Quantum number

Quantum numbers are used in modern physics, the description of measurable parameters that can be determined on a particle, a system or one of its states. They are used on the nuclear physics and particle physics beyond wherever quantum mechanics applies. A quantum number for a given measurable quantity can be assigned to those states in which this quantity is present at a well-defined value, so that would show exactly this value when measured with certainty.

• 4.1 parity quantum number
• 4.2 total angular momentum quantum number
• 4.3 nuclear spin quantum number
• 4.4 total angular momentum quantum number of the atom
• 4.5 Radial quantum number

Introduction

Unlike in classical physics assumed in principle, in quantum mechanics, not all measurable variables in each state a well-determined value. Where, however, a measure in a state of a well-determined value, the condition is called eigenstate of this measure and its well-defined value than the respective eigenvalue. Only such an eigenstate can be attributed to a quantum number, because it gives information which eigenvalue is present with him. The corresponding measurement on the particles and the system would then exactly this eigenvalue supply ( apart from possible measurement errors ). Because of very small systems or particles of many sizes with only discrete eigenvalues ​​show (eg energy levels of an atom ), these values ​​one can simply renumber. An eigenstate is simply attributed to the quantum number, the serial number for the eigenvalue in this collection. If there is a quantity whose eigenvalues ​​are always a multiple of a natural unit (eg, the angular momentum as a unit, the Planck constant ), then there is the quantum number of the numerical factor in front of this unit. In extension to sizes that show in quantum mechanics continuously distributed eigenvalues ​​(such as position and momentum ), the present eigenvalue itself is known as a quantum number. Always, however, to note that according to quantum mechanics in the most possible states of a particle or system for most measurable terms no clear measurement value is to predict. For this size, the states are not eigenstates and then not have the quantum numbers involved. At most, you will find the speech, here have a quantum number a "fuzzy value " or is "not a good quantum number ". The symbols for the quantum numbers are in principle freely selectable, but generally uniformly selected: for example, for the energy of the orbital angular momentum, the spin, small letters for the states of an individual particle, large letters for composite systems.

Full set of quantum numbers

A complete set of quantum numbers characterizing a state as completely as possible using quantum mechanics. In other words, this set contains the information about the (self- ) values ​​of all the variables that could be measured by the system, without any of the measurements would destroy the existence of the exact value of another measure. Thus, for example, a quantum number for the pulse never occur together with a quantum number for the place, because the ability to simultaneously predict reliable measurement results for position and momentum, is excluded by the Heisenberg uncertainty principle.

Bound electron in the hydrogen atom

The quantum numbers are described in detail below, which are for a complete description of the simplest atom, the hydrogen atom, used. The eigenstates of the bound electron and its wave function in the hydrogen atom are described by four quantum numbers:

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

Principal quantum number

The principal quantum number describes the shell, which includes the state of the electron. You can arbitrary natural number values ​​greater than zero accept:

The trays are also known in sequence with K, L, M, ... shell. In the simplest quantum mechanical calculation ( Schrödinger equation with Coulomb potential ), the energy level already is therefore clear:

With the Rydberg energy. Large always correspond to higher excitations, and a very wide one speaks of Rydberg atoms.

Quantum number

The quantum number or angular momentum quantum number characterizing the shape of the atomic orbitals in an atom. Given its value can be any smaller natural number:

The name " angular momentum quantum number " refers to the fact that the eigenvalue of the square of the angular momentum operator is.

Within the text, the value of is often characterized by specific, historically determined letter:

• S for (originally for ' sharp ', such as " s- state" )
• P (originally for engl. ' principal ', ' top' state)
• D for (originally ' diffuse ' for )
• F ( originally for ' fundamental ') for
• G for

And according to alphabetical further. The same notation is, nuclear reactions, etc. are used, for example, for the partial waves in scattering.

Magnetic quantum number of the angular momentum

The magnetic quantum number of the angular momentum is designated, and describes the spatial orientation of the electron orbital angular momentum, or more accurately the size of its component in the Z- units. You may be in amount not greater than the quantum number, but also assume negative values ​​:

It is the magnetic quantum number, because it characterizes the additional potential energy of the electron, which in the z direction occurs when a magnetic field ( Zeeman effect). Through its motion, the electron produces a magnetic moment. Wherein ( in magnitude ) of maximum z-component of the orbital angular momentum is to be the maximum possible parallel or antiparallel orientation to the z axis, and the magnetic moment associated with it in the applied field causes the maximum power increase or decrease. When is the z - component of the orbital angular momentum is zero and has no effect on the energy of the electron.

See also space quantization.

Spin quantum number

Since the spin ( vector ) of the electron spin quantum number

Is there for its z-component of only two possible values ​​:

The magnetic spin quantum number describing the orientation of its spin to the z -axis:

Other quantum numbers

In addition to the isospin and strangeness quantum numbers in elementary some other examples of other quantum numbers are (usually composed or derived ):

Parity quantum number

The parity quantum number designates the symmetry behavior of the state under space reflection. You can have the values ​​and assume and has no counterpart in classical physics. With few exceptions, all the energy eigenstates of the various quantum mechanical systems have a very good approximation of these two quantum numbers.

Total angular momentum quantum number

The total angular momentum quantum number describes the total angular momentum, which is the sum of two or more separate angular momenta. For example, the electron has an orbital angular momentum ( quantum number ) and a spin ( quantum number ). Therefore, eigenstates can form the total angular momentum quantum number

Further distinguished by the magnetic quantum numbers

In such states, and are still good quantum numbers, and but no more.

Wherein a plurality of electron in the atom can also form the states in which form the sum of the path of rotation pulses a well-defined total orbital angular momentum ( quantum number ) and the sum of a total spin of the spins ( the quantum number ). These states can be further coupled to states with well-defined total angular momentum quantum number of the atomic shell:

This coupling scheme is called LS- coupling and describes in good approximation, the energy eigenstates of light atoms.

Nuclear spin quantum number

The nuclear spin quantum number, also briefly called nuclear spin, describes the angular momentum of an entire nucleus. This is composed of the spins and the orbital angular momenta of the individual protons and neutrons, so he can have the following values:

• Integral if the nucleon number is even, for example,
• Half-integral when the nucleon number is odd, for example,

Total angular momentum quantum number of the atom

The total angular momentum quantum number of the atom describes the total angular momentum of the whole atom. This consists of the total angular momentum J of all the electrons and the nuclear spin I:

For its amount is:

It takes F ( a J) the following values:

Radial quantum number

The radial quantum number is the number of zeros (nodes) in the radial part of the wave function of a bound particle:

With

• : Principal quantum number
• : Total number of nodes
• : Quantum number = number of nodes in the angle-dependent part of the wave function:

Etc.