Quantum numbers - Structure of matter

Question 1

How is electron motion described in quantum mechanical physics?

Answer 1

According to the quantum mechanical view, the motion of an electron in the force filed of the atomic nucleus is not represented by a certain trajectory but the electron can be found in a ‘cloud.’

It’s form and distance from the nucleus being determined by other parameters as the orbital angular momentum, magnetic moment and spin.

The space of the electron motion is called an orbital.

The electron state is described by the wave function involving a number of dimensionless parameters, which equal the number of degrees of freedom.

If we also consider the rotation then the number of degrees of freedom of electron equals 4.

Therefore, the state of electrons can be completely described by four quantum numbers.

These numbers are, with the exception of spin, natural integers and they determine the geometry and the symmetry of the electron energy.

It follows from the quantum theory of hydrogen atom that electron may exist at various energy levels E_n given by equation:

E₀ = - (me⁴) / (8ε²₀h²) * 1/n²

• m=9.11*10^-31 kg is the rest mass of electron
• ε₀ = 8.854*10^-12 F.m^-1 is the permittivity of vacuum
• e = 1.6 * 10^-12 C is the electron charge

Question 2

Describe the principal quantum number, n.

Answer 2

Principle quantum number, n, is a natural number, which can possess values ι = 1, 2, 3 …

More over, its values estimates the shell in which electron appears.

The shells K, L, M, N, O, P and Q correspond to values n = 1, 2, 3, 4, 5, 6, and 7 respectively.

Question 3

Describe orbital quantum number, l.

Answer 3

Orbital quantum number, l, of electron in the shell determined by n my possess values l = 0, 1, 2… (n-l)

and it determines the form and symmetry of the electron cloud.

It is determined by the angular momentum L.

The magnitude of L is given by the equation

L = ћ√ι(ι+1)

The orbital quantum number is used for notation of the states in spectrometry.

Values of ι are denoted by letters so that values of ι = 0, 1, 2, 3, 4, 5 correspond to the letters s, p, d, f, g, h respectively.

According to the equation L = ћ√ι(ι+1) the value of orbital angular momentum corresponding to the state s equals zero, the value corresponding to the state p is √2.ћ etc.

Notation of states is a combination of principal quantum number and of this letter.

Thus, the state with n= 2, ι = 0 is 2s, the state with n = 4, ι = 2 is denoted by 4d etc.

Question 4

Describe magnetic quantum number, m_ι.

Answer 4

Magnetic quantum number m_ι can possess the values m_ι. = 0, ±1, ±2 … ±ι for given ι.

It determines the spatial position of the orbital.

The magnitudes of the orbital angular momentum can only have discreat values given by equation

L = ћ√ι(ι+1)

Moreover, the direction of the angular momentum is also not arbitrary and it is limited with respect to the orientaiton of the external magnetic field.

Magnetic quantum number estimates the direction of the vector L by determining its component into the direction of the external magnetic field.

Question 5

Describe the orbital magnetic moment.

Answer 5

Orbital magnetic moment μ_orb = -(e/(2m_e) L

• m_e is mass of electron
• Bolded letters are vectors

Orbital magnetic moment of electron is given in unit called Bohr’s magneton u_{<span>B</span>} = eћ/2m_e = 9.28*10^-24 A.m² (or J/T^-1 since J/T = J/(Wb.m^-2) = J.m²/Wb = J.m²/V.s) = C.V.m²/(V.s) = A.m²).

For given ι, the number of possible orientation of the orbital angular momentum in external magnetic field equals 2 ι+1 since values of m may vary within the range from - ι over 0 to + ι.

Thus, the magnetic quantum number estimates the magnitude of the projection of the mechanical angular momentum L and of the magnetic moment μ into the given direction.

For the z-component of the orbital magnetic moment, μ_{orb, z} it holds μ_{orb, z} = m_ι.μ_B.

Orbital magnetic dipole moments are multiples of μ_B.

Question 6

Describe the spin quantum number, s.

Answer 6

Electron possesses its own, internal angular momentum, which does not depend on its orbital angular momentum.

However, it also possesses its certain own magnetic momentum related to its internal angular momentum.

The magnitude of the angular momentum, S, due to spin of electron is given for any electron, bout or free, by

S = ћ√(s(s+1))

Where s=1/2 is spin quantum number of electron

In an external magnetic field, the vector of the spin angular momentum may possess two orientations.

The component S_z of the spin angular momentum of an electron along the external magnetic field of direction z is determined by spin angular momentum of an electron along the external magnetic field of z is determined by spin magnetic quantum number m_s, which has two values ±1/2 and thus, the value of this component is ±1/2ћ.

Similarly to orbital angular momentum, also spin dipole magnetic momentum μ_s is related to spin angular momentum S by μ_s=-(e/m)S

• e is the charge
• m is the mass of electron

Spin di[ole moments of electrons (and of other elementary particles) are multiples of u_B.

Question 7

Explain the Pauli exclusion principal.

Answer 7

The state of an electron in atom is completly determined by a set of 4 quantum numbers n, l, m and s.

Electron configurations of atoms with more electrons are governed by the principle of Pauli, which states that any two electrons in an atom cannot exist in an identical quantum state.

There is a different set of quantum numbers for each electron in the given atom.

During electron transitions from onet o another state due to absorption or emission of energy, only these transitions occur, during which the principal quantum number may vary arbitrarily where as orbital quantum number only by ±1.

These transitions are called allowed while the other are called forbidden ones.

Thus, out of 3 * 2 = 6 possible transitions from the shell M (n=3, l= 0, 1, 2) into shell L (n=2, l=0,1) only those from 3d into 2p, from 3p into 2s and finally from 3s into 2p are allowed.