Magnetic quantum number

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers (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 m. The magnetic quantum number denotes the energy levels available within a subshell. This quantum number indicates the possible orientation of orbital in space. The value of 'm' for a particular value of 'l' varies from +l to -l including zero.

Derivation

There is 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 wavefunction 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 3 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 momentum vector. The magnetic quantum number m only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of m are equivalent. "M" also affects 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 number associated with angular momentum), there are 2ℓ+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 Stern 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 $\mathbf{L}$ 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 m you begin with an atomic electron's angular momentum, L, which is related to its quantum number ℓ by the following equation:

$\mathbf{L} = \hbar\sqrt{\ell(\ell+1)}$

where $\hbar = h/2\pi$ 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, 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:

$\mathbf{L_z} = m\hbar.$

Another way of stating the formula for the magnetic quantum number $(m_l = -\ell,-\ell + 1,..., 0, ..., \ell - 1, \ell)$ is the eigenvalue, J_z=m_ℓh/2π.

Where 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	$\ell=0,\quad m=0$	1
p	$\ell=1,\quad m=-1,0,+1$	3
d	$\ell=2,\quad m=-2,-1,0,+1,+2$	5
f	$\ell=3,\quad m = -3,-2,-1,0,+1,+2,+3$	7
g	$\ell=4,\quad m = -4,-3,-2,-1,0,+1,+2,+3,+4$	9

The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field, hence the name magnetic quantum number (Zeeman effect).

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.

See also

• Quantum number
  • Azimuthal quantum number
  • Principal quantum number
  • Spin quantum number
  • Total angular momentum quantum number
• Basic quantum mechanics
• Bohr atom
• Schrödinger equation