7 Absorption and Emission of light by atomic electrons

N. Panchapakesan and J.D. Anand

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    1. Learning Outcomes

  • Learn how the probabilities for absorption and emission of light by electrons in an atom are calculated
  • Understand how the 1 in ( ??,? +1) corresponds to spontaneous emission and ??,? corresponds to induced or stimulated emission parts.
  • Calculate the quantum expressions for the radiated energy and compare it with that of classical energy.
  • Learn to evaluate the binding energy of hydrogen atom in ev.
  • Learn to calculate in natural units , where ħ = 1, c=1.

    2. Introduction

 

In the previous module we had studied the induced emission, absorption and spontaneous emission of photon by the non-relativistic electron in a broader sense. We have so far not considered the details of the approximations involved in obtaining transitions probabilities per unit for these processes. We will now study in details the spontaneous emission process in the Dipole approximations. We shall study other interactions like spin interaction term, the quadrupole and magnetic dipole approximations later on. In studying these processes we shall see how certain selection rules emerge which allow some transitions and forbid other transitions. Closely related to the transition probability per unit time is the life time of emitted states and we shall take up an example of the same.

 

We are treating electron non-relativistically , so the kinetic energy of the electron is less than its rest mass. Also the radiation energy of photons is in the visible range. Thus, we may give some justification in retaining some terms and ignoring others. In the dipole approximation we expand,

 This is justified as the value of radius of atom R is 10−8 cm while the wave length of the photon is , in the visible range is ~ 10−5 cm . k ~ 1/ and so kR. the maximum value of kx is ~ 10−3 << 1. So the terms in eq. 2.1 are successively smaller.

|???| ≃ 105 × 10−8 ≃ 10−3 ≪ 1. − − − − − − − − − − − − − − (2.2)

 

On the other hand if a certain transition is not allowed in dipole approximation due to the selection rules then we shall include the ( ?? ⃗ ∙ ? ) term also.

 

In section 3, we study the absorption and emission of Light. We study the spontaneous emission process in the dipole approximation in section 4. We then consider the classical formula for radiated energy of an electron and compare it with the quantum one. In section 5 we learn to calculate in natural units in which ħ=1, c=1. These are very useful in quantum field theory.

 

3.    Absorption and Emission of Light

 

We now have the requisite mechanism to deal with the absorption of photons by non-relativistic atomic electrons. In absorption process an atom in a state A absorbs a is absorbed the term 2 ∙ in the interaction term does not contribute as it involves the participation of two photons. We shall assume that photons of one kind (  ? ⃗ , ℰ ? ) are present. If there are  ,   photons in the initial state, then there are , ??,? − 1 photons in the final state. Thus  ,   in the expression for ? (? , ?) contributes to the matrix element of HI. Denoting the initial state by | |?; ??,?〉 and the final state by |?; ??,? − 1〉, we have from equation (3.5) of Module 6, (dropping 2 term) From Module 6 eq. 3.14 and 3.15 , we have the matrix element for absorption

 

For emission the procedure is quite similar and we have using eq. (3.2) the transition probability

 

4. Spontaneous Emission in Dipole approximation

 

In spontaneous emission there is no radiation present initially and in the absence of the any initial radiation i.e. = 0, the transition probability per unit time into the solid angle  Ω is given by

As explained in the introduction using dipole approximation we can put  ?−?? ⃗ ∙? = 1.

4.1 Dipole Approximation

 

When the wavelength of the emitting photon (? ⃗ , ?) is much larger than the range R, we may use the equation (4.2) for the transition probability. We can simplify this equation further by the following procedure. From uncertainty relation

Let us take ⃗ vector along z axis and let ? makes an angle ? with respect to ? ⃗ axis. As ℰ   1 and ℰ 2 polarization vectors are in theplane, let us choose them along x-axis and y-axis respectively as shown in the diagram

 

 

 

If we want the time corresponding to this we have to divide by c which is length by time. This time , which is the time taken by light to cross /2π , is equal to 1.29 x 10−21 s or sec. We have taken ħ and c in CGS units. For MKS (Metre, Kilogram Second ) units we have to take the values  h = 6.6 x 10=34 J s and c = 3x 108  m/sec . We shall mostly use CGS units.

 

5.2 Hydrogen spectral energy values / Binding Energy

 

For atoms with nuclear charge Ze the energy level is multiplied by 2. So the electron in Helium is much stronger, 54.4 ev. We shall use some of these values in the next module.

 

Often mass of electron or proton is quoted in Mev. This is not mass but energy units. The rest mass is multiplied by 2 to get th energy unit. So if energy unit is used to get length we have to multiply by ħc. ( For mass we multiply by ħ/c. ). Energy units are often more convenient to use.

  1. Summary

We have obtained the matrix elements for the absorption and emission of light by electrons in atoms by using the plane wave expansion for the vector potential and identifying the number eigen states. In the case of emission there is emission even in the absence of radiation (?? = 0). This is called Spontaneous emission and was firsr identified by Einstein as we shall discuss later in module 9.

 

We then expressed the matrix element in the dipole approximation where it becomes the expectation value of x. ? . We recall the classical expression for radiation by an accelerated electron and compare it with the dipole approximation expression. Finally we discussed how to work in natural units with ħ =c = 1.

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Learn More

 

Books:

  1. Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)
  1. Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)
  1. Quantum Electrodynamics, Vol. 4 by V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford, 1982)

    Web Links

 

Lifetime of excited state;  https//en/wikipedia.org/wiki/Excited state

 

farsideph.utexas.edu/teaching/qmech/Quantum/node122.html

 

Interesting Facts

 

The calculations in quantum mechanics are largely in perturbation theory. In classical mechanics we solve differential equations and imposing initial and boundary conditions predict the actual path of a particle. Perturbation method would take an actual path and calculate the perturbation or small change in path due to small perturbations.

 

In quantum mechanics also, the differential equation is solved for getting the unperturbed eigen functions, which are then used for calculating the change due to perturbations.