As the simplest possible atom, hydrogen had been a steady focus of attention from experimental and theoretical physicists in the half-century from 1900 to 1950. John Dalton had already speculated in the early 19th Century that hydrogen was the basic building block of all atoms. The first physically realistic description of hydrogen was obtained by Schrödinger, by solving his equation in spherical polar coordinates, with the electron and proton interacting by the Coulomb interaction, V(r) = −ke2/r. The resulting solutions are incredibly degenerate, with the energies En depending on a single principal quantum number, n.
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A few years later, Paul Dirac found a relativistically correct quantum equation of state and applied it to hydrogen, finding that it automatically included a property of the electron known as intrinsic spin. But it was not until the middle of the 20th Century that Feynman, Schwinger and Tomonaga independently managed to include the interaction of the electron with the vacuum. Also including the intrinsic spin of the proton in the analysis created a situation in which every known detail of the hydrogen spectrum was correctly predicted. The convenient unit of energy for atomic systems is the eV (about 1.6 × 10−19 Joules) , while the convenient unit of distance is the Bohr radius, which is about 5.3 × 10−11 meters. The Bohr radius is the simplest quantity with units of distance that can be constructed from ℏ, c, the mass of the electron, and the fine structure constant α. [The fine structure constant is the simplest dimensionless number that can be constructed from ℏ, c, ε0 and the electron charge. In quantum field theory, it is the “coupling probability” describing how charge couples to the electromagnetic field, and has a value of roughly 1/137.]
 Spherical polar coordinates |
 Radial probability distribution of the electron in H. |
In spherical polar coordinates the equation for Ψ(r,θ,φ) can be separated into the form Rn,ℓ(r)Yℓ,mℓ(θ, φ). Since we are in three dimensions, there are three quantum numbers, the principal quantum number n, an eigenvalue associated with the operator for angular momentum squared, and an eigenvalue associated with the operator for the z component of the angular momentum operator. The set of functions Yℓ,mℓ(θ, φ) are called spherical harmonics, and they invariably occur in the solution for any three dimensional system that is spherically symmetric. The so-called radial probability distribution for the electron is obtained from the expression r2|Rn,ℓ(r)|2. Some relevant eigenvalue equations are L2Yℓ,mℓ = ℏ2ℓ(ℓ + 1)Yℓ,mℓ and LzYℓ,mℓ = ℏmℓYℓ,mℓ. The value of n ranges from 1 to infinity, in steps of 1, while the values of ℓ range from 0 to n-1, in steps of 1, and the values of mℓ range from −ℓ to ℓ in steps of 1.
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Following in the tradition of 1900-era spectroscopy, the values of ℓ have symbols! A state with value 0 is called an s state, a state with value 1 is called a p state, a state with value 2 is called a d state, and the others in order are alphabetical, f, g, h, etc. The ground state (lowest energy state) of ANY spherically symmetric system is always a 1s state, in other words it has n = 1 and ℓ = 0. If the only interaction needed were the Coulomb potential, the states of hydrogen would be extremely degenerate, with energies depending ONLY on the principal quantum number n. But a relativistically correct treatment requires a so-called spin-orbit potential, which involves the dot product of the operator for angular momentum with the operator for intrinsic spin. This breaks the degeneracy, but still does not describe all aspects of the H system's energy levels. The spin-orbit effect is called fine structure. In the next step of realism, the intrinsic spin of the proton is taken in to account, and there is a spin-spin interaction between electron and proton intrinsic spins. The results are called hyperfine structure. There is still a famous discrepancy, the Lamb Shift, which can be understood only by taking account of the quantum nature of the electromagnetic field, and the interaction of the electron with the vacuum. With all this care, the agreement between theoretical predictions and actual energy levels is entirely satisfactory. Because of the spin-orbit term in the potential a new quantum number is required, which looks like j = ℓ + s, with s restricted to ±(1/2). Thus a level 2p in the Schrödinger spectrum splits into two levels, in the notation nℓj, namely 2p1/2 and 2p3/2. And so on. The spin-spin interaction splits the states even further and requires another new quantum number, F. From the point of view of astrophysics, one of the most important transitions seen in nature is the transition from F = 1 to F = 0 in neutral hydrogen. This is a transition from the state where proton and electron spins are aligned, to the state where they are antiparallel.
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Of course solutions to the various dynamical equations exist for E ≥ 0. These, with appropriate boundary conditions, are states describing scattering of an electron from a proton. The energy eigenvalue is continuous from 0 to infinity.
The intrinsic spin is a fundamental property of all fundamental pointlike particles that are charged. Such particles have an intrinsic magnetic moment, and Dirac found that his equation required the association of the magnetic moment with an intrinsic property that is called spin, and predicted the relation between them. The associated operators are S2 and Sz, in analogy with the quantum angular momentum operator L. The S2 operator has eigenvalues ℏ2s(s+1) where for all fermions s = 1/2. The Sz operator has eigenvalue msℏ where for all fermions ms can only be ±(1/2).
For reasons tracing back to
classical electrodynamics, the magnetic moment of fundamental
particles is written as μs = (−eg)/(2m)S.Here
g
is a fudge factor that can be adjusted to give a magnetic moment
in agreement with experiment. Dirac's equation predicts the
value of g as precisely 2. Because of the electron's
interaction with the vacuum, the measured value of g differs
very, very slightly from 2. The difference is correctly
predicted by quantum electrodynamics. Every pointlike
fundamental particle with spin, that is charged, has a magnetic
moment.
The excited states of an electron in an atom are not stationary states, because the electron couples to the electromagnetic field, and thus can always lose energy by emitting a real photon. Photons have no mass or charge, but do have an intrinsic spin of 1. [All bosons have a spin of 1 or 0.] Thus, to conserve angular momentum, there is a selection rule involving change of angular momentum in these electromagnetic transitions. The obvious selection rule is Δℓ = ±1. Other rules are summarized in the text.
How do we break the degeneracy in mℓ? By using an external magnetic field B to break the spherical symmetry of the system. This was discovered by Pieter Zeeman as far back as 1896. There is a magnetic moment operator associated with the angular momentum operator of any charged particle, so the classical interaction potential −μ·B breaks spherical symmetry since the magnetic field defines a specific direction in space. Call this direction the z axis. There are two contributions to the magnetic moment, one from the angular momentum eigenstate of the electron, the other from its intrinsic spin, so μz = (e/(2m))[Lz + 2Sz]. Notice that for s states only the spin magnetic moment causes a splitting, whereas in general the splitting depends both on ms and mℓ.
For clarity, only the mℓ splitting is shown in this diagram.
