Quantum number
To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed.
For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks, which have no classical correspondence.
In the era of the old quantum theory, starting from Max Planck's proposal of quanta in his model of blackbody radiation (1900) and Albert Einstein's adaptation of the concept to explain the photoelectric effect (1905), and until Erwin Schrödinger published his eigenfunction equation in 1926,[1] the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints.
[2]: 106 Many results from atomic spectroscopy had been summarized in the Rydberg formula involving differences between two series of energies related by integer steps.
The model of the atom, first proposed by Niels Bohr in 1913, relied on a single quantum number.
Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the Balmer series portion of Rydberg's atomic spectrum formula.
[3] As Bohr notes in his subsequent Nobel lecture, the next step was taken by Arnold Sommerfeld in 1915.
[4] Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them.
[6]: 152 Karl Schwarzschild and Sommerfeld's student, Paul Epstein, independently showed that adding third quantum number gave a complete account for the Stark effect results.
A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized.
[7] The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the Zeeman effect.
Wolfgang Pauli's solution to this issue was to introduce another quantum number taking only two possible values,
[8] This would ultimately become the quantized values of the projection of spin, an intrinsic angular momentum quantum of the electron.
In 1927 Ronald Fraser demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum.
[7] Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century.
[9] When Schrödinger published his wave equation and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.
Beginning with Heisenberg's initial model of proton-neutron binding in 1932, Eugene Wigner introduced isospin in 1937, the first 'internal' quantum number unrelated to a symmetry in real space-time.
Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by Emmy Noether and Hermann Weyl to the electromagnetic field.
By 1953 Chen Ning Yang had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons.
[11]: 202 With Robert Mills, Yang developed a non-abelian gauge theory based on the conservation of the nuclear isospin quantum numbers.
Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian, quantities that can be known with precision at the same time as the system's energy.
and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity.
In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases.
Four quantum numbers can describe an electron energy level in a hydrogen-like atom completely: These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons).
The average distance between the electron and the nucleus increases with n. The azimuthal quantum number, also known as the orbital angular momentum quantum number, describes the subshell, and gives the magnitude of the orbital angular momentum through the relation
The occupation of the electron states in such an atom can be predicted by the Aufbau principle and Hund's empirical rules for the quantum numbers.
The Aufbau principle fills orbitals based on their principal and azimuthal quantum numbers (lowest n + l first, with lowest n breaking ties; Hund's rule favors unpaired electrons in the outermost orbital).
If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be 1/2 again (see note)), then the nuclear angular momentum quantum numbers I are given by:
It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, I, of any odd-A nucleus and integer values for any even-A nucleus.
Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of hydrogen (H), carbon (C), and sodium (Na) are;[20] The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons.
