CHEMISTRYWIDE: ELECTRON SPIN

Electron spin

In chemistry, familiar with the yellow color imparted to a flames by sodium elements. The very strongest yellow color line the D line in the sodium spectrum is actually two closely spaced lines. The sodium atoms D lines arises from a transition from the excited state configuration 1s² 2s² 2p⁶ 3p to the ground state. The doublet nature of this and other lines in the Sodium spectrum indicating a doubling of the expected number of states available to the valence electron.

The explained this fine structure of atomic spectra, Goudsmit and Uhlenbeck proposed in 1925 that the electrons has an intrinsic - built in angular momentum in additions to the orbital angular momentum due to it's motion about the any atoms nucleus. If we pictures the electron as a sphere of charge Spinning about one, it's diameter and we can look how such an intrinsic angular momentum can arises. Hence we have to the term is spin angular momentum or, more simply, spin. However, the electron "spin" is not a classical effect, and pictures of an electron rotating about an axis has no physically reality. The intrinsic angular momentum is real, but not easily visualization model can properly explain it's origin. We can't hope to understand microscopic particles based on models taken from our experience in the macroscopic world, other elementary particles besides the electron have a spin angular momentum.

Scientist, Dirac (1928) developed the relativistic quantum mechanics of electrons, and in his treatment electron Spin arises naturally.

In the nonrelativistic quantum mechanics to which is confining ourselves, electron spin must be introduced as additional hypothesis. We have to learn that each physical property has it's corresponding linear Hermitian operator in quantum mechanics. For such property as orbital angular momentum, we can construct the quantum - mechanical operators from the classical expression by replacing pₓ, pᵧ, pz by the appropriate operators. The inherent the spin angular momentum of a microscopic particles have no analog in classical mechanics, so we can't use this method to construct operators for spin. For our purposes, we can simply use symbols for the spin operators, without giving an explicit from for them.

Analogues to the angular orbitals momentum operator, L̂², L̂ₓ, L̂ᵧ,ʐ, we have the spin angular - momentum operators Ŝ², Ŝₓ, Ŝᵧ, Ŝʐ, which are quantum postulated to be linear and Hermitian. Ŝ² is the operator for the square of the magnitude of the total spin angular momentum of a particles. Ŝʐ is the operator for the ʐ component of the particles spin angular momentum. We have

Ŝ² = Ŝ²ₓ + Ŝ²ᵧ + Ŝ²z...............(1)

We can postulate that the spin angular - momentum operators obeyed the same commutation relations as the orbital angular - momentum operators. Analogous to [ L̂ₓ, L̂ᵧ ] = iħL̂ʐ, [ L̂ᵧ, L̂ʐ ] = iħL̂ₓ, [ L̂ʐ ,L̂ₓ ] = iħL̂ᵧ we have,

[ Ŝₓ, Ŝᵧ ] = iħŜʐ, [ Ŝᵧ, Ŝʐ ] = iħŜₓ, [ Ŝʐ, Ŝₓ ]

=iħ Ŝᵧ ................(2)

From equation (1) and (2), it followed, by the same operator algebra used to obtained that,

[ Ŝ², Ŝₓ ] = [ Ŝ², Ŝᵧ ] = [ Ŝ², Sʐ ] = 0.......(3)

Since Equation (1) and (2) it's follow from the, which is depended only on the communication relations and not on the specific forms of the operator that the eigenvalue of Ŝ² are,

s ( s + l) ħ², s = 0, ¹/₂, 1 , ³/₂........(4)

and the eiɡenvalues of Ŝ are

mₛħ , mₛ = − s, −s + 1 ,..., s − 1, s .....(5)

The quantum numbers "s" is called spin of the particles. Although restrict electrons to a single value for s, experiment showing that all electrons do have a single value for s, = ¹/² protons and neutrons also have

s = ¹/².

Pions have s= 0, photons have s=1. However equation (5) doesn't hold for photons. The photons travelling speed c in vacuum. Because of their relativistic nature, it turns out that photons can having either mₛ = + 1 or mₛ = − 1, but not ms = 0.

These two mₛ values corresponding to left circularly polarized and right circularly polarized light.

With s=¹/², the magnitude of the total spin angular momentum of an electron is given by the square root of equation (4) as

[ ¹/₂ (3/2)ħ² ] ¹/² = 1/2 √3 ħ.......(6)

For s= 1/2 equation (5) gives the possible eigenvalue of Ŝʐ of an electron as + 1/2 ħ and − 1/2 ħ.

The electron spin eigenvalue that correspond to these Ŝʐ eigenvalue are denoted by α and β :

Ŝʐα = + 1/2 ħα ......... (7)

Ŝʐβ = + 1/2 ħβ ..........(8)

Since Ŝʐ commutes with Ŝ², we can taken the eigenfunction of Ŝʐ to be eigenfunction of Ŝ² also, with the eigenvalue given by equation (4) with s = 1/2

Ŝ²α = 3/4 ħ² α, Ŝ²β = 3/4 ħ²β .... (9)

Ŝʐ doesn't commute with Ŝₓ or Ŝᵧ, so alpha and beta are not eigenfunctions of this operator. The terms spin down or spin up refers to mₛ = + 1/2 and mₛ = − 1/2 respectively.

Possible orientation of the electron spin vector with respect to the z axis. In all case, s lies on the surface of a cone whose axis is the z axis.

The wave function of the spatial coordinates of the particles :

ψ = ψ ( x, y, z). We might ask, what is the variable for the spin eigenfunctions α and β?! Sometimes one talks of the a spin coordinate ω, without really specifying what's the coordinate is. Most often, one takes the spin quantum number mₛ as being the variable on which the spin eigenfunctions depend. This procedure is quietly unusual as comparing with the spatial wave functions, but because we have only two possible electronic spin eigenfunctions and eigenvalues, this is a convenient of choice. We have

α = α ( mₛ), β = β (mₛ) .........(10)

Usually we want the eigenfunctions to be normalized. The three variables of a one- particle spatial wave function ranges continously from − ∞ to + ∞ , so normalization means,

The variable mₛ of the electronic spin and eigenfunction takes ony to the two discrete values is, + 1/2 and - 1/2 normalization of the one - particle spin eigenfunction therefore,

1/2

∑ ∣ α (mₛ) ∣ ² = 1,

mₛ = - 1/2

1/2

∑ ∣ β (mₛ) ∣ ² = 1

mₛ =- 1/2

..................................... (11)

Since the eigenfunction α and β corresponding to different eigenvalues of the Hermitian operator Ŝʐ , they are orthogonal :

1/2

∑α* (mₛ) β (mₛ) =0

mₛ =- 1/2

.................. (12)

Taking α (mₛ) = δmₛ, ¹/² and β (mₛ) = δmₛ, − 1/2 , where δ ⱼₖ is the Kronecker delta function,

Then we can satisfy equation (11) and (12)

When we can consider the complete wave function for an electron including the both spaces and spin variables, we shall normalized it accordingly to,

...... (13)

The notation is,

∫ ∣ ψ ( x, y, z, mₛ ∣ ² dτ

We can denote summation over the spin variable and integration over the fully range of the spatial variables as equation (13). The symbol ∫ dυ will denoted integration over the fully range of the system's spatial variables.

The electrons is currently considering to be a pointlike elementary particles with no substructure. Higher-energy electron - positron collision experiments showing no evidence for a nonzero electron size and put an upper limits of 3 ×10 ⁻¹⁹ m on the radius of an electron density.

Neutrons and protons are made of quarks, and aren't elementary particles. The radius mass size (rms) charge radius is 0.88 × 10⁻¹⁵m.

ELECTRON SPIN

1 comment:

Label

Author