INTERACTION OF RADIATION AND MATTER

Now, we can consider the interaction of molecule or an atom with electromagnetic radiation.

                        The proper study of quantum - mechanical approach would treat both radiation and atom quantum mechanically, but we can simplify things by using the classical picture of the light as a electromagnetic waves of oscillating magnetic and electric fields.

                                       The detailed investigation, which we can omit, shows that actually interaction the between the radiation's magnetic fields and the atom's charges is much more weaker than the interaction between the radiation's electric field and charges, so we can consider only the latter interaction. Especially, in NMR spectroscopy the important interaction between the magnetic dipole moments of the nuclei and radiation's magnetic field. But we can't consider this case.

                                    Let the electric field denoted by ɛₓ of the electromagnetic waves point in the x direction only. The is called plane - polarised radiation.

The electric field is define as the force per unit charges, so the force on charge Qᵢ is   F = Qᵢɛₓ = dV/dx,  integration giving the potential energy of interaction between the radiation's electric field and the charges as V = Qᵢₓ x, Where's the arbitrary interaction constant was taken as a zero. For a system of several charges, V = ∑ᵢ Qᵢ xᵢ ɛₓ. This is the TIME - DEPENDENT PERTURBATION Ĥ’ (t).

                       The space and time dependent of the electric field of an electromagnetic wave traveling in the ʐ- direction with wavelength λ and frequency ʋ is given by ɛₓ = ɛ₀ sin (2πυt - 2πʐ/ λ), where ɛ₀ is the maximum value of ɛₓ (The amplitude) Therefore,

Ĥ (t ) = − ɛ₀ ∑ Qᵢ xᵢ sin (2πʋt −2πʐᵢ /λ)

                     i

                  Where the sum goes over all the nuclei and electrons of the molecule or atom.

Defining ω and ωₘₙ as

 ω ≡ 2πυ,     ωₘₙ ≡ (E⁰ₘ - E⁰ₙ) /ħ............(1)

and substituting Ĥ (t) and we can get the coefficient in the expansion of the state function ψ as

bₘ ≈ δₘₙ + i ɛ₀/ħ ∫ ₀ᵗ´ exp (iωₘₙᵗ)   ⟨ ψ⁰ₘ ∣ ∑                                                                          i       Qᵢxᵢ sin (ωt- 2π ζᵢ/ λ) ∣ ψₙ⁰ ⟩ dt

              The integral ⟨ ψ⁰ₘ ∣ ∑ ᵢ.....∣ ψ⁰ₙ ⟩ in this equation is over all space, but significantly contribution to it's magnitude come from the regions where ψ⁰ₘ and ψ⁰ₙ are vanishingly small, and that regions well outside the molecule or atom, ψ⁰ₘ and ψ⁰ₙ and such regions ignored.

                                    Let the coordinate origin be chosen within the molecule or atoms. But since regions are well outside the atom can be ignored, the coordinate ζᵢ can be consider to having a maximum magnitude of the order of 1 nanometer. For UV light, that wavelength λ is one of the order of 10² nanometer. For microwave, visible, infrared and radio frequency radiation, λ is even larger. Hence 2πζᵢ/ λ is very small amount and can be neglected, and this leaves ∑ᵢ Qᵢxᵢ sin ωt in the integral.

Use of the identity sin ωt = ( eiωᵗ⁻ e⁻ ᵢωᵗ) ᵢ gives,

bₘ ( t') ≈ ₘₙ + ɛ⁰/2ħ ⟨ ψ⁰ₘ ∣  ∑Qᵢxᵢ ∣ ψ⁰ₙ ⟩ ∫ ₀ᵗ'   

                                                i                                    [ eᵢ ( ωₘₙ+ω) t −ei( ωₘₙ-ω)ᵗ] dt

Using ∫ ₀ᵗ' eᵃᵗ dt = α⁻¹ ( eᵃᵗ' - 1), we get

bₘ (t') ≈ ₘₙ + ɛ₀/2ħi    ⟨ ψ⁰ₘ ∣ ∑ᵢxᵢ ∣ ψ⁰ₙ⟩

...........(2)

For m ≠ n, the δₘₙ terms equals zero.

                 ∣ bₘ (t') ∣ ² gives the exact probability of a transition to state "m" from state "n". There is two cases where this probability becomes of significantly magnitude. If ωₘₙ = ω, the denominator of the second fraction in brackets is zero and this fraction's absolute value is larger. If ωₘₙ = - ω, the first fraction has a zero denominator and larger absolute value.

                      For ω ₘₙ = ω, equation (1) gives E⁰ₘ - E⁰ₙ = hυ. Exposure of the atom to radiation of frequency υ has produced a transition from the stationary state "n" to stationary state "m", where since υ positive, E⁰ₘ > E⁰ₙ. We might suppose the energy for this transition came from the system's and absorption of a photon of energy hυ. This supposition is confirmed by a fully quantum - mechanical treatment as a called " Quantum field theory" in which the radiation is treated quantum mechanically rather than the classically. We have absorption of radiation with a consequent increasing in the system's energy.

              For ωₘₙ = - ω, we get E⁰ₙ - E⁰ₘ = hυ. Exposure to radiation of frequency υ has induced a transition from stationary state n to stationary state m, where since υ is positive E⁰ₙ > E⁰ₘ. The system has gone to a lowest energy level, and a quantum field theory treatment shows that a photon of energy hυ is emitting in this process. This is called stimulated emissions of radiation. Stimulated emission process occurs in lasers.

                                    A defect of our treatment is that it's doesn't predict spontaneous emission, the emission of a photons by system is not exposed to radiation, the system falling to a lower energy level in the process. Actually Quantum field theory doesn't predict the spontaneous emission.

          Note from equation (2) that the probability of absorption is proportional to,

               ∣ ⟨ψ⁰ₘ∣ ∑ᵢ Qᵢ ₓᵢ ∣ ψ⁰ₙ ⟩²

      The quantity  ∑iQixi     is the x component of the system's and electric dipole - moment operator  μ̂ which is

μ̂ = i ∑i Qixi + j ∑i Qiyi + k ∑i Qiʐi = iμ̂ₓ + jμ̂ᵧ + kμ̂ʐ, where i, j, k are unit vectors along with axis and μ̂ₓ, μ̂ᵧ, μ̂ʐ are the components of μ̂.

We assumed that polarized radiation with the electric field in the x direction only. If the radiation has electric - field component in the y and ʐ direction also, then the probability of absorption will be proportional to,

∣ ⟨ ψ⁰ₘ∣ μ̂ₓ∣ ψ⁰ₙ⟩ |²  + ∣ ⟨ψ⁰ₘ∣ μ̂ᵧ∣ ψ⁰ₙ⟩ ∣ ² + 

∣ ⟨ψ⁰ₘ∣ μ̂ʐ∣ ψ⁰ₙ⟩ |² = ∣ ⟨ψ⁰ₘ∣ μ̂∣ ψ⁰ₙ⟩ |²

The integral     ⟨ψ⁰ₘ∣ μ̂∣ ψ⁰ₙ⟩ = μₙₘ   is the

transition dipole moment.

                   

                         When μₙₘ = 0, the transition between the states m and n with absorption or emission of radiation is said to be forbidden. The allowed transition have μₙₘ ≠ 0, because of approximation made in equation of (3), forbidden transition may have some small probability of occurs.

         For example, consider the particle in a one-dimensional box, the transition dipole moment is    ⟨ψ⁰ₘ∣ Qx∣ ψ⁰ₙ⟩  , where Q is the particle's charge and x is it's coordinate and where ψ⁰ₘ = (2/l) ¹/² sin ( mπx/l) and ψ⁰ₙ =

(2/l) ¹/²  sin (nπx/ l). The evaluation of this integral shows it's nonzero only when,

m− n = ± 1, ± 3, ± 5,..............

and is zero when m −n = 0, ± 2,.......The selection rule for a charged particles in a one-dimensional box is that the quantum numbers must change by an odd integer when radiations is emitted or absorbed.

                                Evaluation of the transition moment for the harmonic oscillator and for the two particles rigid rotor giving the selection rules, 

∆υ = ±1 and ∆j = ± 1.      The quantity of

∣ bₘ ∣ ² in equation (2) is sharply peaked at,

ω = ωₘₙ and ω = − ωₘₙ, but there's a nonzero probability that transition will occurs when ω is not precisely equal to,

∣ ωₘₙ ∣ , that's, when hυ is not precisely to

∣ E⁰ₘ - E⁰ₙ ∣ . This fact is relating to the energy - time uncertainty relations. The states with a finite lifetime have an uncertainty in their energy.

                                 Actually, radiation is not the only time - dependent perturbation that produces transitions between states. When molecule or atom comes close to another atom or molecule, it suffersa time - dependent perturbation that can change it's states. But selection rules derived for the radiative transition need not apply to collision processes, since Ĥ ( t) differs for the two processes.

                                 Thanks for reading, 

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