9 Higher Order Interactions_Landmarks on the way to Quantum Field Theory

N. Panchapakesan and J.D. Anand

    Learning outcomes

• Estimate the value of higher order interactions like spin, quadrupole and magnetic dipole.
• Learn to estimate the power by which each order of perturbation decreases.
• Learn the derivation of Planck’s Law from QED with hermiticity playing a crucial role.
• Learn about two landmarks before quantum theory of radiation :  Planck’s
• Law derivation by (1) Einstein in 1917 which first identified Spontaneous Emission and (2) by Satyendra Nath Bose in 1924 which led to Bose-Einstein statistics.
• Understand , that the principle of detailed balance used by Einstein is a consequence of hermiticity of the Hamiltonian used in Quantum mechanics.

    2. Introduction

The interaction due to the magnetic moment of the electron, not included in our interaction Hamiltonian so far is next studied and shown to be of higher order and hence smaller by thousand times in the amplitude . The momentm k of photon in k.x is of the order of (~1/ ) and x is limited in the integral over the radius of the atom a . The amplitude is smaller by |kx| (~a/ ~ 10−3). and the transition probability is smaller by a factor of , square of that (10−6). where a is the radius of the atom and is the wave length of light emitted. We shall study this in section 3.

Some times the first term 1 , in the expansion of plane wave , leading to dipole emission or absorption is forbidden by selection rules. The next term in the expansion the leads to the quadrupole and the magnetic dipole terms. These are of higher order and hence smaller as seen above . In studying them it is convenient to expand the plane wave part    .   in terms of spherical Bessel function and Legender polynomials. We do this in section 4.

We can apply our results obtained to calculate Planck’s law for black body radiation, the law which started off the quantum revolution and brought Planck’s constant h into physics. We make use of the hermiticity of the interaction Hamiltonian in doing this. A charged particle like electron emits radiations as well as absorbs electromagnetic radiations. So in deriving Planck’s Law inside a back body, one can consider a system of atoms in thermodynamic equilibrium. In emitting and absorption of photons the electrons of the atoms can be treated non-relativistically. The atom in a state A emits a photon of energy ω and goes to state B. The atom in state A absorbs a photon of energy and goes to state B.  By treating these system of atoms in thermodynamic equilibrium, we derive the Planck’s Law of radiation in section 5.

In section 6 and 7 we discuss two land mark works by Einstein and Satyendra Nath Bose. The first is Einstein’s derivation of the Planck Law by considering trasitions between two states of an atom A and b as mentioned earlier. Apart from two terms for absorption and emission in the presence of radiation he had to consider a term for spontaneous emission even in the absence of radiation . This is the 1 part of + 1, we have already seen. This was not known in 1917 when Einstein did his calculation. This discussed in section 6.

In 1924, Bose derived the Planck’s law in a totally revolutionary way. He could not get it published and so took the unusual step of sending it to Einstein, who recognizing its importance got it translated into German, and published. Einstein also showed that it had applications to other particles than photons, which could be having a mass unlike photon, which is massless. This we know now as Bose-Einstein statistics applicable to particles of integral, including zero, spin. Such particles are now called Bosons, a name given by Dirac.

3. Spin Interaction Term:

So far we have neglected the spin and hence the magnetic moment of the electron in its interaction with the radiation field. If we consider the spin of the electron then we have one additional term in the interaction ??;

4.1 Magnetic Dipole and Quadrupole Transitions

Occasionally we may have (?̅)BA = 0 , for every state B with energy lower than that of the state A. This may be due to the selection rules. In these cases we then go back to the plane wave expansion

The first term of equation ( 4.3 ) has (k⃗⃗ ×ℇ⃗⃗⃗λ)) which is the first term in the expansion of curl A and hence denotes the magnetic field B. L is related to the angular momentum and hence can be expressed in terms of the magnetic moment μ also. The first term becomes

where ?⃗ is the magnetic moment of the electron. Thus ( 4.4 ) is the interaction of magnetic moment of the charged particle with the magnetic field ?⃗⃗. This term is called M1 or magnetic dipole term.

The second term of equation (4.3 ) is called E2 or Electric quadrupole term. To simplify this term we now use the relation [?⃗2, ?⃗] = −2??⃗ ??, 2?[?0, ?⃗] = −2??⃗ , where ?0 is the free Hamiltonian.

Thus ?⃗ = ??[?0, ?⃗] − − − − − −(4.5)

Using (4.5) we write the second term of eq. (4.3) as

This term gives rise to the electric quadrupole or E2 term. To see this in another way we go back to another expansion for plane wave .

4.2 Another Expansion of Plane Wave

We can write

     with the p. term this will give a product of two  l=1 terms. The resulting term will have l=2,1 or 0. The quadrupole term is the one with  l=2. This results in emission of quadrupole term E2.

5.  Derivation of Planck’s Law from field theory.

Consider a system of atoms in thermodynamic equilibrium. . The atom in a state A emits a photon of energy ω and goes to state B. The atom in state A absorbs a photon of energy   and goes to state B.   The processes are

    ? → ? + ?, ? + ? → ? , ?? ? ⇄ ? + ? − − − − − − − − − − − − − − − −(5.1)

Let EA be the energy of the atom A and EB that of B then

?? = ?? + ?

Let the population (number) of atoms in state A and B be N(A) and N(B) respectively. The transition probability ??−? is given by Golden rule ( Module 6) which for a transition is

??→? = 2?|??? |² δ(?? − ?? − ?) − − − − − − − − − − − − − − − − − − (5.2)

In the above expression ??? is the matrix element of the interaction Hamiltonian (i.e. perturbation theory is assumed to be valid ). In the equilibrium state

In the above ?? is Boltzman Constant and T the equilibrium temperature. If ℰ⃗? is the polarization vector of the photon of energy ? and momentum k then,

The number of states dn in a black body filled with radiation is obtained using the same arguments as for ?? and is

? 2.4??² ??/(2?)³. = 8??²???/(2?)³ − − − − − − − − − − − −(5.9)

The factor 2 is due to the availability of two possible directions in which light can be polarized. Thus the  number of photons with an energy from ? and ? + ?? in the volume V is

which is Planck’s radiation Law for determining the energy density ?( ω, ?) or U (ν, ?) of black body radiation,

6.    Einstein’s Derivation of Planck’s Law and Spontaneous emission

This derivation was given by Einstein in 1917 where he first identified the factor for spontaneous emission. Let the initial state of the electron of the atom be n with energy En . After absorbing a photon of energy it has a state m with energy Em , then

?? = ?? + ? − − − − − − − − − − − − − − − − − − − −(6.1)

Let population (number) of the atoms in state m and n be Nm and Nn respectively. Einstein realized that the probability per unit time per atom for the emission transition must have two parts. We denote the spontaneous emission by A, which does not need any radiation to be present and the stimulated emission by ??(?), which is proportional to the amount of radiation ?(?) present. The absorbed emission is denoted by ?′?(?) and depends on the radiation present. In thermodynamic equilibrium we have

???′?(?) = ??[? + ??(?)] − − − − − − − − − − − − − (6.2)

 

Einstein argued that from the principle of detailed balance ?′ = ?. To derive the law from field theory we had used hermiticity of the Hamiltonian . Comparing (6.5) with Planck’s law (5.10) we get A = B ( ?3 / ?2 )

7. Bose Derivation of Planck Law

Using statistical mechanics Bose calculated the probability for distributing ?? photons over ?? cells, all having energy ?? . Since there can be any number of photons in one cell this corresponds to inserting ?? -1 partitions between ?? photons. If we have ?? =4 we have

…….|…|…..|……….

?1 ?2 ?3

with dots showing photons and vertical lines partitions ?1 , ?2and ?3.

Bose’s paper was rejected by a journal for publication. So he sent it in 1924 to Prof. Einstein requesting him to forward it for publication if he thought it was correct. Einstein translated the paper into German language and sent it for publication and wrote a few more papers applying the Bose method of counting to normal gases. This was the birth of systems obeying Bose-Einstein statistics. We now know that integral (including zero spin ) particles obey Bose statistics and are called Bosons . We saw in the earlier module that bosons, scalar particles satisfy commutation relation when quantized. Photons have spin 1 and so are bosons. They do not have 2S + 1 or 3 components, but have only two components or polarisations as they have zero mass. Zero mass implies gauge constraints and we have seen how gauge constraints reduce the degrees of freedom to two polarisations.

There were criticisms that the counting of states by Bose and Einstein violated the independent nature of particles. Einstein accepted this criticism but said that as the result was right they have to find out the reason for this mysterious attraction between photons . We now know that symmetry of boson wave function brings in an attraction just as that of anti-symmetry of fermions (particles with half integral spins) brings in a repulsion leading to Pauli principle. This was finally explained by P.A.M.Dirac in 1926. The statistics obeyed by Fermions is called Fermi-Dirac Statistics.

Planck introduced the photon in 1900 and Einstein gave the photo electric effect in 1905. But even in 1924, photon was still not accepted as a particle and Einstein expressed surprise at the way Bose had derived the number of states . It took a long time 27 years for dual nature of light to be accepted. Abraham Pais in his book on Eistein and his work mentions four path breaking papers published before quantum revolution in 1926. They are by Max Planck (1900), Albert Einstein (1905), Niels Bohr (1911) and Satyendra Nath Bose (1924).

8. Summary

In this module we have studied the higher order interactions like spin interaction, electric quadrupole and magnetic dipole interactions. We find these higher order terms are thousand times smaller as they are proportional to the ratio of the radius of the atom to the wave length of light. We have learnt how Planck’s Law can be derived from quantum field theory. We have also learnt how Einstein derived the Planck’s law from Bohr’s theory in 1917, and Satyendra Nath Bose derived it from statistical arguments in 1924 before the discovery of quantum mechanics in 1926, and became the first author on quantum statistics.

    Learn More Books

1.  Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)

2.  Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted

3.  “Subtle is the Lord , The science and Life of Albert Einstein” , Oxford University Press, 1982

Web sites

For history see

www.physics.udel.edu/~msafrono/626/Lectures%2013-14.pdf

www.colorado.edu/entities/quantum-field-theory/qft.history.html

Photons not being independent

Dirac, in his book on Quantum Mechanics (1947 edition) explains this very simply.

I describe this in my own words. If you have two coins and toss them. The probability for two heads to appear is one quarter or ¼ . This is because it is one possibility among four.

2 heads, 2 tails, one head one tail or one tail and one head.

When they have to be symmetric wave functions, then only the sum of the last two is symmetric. The difference is antisymmetric.

So there are only three possibilities. Hence the probability for two heads or two tails is one third or 1/3. This is more than ¼, what it was earlier. The probability has gone up due to attraction due to Bose-Einstein Statistics.

True Story

When S.N.Bose was taking Mr. and Mrs, Dirac, in car , home for lunch, the Diracs sat at the back and Bose and driver were in front. When another student wanted also to come Bose took him in front. Mrs Dirac objected and said he should come back. Dirac remarked : it is all a question of statistics. ( Dirac statistics repels and Bose statistics attracts.)