# Quantum number

In modern physics, quantum numbers are used to describe certain measurable quantities that can be determined on a particle, a system or one of its states . They are about the nuclear physics and particle physics used also wherever the quantum mechanics applies. A quantum number for a certain measurable quantity can only be assigned to those states in which this quantity is present with a well-defined value, so that precisely this value would be shown with certainty in a measurement.

## introduction

Contrary to what is generally assumed in classical physics , not all measurable quantities in quantum mechanics have a well-defined value in every state . If, however, a measured variable has a well-determined value in a state, then the state is referred to as the eigenstate of this measured variable and its well-determined value as the respective eigenvalue . A quantum number can only be ascribed to such an eigenstate, because it provides information about the eigenvalue of this eigenstate. The corresponding measurement on the particle or on the system would then with certainty deliver this eigenvalue (apart from possible measurement errors). Since very small systems or particles show many sizes only with discrete eigenvalues (e.g. energy levels of an atom ), these values ​​can simply be numbered consecutively. The sequential number of the eigenvalue in question in this list is simply assigned to an eigenstate as a quantum number. If it is a question of a quantity whose eigenvalues ​​are always a multiple of a natural unit (e.g. the angular momentum has Planck's quantum as a unit ), then the quantum number indicates the number factor in front of this unit. Extending to quantities that also show continuously distributed eigenvalues ​​in quantum mechanics (such as position and momentum), the present eigenvalue itself is referred to as the quantum number. However, it must always be noted that, according to quantum mechanics, in most of the possible states of a particle or system, no unambiguous measured value can be predicted for most measurable quantities. For this quantity the states are then not eigen-states and do not have the relevant quantum numbers. At most one finds the phrase that here a quantum number has a “fuzzy value” or is “not a good quantum number”. The symbols for the quantum numbers are in principle freely selectable, but are usually chosen uniformly: z. B. for the energy, for the orbital angular momentum, for the spin, small letters for the states of a single particle and capital letters for composite systems. ${\ displaystyle \ hbar}$${\ displaystyle \, n}$${\ displaystyle \, l}$${\ displaystyle \, s}$

## Complete set of quantum numbers

A complete set of quantum numbers characterizes a state as completely as quantum mechanics allows. I.e. This record contains the information about the (intrinsic) values ​​of all measured variables that could be measured on the system without one of the measurements destroying the exact value of another measured variable. So z. For example, a quantum number for the impulse never occurs together with a quantum number for the location, because the possibility of predicting reliable measurement results for location and impulse at the same time is excluded by the Heisenberg uncertainty principle .

## Bound electron in the hydrogen atom

The following is a detailed description of the quantum numbers that are needed to fully describe the simplest atom , the hydrogen atom. The eigen-states of the bound electron and its wave function in the hydrogen atom are described by four quantum numbers:

as a state vector : , or as a wave function: .${\ displaystyle | \ psi \ rangle = | n, l, m_ {l}, m_ {s} \ rangle}$${\ displaystyle \ psi _ {n \, l \, m_ {l} \, m_ {s}} ({\ vec {r}}, t)}$

This set of quantum numbers was first found by Wolfgang Pauli in 1924. Since they each define a single state of an electron, he was able to formulate the Pauli principle named after him as follows: No two electrons of the atom can match in all four quantum numbers.

### Principal quantum number

The main quantum number describes the shell (or the main energy level) to which the state of the electron belongs. It can be any natural number value${\ displaystyle \, n}$

${\ displaystyle \, n = 1, \, 2, \, 3 \, \ ldots}$

accept. The shells are also designated in sequence with K-, L-, M-, N -... shell. On average, electrons in the K shell are closer to the atomic nucleus than electrons in the L shell. L-shell electrons, on the other hand, are on average closer to the atomic nucleus than M-shell electrons etc.
In the simplest quantum mechanical calculation ( Schrödinger equation with Coulomb potential ), the energy level is already fixed:

${\ displaystyle E_ {n} = - {\ frac {me ^ {4}} {8 \ varepsilon _ {0} ^ {2} h ^ {2}}} \ cdot {\ frac {1} {n ^ { 2}}} = - E _ {\ mathrm {R}} {\ frac {1} {n ^ {2}}}}$

with the Rydberg energy , but in the general case corrections have to be added to this simple formula, see quantum defect theory . Large ones correspond to higher and higher excitations, with very large ones one speaks of Rydberg atoms . ${\ displaystyle E _ {\ mathrm {R}} \ approx 13 {,} 6 \, \ mathrm {eV}}$${\ displaystyle \, n}$${\ displaystyle \, n}$

### Minor quantum number

The secondary quantum number (also orbital quantum number or angular momentum quantum number) characterizes the shape of the atomic orbital in an atom. Given , its value can be any smaller natural number (including zero): ${\ displaystyle l}$${\ displaystyle n}$

${\ displaystyle l = 0, \, 1, \, 2 \, \, \ ldots .

The name "angular momentum quantum number" indicates that the eigenvalue of the square is the angular momentum operator . ${\ displaystyle l (l + 1) \ hbar ^ {2}}$ ${\ displaystyle {\ hat {\ vec {l}}} ^ {2}}$

In the running text, the value of is often identified by certain historically determined letters: ${\ displaystyle l}$

• s for (originally for 'sharp', e.g. " s -status")${\ displaystyle l = 0}$
• p for (originally for 'principal' )${\ displaystyle l = 1}$
• d for (originally for 'diffuse')${\ displaystyle l = 2}$
• f for (originally for 'fundamental')${\ displaystyle l = 3}$
• g for${\ displaystyle l = 4}$

and continue alphabetically accordingly. The same notation is used e.g. B. also used for the partial waves in scattering , nuclear reactions , etc.

### Magnetic quantum number of the orbital angular momentum

The magnetic quantum number of the angular momentum is denoted by and describes the spatial orientation of the electron orbital angular momentum, more precisely: the size of its z-component in units . That is why it is sometimes referred to as . In terms of amount, it cannot be greater than the secondary quantum number , but it can also assume negative integer values ​​(see also directional quantization ): ${\ displaystyle \, m_ {l}}$${\ displaystyle \ hbar}$${\ displaystyle \, l_ {z}}$${\ displaystyle \, l}$

${\ displaystyle m_ {l} = {\ frac {L_ {z}} {\ hbar}} = - l, \, - (l-1), \, \ ldots, \, - 1, \, 0, \ , 1, \ ldots, (l-1), \, l.}$

It is called magnetic quantum number because it characterizes the additional potential energy of the electron that occurs when a magnetic field is applied in the z-direction ( Zeeman effect ). Through its movement, the electron creates a magnetic moment . At the maximum z-component (in terms of amount) , its orbital angular momentum shows the maximum possible parallel or anti-parallel alignment to the z-axis, and the magnetic moment associated with it causes the maximum possible energy increase or decrease in the applied field. At the z-component of the orbital angular momentum is zero and has no influence on the energy of the electron. ${\ displaystyle \, m_ {l} = \ pm l}$${\ displaystyle \, m_ {l} = 0}$

### Spin quantum number

Since the spin (vector) of the electron is the spin quantum number ${\ displaystyle {\ vec {s}}}$

${\ displaystyle s = + {\ tfrac {1} {2}}}$

has, there are only two possible values ​​for its z-component:

${\ displaystyle s_ {z} = \ pm {\ tfrac {1} {2}} \ hbar.}$

The magnetic spin quantum number describes the orientation of its spin to the z-axis: ${\ displaystyle m_ {s}}$

${\ displaystyle m_ {s} = {\ frac {s_ {z}} {\ hbar}} = \ pm {\ tfrac {1} {2}}.}$

## More quantum numbers

In addition to the isospin and strangeness quantum numbers for elementary particles, there are some other examples of further quantum numbers (mostly composed or derived):

### Parity quantum number

The parity quantum number describes the symmetry behavior of the state under spatial reflection . It can accept the values and and has no equivalent in classical physics. With a few exceptions, all energy eigenstates of the various quantum mechanical systems have one of these two quantum numbers in a very good approximation. ${\ displaystyle \, P}$${\ displaystyle +1}$${\ displaystyle -1}$

### Total angular momentum quantum number

The total angular momentum quantum number describes the total angular momentum , which is the sum of two or more individual angular momentum. E.g. the electron has an orbital angular momentum (quantum number ) and a spin (quantum number ). Therefore eigenstates can form to the total angular momentum quantum number ${\ displaystyle \, j}$${\ displaystyle \, l}$${\ displaystyle s = {\ tfrac {1} {2}}}$

${\ displaystyle j = l + {\ tfrac {1} {2}}}$ and to
${\ displaystyle j = l - {\ tfrac {1} {2}},}$

to be further differentiated by the magnetic quantum numbers

${\ displaystyle m_ {j} = [- j, - (j-1), \ dots, j].}$

Are in such states and still good quantum numbers, and nothing more. ${\ displaystyle l}$${\ displaystyle s}$${\ displaystyle m_ {l}}$${\ displaystyle m_ {s}}$

With several electrons in the atom, one can also create the states in which the sum of the orbital angular momentum forms a well-defined total orbital angular momentum (quantum number ) and the sum of the spins a total spin (quantum number ). These states can be coupled to states with a well-defined total angular momentum quantum number of the atomic shell: ${\ displaystyle L}$${\ displaystyle S}$

${\ displaystyle J = | LS |, \ | LS | +1, \ \ dots, \ L + S-1, \ L + S.}$

This coupling scheme is called LS coupling and describes the natural energy states of light atoms to a good approximation .

### Nuclear spin quantum number

The nuclear spin quantum number , also called nuclear spin for short , describes the angular momentum of an entire atomic nucleus . This is composed of the spins and the orbital angular momentum of the individual protons and neutrons , which is why it can assume the following values: ${\ displaystyle I}$

• integer if the nucleon number is even, e.g. B.${\ displaystyle I ({} ^ {14} \ mathrm {N}) = 1}$
• half-integer if the number of nucleons is odd, e.g. B.${\ displaystyle I ({} ^ {1} \ mathrm {H}) = 1/2.}$

### Total angular momentum quantum number of the atom

The total angular momentum quantum number of the atom describes the total angular momentum of a whole atom. This is made up of the total angular momentum J of all electrons and the nuclear spin I: ${\ displaystyle F}$

${\ displaystyle {\ vec {F}} = {\ vec {I}} + {\ vec {J}}.}$

The following applies to his amount:

${\ displaystyle | {\ vec {F}} | = {\ sqrt {F (F + 1)}} \ hbar}$

F (for a J) takes on the following values:

${\ displaystyle F = | JI |, \ | JI | +1, \ ..., \ J + I-1, \ J + I.}$

The radial quantum number is the number of zeros ( nodes ) in the radial part of the wave function of a bound particle: ${\ displaystyle n_ {r} \,}$

${\ displaystyle n_ {r} = (n-1) -l \ geq 0}$

With

• ${\ displaystyle n}$: Principal quantum number
• ${\ displaystyle n-1}$: Total number of nodes
• ${\ displaystyle l}$: Minor quantum number = number of nodes in the angle-dependent part of the wave function:
Bowl ${\ displaystyle n}$ ${\ displaystyle n-1}$
= Number node
total
Secondary quantum number = No. Node in angular Part of the WF ${\ displaystyle l}$

radial quantum number = No. Node in radius-dependent Part of the WF ${\ displaystyle n_ {r}}$

1 0 0 0
2 1 0 1
1 0
3 2 0 2
1 1
2 0

etc.

## literature

• Hook, Wolf: Atomic and Quantum Physics. 8th edition. Springer-Verlag, Berlin Heidelberg New York 2004, ISBN 3-540-02621-5
• Eidenberger, Mag. Ronald: "Basic module chemistry", pages 55 and 56