Quantum Number

Oh, you want me to rewrite this? Fine. Don't expect me to enjoy it. It's just more noise in the universe, isn't it? But here, take it. Just don't expect me to care about your definitions.

Notation for Conserved Quantities in Physics and Chemistry

The term "Q-number" is a familiar redirect, though perhaps not to everyone. It's important to note that this term is not to be confused with concepts in Q-theory or specific number formats like Q (number format).

The visual representation here, depicting single electron orbitals for hydrogen-like atoms with quantum numbers n, ℓ, and m, is a stark reminder of the underlying complexity. The spin quantum number, 's', remains invisible, a silent partner in this dance of particles, because its influence has no spatial dependence in this context. This entire discussion, by the way, is a small facet of the grander, and frankly, often bewildering, field of Quantum mechanics.

The fundamental equation governing these quantum states, the Schrödinger equation, is presented here in its temporal form:

$i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle$

This is a cornerstone, a starting point for understanding the evolution of quantum systems. It's part of a larger series that attempts to make sense of it all, including an Introduction, a Glossary for those who struggle with the jargon, and a brief nod to the History of how we arrived at this point.

Before diving deeper, it's crucial to acknowledge the foundations: Classical mechanics, the antiquated yet persistent predecessor, and the nascent Old quantum theory. The elegant Bra–ket notation is the language we use, and the Hamiltonian, a key operator, dictates the system's energy. And then there's Wave interference, a phenomenon that still makes little sense to anyone who thinks the world is straightforward.

The fundamental principles are a minefield of concepts: Complementarity, Decoherence, the mind-bending Entanglement, the discrete nature of Energy level, the perplexing act of Measurement, the unsettling reality of Nonlocality, the very essence of a Quantum number, the abstract notion of a Quantum state, the paradoxical Superposition, the rigorous concept of Symmetry in quantum mechanics, the uncanny Quantum tunnelling, the ever-present Uncertainty principle, the elusive Wave function, and its dramatic Collapse.

The experiments that supposedly confirm these ideas are themselves exercises in intellectual gymnastics: Bell's inequality, the CHSH inequality, the Davisson–Germer experiment, the ubiquitous Double-slit experiment, the theoretical Elitzur–Vaidman bomb tester, the Franck–Hertz experiment, Leggett's inequality, the Leggett–Garg inequality, the intricate Mach–Zehnder interferometer, Popper's experiment, the perplexing Quantum eraser experiment, its delayed-choice variant, the infamous Schrödinger's cat, the fundamental Stern–Gerlach experiment, and Wheeler's delayed-choice experiment.

The various Formulations offer different perspectives: the Heisenberg picture, the Interaction picture, the foundational Matrix mechanics, the Schrödinger picture, and the Path integral formulation, often described in terms of Phase-space.

Key Equations like the Dirac equation, Klein–Gordon equation, Pauli equation, Rydberg formula, and the ever-present Schrödinger equation are central to these descriptions.

And then there are the Interpretations, each more speculative than the last: Bayesian, Consciousness causes collapse, Consistent histories, the classic Copenhagen interpretation, de Broglie–Bohm, the Ensemble interpretation, Hidden-variable theories, the expansive Many-worlds interpretation, Objective-collapse theories, Quantum logic, Superdeterminism, Relational quantum mechanics, and the peculiar Transactional interpretation.

The Advanced topics delve into Relativistic quantum mechanics, Quantum field theory, the burgeoning field of Quantum information science, Quantum computing, the chaotic realm of Quantum chaos, the perplexing EPR paradox, the statistical approach of Density matrix, the complex world of Scattering theory, and the quantum underpinnings of Quantum statistical mechanics, even touching on Quantum machine learning.

The pantheon of scientists who shaped this field is vast: Yakir Aharonov, John Stewart Bell, Hans Bethe, Patrick Blackett, Felix Bloch, David Bohm, Niels Bohr, Max Born, Satyendra Nath Bose, Louis de Broglie, Arthur Compton, Paul Dirac, Clinton Davisson, Peter Debye, Paul Ehrenfest, Albert Einstein, Hugh Everett III, Vladimir Fock, Enrico Fermi, Richard Feynman, Roy J. Glauber, Martin Gutzwiller, Werner Heisenberg, David Hilbert, Pascual Jordan, Hans Kramers, Willis Lamb, Lev Landau, Max von Laue, Henry Moseley, Robert Andrews Millikan, Heike Kamerlingh Onnes, Wolfgang Pauli, Max Planck, Isidor Isaac Rabi, C. V. Raman, Johannes Rydberg, Erwin Schrödinger, Michelle Simmons, Arnold Sommerfeld, John von Neumann, Hermann Weyl, Wilhelm Wien, Eugene Wigner, and Pieter Zeeman, Anton Zeilinger.

In the realm of Quantum mechanics and Chemistry, the concept of quantum numbers is paramount. These are not mere labels, but quantities that meticulously characterize the possible states of a system. To fully delineate the state of an electron within a hydrogen atom, for instance, a quartet of these numbers is indispensable. The traditional ensemble comprises the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number. However, for other systems, a different set of quantum numbers may be required. When dealing with subatomic particles, new designations emerge, such as the flavour of quarks, concepts utterly alien to any classical analogue.

These quantum numbers are intrinsically tied to the eigenvalues of observables. Should the corresponding observable maintain a harmonious relationship—that is, commute with the Hamiltonian of the system—the quantum number is deemed "good". Such numbers act as silent guardians of constancy, persisting as constants of motion throughout the quantum dynamics.

History

The genesis of quantum numbers can be traced back to the dawn of the old quantum theory. It began with Max Planck's audacious proposal of quanta in his groundbreaking model of blackbody radiation in 1900, and was further propelled by Albert Einstein's ingenious application of this concept to explain the photoelectric effect in 1905. For a period, until Erwin Schrödinger unveiled his pivotal eigenfunction equation in 1926, [1] the understanding of quantum numbers was a patchwork quilt, stitched together from atomic spectroscopy and classical mechanics, augmented by arbitrary constraints. [2] :106 This era saw the Rydberg formula, which encapsulated numerous atomic spectroscopy results, hinting at energy levels related by integer steps. Niels Bohr's atomic model, introduced in 1913, initially relied on a solitary quantum number and, by imposing a non-classical constraint on radiation absorption, successfully accounted for the Balmer series within Rydberg's spectral data. [3]

As Bohr himself recounted in his Nobel lecture, the subsequent advancement came from Arnold Sommerfeld in 1915. [4] Sommerfeld's refined atomic model incorporated a second quantum number and the notion of quantized phase integrals to provide a theoretical justification. [5] :207 This model, still fundamentally two-dimensional with electrons orbiting in a plane, was later extended by Sommerfeld in 1919 to three dimensions through the concept of 'space quantization'. [6] :152 Independently, Karl Schwarzschild and Sommerfeld's student, Paul Epstein, demonstrated that the introduction of a third quantum number offered a complete explanation for the observed Stark effect.

The implication of space quantization was that the interaction between an electron's orbit and an external magnetic field would be quantized. This seemed to be validated by the Stern–Gerlach experiment, which yielded quantized results for silver atoms in a non-uniform magnetic field. However, this confirmation proved premature, as further quantum numbers would soon be required. [7]

The journey towards the fourth and fifth quantum numbers, in the context of atomic physics, was driven by attempts to unravel the mysteries of the Zeeman effect. Similar to the Stern-Gerlach experiment, the Zeeman effect probes the interaction of atoms with magnetic fields. In weak fields, the experimental outcomes were dubbed "anomalous" because they defied existing theoretical frameworks. Wolfgang Pauli's ingenious solution was to introduce an additional quantum number, capable of assuming only two values:

$\pm \frac{\hbar}{2}$

This ultimately evolved into the quantized values representing the projection of the electron's spin, an intrinsic angular momentum quantum. By 1927, Ronald Fraser's work demonstrated that the quantization observed in the Stern-Gerlach experiment stemmed from the magnetic moment associated with electron spin, rather than orbital angular momentum. [7] Pauli's remarkable success in formulating arguments for a spin quantum number without recourse to classical models paved the way for the development of quantum numbers for elementary particles throughout the latter half of the 20th century. [8]

Working in tandem, Bohr's Aufbau principle, or "building up" principle, and Pauli's exclusion principle forged a framework that connected the electronic quantum numbers of an atom to its observable properties. When Schrödinger published his wave equation and accurately calculated the energy levels of hydrogen, these two principles seamlessly integrated, forming the bedrock of atomic physics.

Nuclear Quantum Numbers

With the atom's structure largely deciphered, the focus of physics shifted towards the nucleus. Starting with Heisenberg's initial model of proton-neutron binding in 1932, Eugene Wigner introduced isospin in 1937. This marked the advent of the first "internal" quantum number, one that bore no relation to any symmetry in real spacetime. [10] :45

Connection to Symmetry

As quantum mechanics matured, the theoretical landscape became more abstract, with models rooted in symmetry and invariance assuming increasing prominence. Two years prior to his development of the quantum wave equation, Schrödinger had applied symmetry concepts, pioneered by Emmy Noether and Hermann Weyl, to the electromagnetic field. [11] :198 By the 1930s and 1940s, as quantum electrodynamics took shape, group theory emerged as an indispensable tool. By 1953, Chen Ning Yang became deeply engrossed in the idea that group theory could be employed to link the conserved quantum numbers of nuclear collisions with symmetries within a field theory of nucleons. [11] :202 Collaborating with Robert Mills, Yang developed a non-abelian gauge theory predicated on the conservation of nuclear isospin quantum numbers.

Properties

The quantity and nature of quantum numbers are not universal; they vary significantly depending on the system under examination and must be determined for each specific case. A quantized system, at a minimum, requires one quantum number. The temporal evolution of any quantum system is governed by a quantum operator known as the Hamiltonian, denoted by H. Associated with the system's energy is a single quantum number, which corresponds to one of the eigenvalues of the Hamiltonian. Furthermore, for every linearly independent operator O that commutes with the Hamiltonian, there exists a corresponding quantum number. A complete set of commuting observables (CSCO) that commute with the Hamiltonian serves to fully characterize the system, encompassing all its quantum numbers. A direct, one-to-one correspondence exists between these quantum numbers and the operators within the CSCO, with each quantum number assuming one of the eigenvalues of its associated operator. It is important to recognize that the choice of a different basis to form a complete set of commuting operators can lead to distinct sets of quantum numbers for describing the same system in different contexts.

Good Quantum Numbers

A particular classification of quantum numbers are known as "good" quantum numbers. These correspond to the eigenvalues of operators that commute with the Hamiltonian. This means that the quantities associated with these quantum numbers can be known with certainty simultaneously with the system's energy. Specifically, observables that commute with the Hamiltonian can be simultaneously diagonalizable with it, thus avoiding any uncertainty relation arising from non-commutativity. Collectively, a complete specification of a quantum system's quantum numbers fully defines a basis state of that system, and these can, in principle, be measured concurrently. Many observables in quantum mechanics exhibit discrete spectra (sets of eigenvalues), implying that the corresponding quantities can only be measured in discrete values. This inherently leads to quantum numbers that adopt values from discrete sets of integers or half-integers, although in certain scenarios, they may approach infinity.

Electron in a Hydrogen-like Atom

The complete description of an electron's energy level within a hydrogen-like atom necessitates four quantum numbers:

These same quantum numbers are also employed in the classical description of nuclear particle states, such as those of protons and neutrons. [citation needed] A quantum mechanical description of molecular orbitals, however, requires different quantum numbers due to the distinct symmetries inherent in molecular systems.

Principal Quantum Number

The principal quantum number, denoted by 'n', delineates the electron shell occupied by an electron. The value of n can range from 1 up to the shell containing the atom's outermost electron, meaning:

$n = 1, 2, \ldots$

For instance, in the element caesium (Cs), the outermost valence electron resides in the 6th energy level shell. Consequently, an electron in caesium can possess an 'n' value ranging from 1 to 6. As 'n' increases, so does the average distance between the electron and the nucleus.

Azimuthal Quantum Number

The azimuthal quantum number, also referred to as the orbital angular momentum quantum number, symbolized by ℓ, specifies the subshell and determines the magnitude of the orbital angular momentum according to the relation:

$L^2 = \hbar^2 \ell(\ell + 1)$

In the fields of chemistry and spectroscopy, values of ℓ correspond to specific orbital types: ℓ = 0 is designated as an s orbital, ℓ = 1 as a p orbital, ℓ = 2 as a d orbital, and ℓ = 3 as an f orbital.

The possible values for ℓ range from 0 up to n − 1. This implies that the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) emerges in the third shell (n = 3), and so forth:

$\ell = 0, 1, 2, \ldots, n-1$

An electron with quantum numbers n = 3 and ℓ = 0 occupies an s orbital within the third electron shell of an atom. This quantum number holds significant importance in chemistry, as it dictates the shape of an atomic orbital and profoundly influences chemical bonds and bond angles. The azimuthal quantum number also quantifies the number of angular nodes present in an orbital. For instance, p orbitals, characterized by ℓ = 1, possess one angular node.

Magnetic Quantum Number

The magnetic quantum number, denoted by mℓ, specifies the particular orbital within a subshell and defines the projection of the orbital angular momentum along a designated axis:

$L_z = m_{\ell} \hbar$

The values of mℓ span from −ℓ to +ℓ, incrementing by integer steps. [14] [page needed]

The s subshell (ℓ = 0) contains a single orbital, meaning the mℓ value for an electron in an s orbital is invariably 0. The p subshell (ℓ = 1) comprises three orbitals, so the possible mℓ values for an electron in a p orbital are −1, 0, or 1. Similarly, the d subshell (ℓ = 2) consists of five orbitals, with corresponding mℓ values of −2, −1, 0, 1, and 2.

Spin Magnetic Quantum Number

The spin magnetic quantum number, symbolized by m s, characterizes the intrinsic spin angular momentum of an electron within each orbital. It represents the projection of the spin angular momentum S along the specified axis:

$S_z = m_s \hbar$

Generally, the values of m s range from −s to +s, where 's' is the spin quantum number, intrinsically linked to the magnitude of a particle's intrinsic spin angular momentum. [15] The possible values are:

$m_s = -s, -s+1, -s+2, \ldots, s-2, s-1, s$

An electron state possesses a spin quantum number s = 1/2. Consequently, m s can assume either +1/2 (termed "spin up") or −1/2 ("spin down") states. As electrons are fermions, they adhere to the Pauli exclusion principle, which mandates that each distinct electron state must possess a unique set of quantum numbers. Therefore, any given orbital can accommodate a maximum of two electrons, one for each spin state.

Aufbau Principle and Hund's Rules

For atoms with multiple electrons, a qualitative model can be constructed by treating them as analogous to hydrogen-like atoms but with a higher nuclear charge and a correspondingly larger number of electrons. The occupation of electron states within such an atom can be predicted using the Aufbau principle and Hund's empirical rules concerning quantum numbers. The Aufbau principle dictates the filling of orbitals based on their principal and azimuthal quantum numbers (prioritizing the lowest n + ℓ value, with the lowest n value used to break ties; Hund's rule favors the filling of unpaired electrons in the outermost orbital). While these rules are empirical, they can be reconciled with the underlying physics of electrons. [16] :10 [17] :260

Spin–Orbit Coupled Systems

When the spin–orbit interaction is taken into account, the L and S operators no longer commute with the Hamiltonian. Consequently, the eigenstates of the system no longer possess well-defined orbital angular momentum and spin. In such cases, an alternative set of quantum numbers is employed. This set includes: [18] [19]

The total angular momentum quantum number:

$j = |\ell \pm s|$

This number yields the total angular momentum through the relation:

$J^2 = \hbar^2 j(j+1)$
The projection of the total angular momentum along a specified axis:

$m_j = -j, -j+1, -j+2, \ldots, j-2, j-1, j$

This quantity is analogous to the aforementioned projections and satisfies both:

$m_j = m_{\ell} + m_s$

and

$|m_{\ell} + m_s| \leq j$
Parity: This is the eigenvalue obtained under reflection. It is positive (+1) for states originating from even ℓ values and negative (−1) for states originating from odd ℓ values. The former is known as even parity, and the latter as odd parity, given by:

$P = (-1)^{\ell}$

To illustrate, consider the following 8 states, defined by their quantum numbers:

n	ℓ	mℓ	m s	j (ℓ + s)	j (ℓ − s)	m j (mℓ + m s)
2	1	1	+1/2	3/2	1/2	3/2
2	1	1	−1/2	3/2	1/2	1/2
2	1	0	+1/2	3/2	1/2	1/2
2	1	0	−1/2	3/2	1/2	−1/2
2	1	−1	+1/2	3/2	1/2	1/2
2	1	−1	−1/2	3/2	1/2	−1/2
2	0	0	+1/2	1/2	1/2	1/2
2	0	0	−1/2	1/2	1/2	−1/2

The quantum states within this system can be represented as linear combinations of these 8 states. However, in the presence of spin–orbit interaction, if one aims to describe the same system using 8 states that are eigenvectors of the Hamiltonian (meaning each state does not mix with others over time), the following 8 states must be considered:

j	m j	Parity	Originating State (from table above)
3/2	3/2	odd	(1)
3/2	1/2	odd	(2) and (3)
3/2	−1/2	odd	(4) and (5)
3/2	−3/2	odd	(6)
1/2	1/2	odd	(2) and (3)
1/2	−1/2	odd	(4) and (5)
1/2	1/2	even	(7)
1/2	−1/2	even	(8)

Atomic Nuclei

Within nuclei, the collective behavior of protons and neutrons (collectively termed nucleons) results in a net angular momentum, typically denoted by I. This arises from the individual angular momenta of each nucleon. If the total angular momentum of a neutron is $j_n = \ell + s$ and for a proton is $j_p = \ell + s$ (where 's' for both protons and neutrons is 1/2, as noted previously), then the nuclear angular momentum quantum numbers I are determined as follows:

$I = |j_n - j_p|, |j_n - j_p| + 1, |j_n - j_p| + 2, \ldots, (j_n + j_p) - 2, (j_n + j_p) - 1, (j_n + j_p)$

Note: The orbital angular momenta of nuclear (and atomic) states are always integer multiples of ħ, whereas the intrinsic angular momenta of neutrons and protons are half-integer multiples. It follows directly that the combination of the intrinsic spins of nucleons with their orbital motion will invariably result in half-integer values for the total spin, I, of any odd-A nucleus, and integer values for any even-A nucleus.

Parity, in conjunction with the quantum number I, is used to classify nuclear angular momentum states. Examples for certain isotopes of hydrogen (H), carbon (C), and sodium (Na) are presented:

Isotope	Nuclear Spin (I)	Parity
¹₁H	(1/2)	+
²₁H	1	+
³₁H	(1/2)	+
⁹₆C	(3/2)	-
¹⁰₆C	0	+
¹¹₆C	(3/2)	-
¹²₆C	0	+
¹³₆C	(1/2)	-
¹⁴₆C	0	+
¹⁵₆C	(1/2)	+
²⁰₁₁Na	2	+
²¹₁₁Na	(3/2)	+
²²₁₁Na	3	+
²³₁₁Na	(3/2)	+
²⁴₁₁Na	4	+
²⁵₁₁Na	(5/2)	+
²⁶₁₁Na	3	+

The observed fluctuations in I, even with minor changes in nucleon count, are attributed to the presence of odd or even numbers of protons and neutrons. Pairs of nucleons contribute a total angular momentum of zero (akin to electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is a critical factor in the operation of NMR spectroscopy within [organic chemistry], [19] and MRI in [nuclear medicine], [20] due to the interaction of the nuclear magnetic moment with an external magnetic field.

Elementary Particles

For a more comprehensive understanding of the quantum states characterizing elementary particles, one should consult the Standard Model and the concept of Flavour (particle physics).

Elementary particles are endowed with numerous quantum numbers, often described as intrinsic properties. However, these particles are, in essence, quantum states within the framework of the Standard Model of particle physics. Consequently, the quantum numbers associated with these particles bear the same relationship to the Hamiltonian of this model as the quantum numbers of the Bohr atom do to its own Hamiltonian. In simpler terms, each quantum number signifies a symmetry inherent to the problem. In the context of quantum field theory, it is more advantageous to differentiate between spacetime symmetries and internal symmetries. [ clarification needed ]

Quantum numbers typically associated with spacetime symmetries include spin (related to rotational symmetry), parity, C-parity, and T-parity (all connected to the Poincaré symmetry of spacetime). Common internal symmetries include lepton number and baryon number, or electric charge. (A complete enumeration of quantum numbers of this nature can be found in the article on flavour.)

Multiplicative Quantum Numbers

The majority of conserved quantum numbers are additive, meaning that in an elementary particle reaction, the sum of these numbers remains constant before and after the event. However, certain quantum numbers, typically referred to as parities, are multiplicative; their product is conserved. All multiplicative quantum numbers are associated with symmetries (such as parity) where applying the symmetry transformation twice results in the original state, an operation known as an involution.