should be included
and additional properties for the electron proposed.
4.2 ELECTRON SPIN
The l-degeneracy of the energy levels of the one-electron atom is removed by
any additional interaction which is noncoulombic in form. Such an interaction
may be provided by the relativistic corrections to the Schrödinger equation, or
by the non-point structure of the nucleus. As a consequence, the energy levels
for a given n split into several closely-spaced levels corresponding to different
l values. This leads to multiplets of spectral lines which are described as the
fine structure of the spectrum.
There are some fine-structure lines which cannot be described in terms of
the l-multiplets. Examples of these are the alkali metal spectra which show
doublets of closely spaced lines. Prominent among these is the sodium yellow
line for the (n = 3, l = 1) transition to (n = 3, l = 0), which actually consists of two
closely-spaced lines of wavelengths 5890 Å and 5896 Å. To explain these splitting,
Goudsmit and Uhlenbeck (1925) proposed that the electron has an intrinsic
angular momentum called the spin (the origin of spin is not the spatial motion of
the electron), and an associated magnetic moment. If S describes the spin of the
electron, the associated magnetic moment is
µ = –b S
(4.33)
where, since the electron has negative charge, it is assumed that b is positive
constant of proportionality. It is the interaction of this magnetic moment with the
magnetic field seen by the electron due to the motion of the nucleus around it,
that contributes to the fine structure of the atomic energy levels.
A definitive support to the hypothesis of spin is provided by the experiment
of Stern and Gerlach (1922). In this experiment, a beam of neutral silver atoms
was passed through an inhomogeneous magnetic field in the z direction
(Fig. 4.1). If the atom has a magnetic moment µ, it has a potential energy
U = –µ ⋅
µ ⋅
µ ⋅
µ ⋅
µ ⋅ B
(4.34)
in this field, and is subjected to a force in the z-direction,
F z = x
B
z
∂
µ ∂
(4.35)