Magnetic quantum number

Encyclopedia

In atomic physics

, the

s (the principal quantum number

, the azimuthal quantum number

, the magnetic quantum number, and the spin quantum number

) which describe the unique quantum state of an electron and is designated by the letter

or orbital

. The wavefunction of the Schrödinger wave equation

reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The magnetic quantum number arose in the solution of the azimuthal part of the wave equation as shown below.

The magnetic quantum number associated with the quantum state is designated as

vector. The magnetic quantum number

associated with angular momentum), there are 2

and Walther Gerlach.

Since each electronic orbit has a magnetic moment in a magnetic field the electronic orbit will be subject to a torque which tends to make the vector parallel to the field. The precession of the electronic orbit in a magnetic field is called the Larmor precession

.

To describe the magnetic quantum number

where is the reduced Planck constant. The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta

. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.

To show that only certain discrete amounts of angular momentum are allowed,

Another way of stating the formula for the magnetic quantum number is the eigenvalue,

Where the quantum number

The magnetic quantum number determines the energy shift of an atomic orbital

due to an external magnetic field, hence the name

).

However, the actual magnetic dipole moment of an electron in an atomic orbital arrives not only from the electron angular momentum, but also from the electron spin, expressed in the spin quantum number

.

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, the

**magnetic quantum number**is the third of a set of quantum numberQuantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

s (the principal quantum number

Principal quantum number

In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

, the azimuthal quantum number

Azimuthal quantum number

The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...

, the magnetic quantum number, and the spin quantum number

Spin quantum number

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

) which describe the unique quantum state of an electron and is designated by the letter

*m*. The magnetic quantum number denotes the energy levels available within a subshell.## Derivation

There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers*n*,*ℓ*,*m*, and*s*specify the complete and unique quantum state of a single electron in an atom called its wavefunctionWavefunction

Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

or orbital

Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

. The wavefunction of the Schrödinger wave equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The magnetic quantum number arose in the solution of the azimuthal part of the wave equation as shown below.

The magnetic quantum number associated with the quantum state is designated as

*m*. The quantum number*m*refers, loosely, to the direction of the angular momentumAngular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

vector. The magnetic quantum number

*m*does not affect the electron's energy, but it does affect the probability cloud. Given a particular*ℓ*,*m*is entitled to be any integer from -*ℓ*up to*ℓ*. More precisely, for a given orbital momentum quantum number*ℓ*(representing the azimuthal quantum numberAzimuthal quantum number

The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...

associated with angular momentum), there are 2

*ℓ*+1 integral magnetic quantum numbers*m*ranging from -*ℓ*to*ℓ*, which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values*m*. This phenomenon is known as space quantization. It was first demonstrated by two German physicists, Otto SternOtto Stern

Otto Stern was a German physicist and Nobel laureate in physics.-Biography:Stern was born in Sohrau, now Żory in the German Empire's Kingdom of Prussia and studied at Breslau, now Wrocław in Lower Silesia....

and Walther Gerlach.

Since each electronic orbit has a magnetic moment in a magnetic field the electronic orbit will be subject to a torque which tends to make the vector parallel to the field. The precession of the electronic orbit in a magnetic field is called the Larmor precession

Larmor precession

In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...

.

To describe the magnetic quantum number

*m*you begin with an atomic electron's angular momentum, L, which is related to its quantum number*ℓ*by the following equation:where is the reduced Planck constant. The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta

Quantum

In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.

To show that only certain discrete amounts of angular momentum are allowed,

*ℓ*has to be an integer. The quantum number*m*refers to the projection of the angular momentum for any given direction, conventionally called the z direction. L_{z}, the component of angular momentum in the z direction, is given by the formula:Another way of stating the formula for the magnetic quantum number is the eigenvalue,

**J**._{z}=m_{ℓ}h/2πWhere the quantum number

*ℓ*is the subshell, the magnetic number*m*represents the number of possible values for available energy levels of that subshell as shown in the table below. Relationship between Quantum Numbers |
||
---|---|---|

Orbital | Values | Number of Values for m |

s | 1 | |

p | 3 | |

d | 5 | |

f | 7 | |

g | 9 |

The magnetic quantum number determines the energy shift of an atomic orbital

Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

due to an external magnetic field, hence the name

*magnetic*quantum number (Zeeman effectZeeman effect

The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...

).

However, the actual magnetic dipole moment of an electron in an atomic orbital arrives not only from the electron angular momentum, but also from the electron spin, expressed in the spin quantum number

Spin quantum number

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

.

## See also

- Quantum numberQuantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

- Azimuthal quantum numberAzimuthal quantum numberThe azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...
- Principal quantum numberPrincipal quantum numberIn atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...
- Spin quantum numberSpin quantum numberIn atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...
- Total angular momentum quantum number

- Azimuthal quantum number
- Basic quantum mechanics
- Bohr atom
- Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....