5 Quantum Electrodynamics-particle or photon interpretation

N. Panchapakesan

    1. Learning Outcomes

(a) Learn to expand the vector potentials as harmonic oscillator plane waves in momentum space and identify the coefficients as creation and destruction operators.

(b) Evaluate the Hamiltonian in terms of creation and destruction operators.

(c) Learn about normal ordering and how they remove the infinite part of the zero point energy by their definition.

(d) Calculate the commutation relation for electric and magnetic fields.

(e) Understand the relation between wave function and the vector potential.

(f ) Learn about  “Aharanov Bohm Effect” and the significance of vector potential in quantum mechanics.

(g) Understand how single valuedness of wave function leads to quantisation of the flux in a superconducting ring.

(h) Evaluate the interaction Hamiltonian for non relativistic electron and the e.m. field from the Scrodinger equation with covariant derivative.

2. Introduction

In the last module we got over the difficulty in quantizing the e.m field by choosing to use the coulomb (or radiation or transverse ) gauge for quantisation along with the modification of the Dirac delta function on the right hand side of the commutation relation.

In this module we go ahead and expand the vector potentials (the scalar potential is zero) as plane waves in momentum space and identify the creation and destruction operators. Their commutation relation is similar to the one in scalar field. We evaluate the Hamiltonian and find it has infinite zero point energy. We subtract this infinite energy not directly but by using a normal ordered Hamiltonian.

In the radiation gauge the four components of the four vector are reduced to two. 0 is made to vanish using the gauge freedom and .A =0 constraint reduces the four degrees of freedom to two. Fermi showed that the longitudinal part of A and 0 combine to form the instantaneous coulomb interaction between the electrons , when non relativistic electrons interact with the radiation field with two degrees of freedom (the two transverse polarisations).

We also see the special importance given to the vector potential in quantum theory and discuss the new effect produced . The explanation of this using magnetic field requires non local interaction. So we need the vector potential in quantum mechanics if we want a local theory.

3. Momentum Space Expansion of the field as plane waves

Zero Point Energy and Normal ordering

We define the zero of energy so that the energy in vacuum is zero. For this we subtract the energy of vacuum, that is the zero point energy from the total energy value.

The same thing is achieved by normal ordering. In normal ordering the annihilation operators are always moved to the right. The switching is done omitting the 1 on the R.H.S (right hand side ) of the commutation relation. Normal ordering is denoted by colons ,: , on the left and right.

Thus ∶ ?_?⁺?_? + ?_??_?⁺: = ?_?⁺?_? + ?_?⁺?_? = 2?_?⁺?? .

Notice we have switched the order in the second term in eq. (3.9a) but have not brought in the non vanishing right hand side of the commutator like in (3.9b).

For a field we separate the positive and negative energy (or frequency) parts and put frequency parts on right.  ( Positive frequency ( ?^−???) part always goes with a or annihilation operator).

Thus ∶ ??: = ?⁽⁻⁾ ?⁽⁻⁾ + 2?⁽⁻⁾?⁽⁺⁾ + ?⁽⁺⁾?⁽⁺⁾

5. Significance of Vector potential (Aharanov-Bohm Effect)

In classical electrodynamics the vector potential has no separate physical significance. It plays its role only through electric and magnetic fields. However when we go to quantum mechanics this is no longer true. The vector potential plays a special role and has its own significance. This was shown first by Aharanov and Bohm in 1959 and is called “Aharanov Bohm” effect.

Before discussing the details of the effect, we should note that classical electrodynamics is a local theory of electric and magnetic fields. If there are no fields at a point in space the effect of e.m. fields is zero. However the vector potential can be finite or non zero at a point where the magnetic field is zero. This effect shows that an electron passing through a point where magnetic field is zero will still feel an effect if vector potential is non zero at that point in quantum theory.

In terms of magnetic field it looks like a non local theory, but in terms of vector potential, it is a local theory. This would mean that to have a local quantum theory we must give more importance to vector potential than the magnetic field. Before discussing the effect we need the expression for wave function in terms of the vector potential.

4. Relation between solutions for zero and finite vector potential

Let us start with the non relativistic Schrodinger equation

? = ?(0) ?−?? ∫ ?.??′?0A.ds

6. Aharanov – Bohm effect:

Consider the double slit geometry with a Solenoidal magnetic field for electron interference. A coherent beam from a source on the left of the figure is sent around two sides of a solenoid by a double slit arrangement. It interferes on the right hand side where two beams meet.

magnetic flux enclosed by paths 1 and 2. Here the closed line integral is along path 1 and then along path 2 in the opposite direction.

So interference pattern depends on total flux even though the electron paths transverse only regions with B = 0. Classically the dynamical behavior of electron is given by the Lorentz force, which is zero when electron path is through points where there is no magnetic field. But in quantum mechanics there are observable effects due to magnetic field in regions inaccessible to electron. A is however non zero along the path. So any description in terms of B has to be non-local. A local description requires use of A, the vector potential. So in Quantum Mechanics A plays an essential role.

In the A.B. effect for the paths at maximum of the interference pattern the wave function is single valued.   (that is why it is a maximum.) Paths with phase, 3 are minimum. The change in interference pattern with change in flux has been verified experimentally.

7. Quantisation of Flux

We next consider the magnetic flux enclosed by a super conducting ring, shown in the figure. At the center there is magnetic field due to a solenoid (not shown in figure).

There is no magnetic field in the ring. This is due to Meissner effect. If the ring encloses magnetic flux , then the wave function inside the ring is given by

? = ?(0) ?−2?? ∫ ?.??′?0A.ds’ .

The factor ‘2’ in the exponent is due to particle in a superconductor not being an electron with charge e but a Cooper pair with charge 2e. We assume that a Cooper pair behaves like a charged bound particle, even though the pairing is in momentum space .

Figure. 7.2 Superconducting ring. Enclosed flux in the centre. Ther is no magnetic field in the ring.

The factor ‘2’ in the exponent is due to particle in a superconductor not being an electron with charge e but a Cooper pair with charge 2e. we assume that a Cooper pair behaves like a particle. Now if we start from a point in the ring and come back to the same point the wave function must be single valued and so same, for all closed paths. This should be so whether or not the closed path includes any flux. This requires the phase change to be a multiple of 2π. So we require

Book

Sakurai J.J, “ Advanced Quantum Mechanics”, Pearson Education Inc. 2006.

Web links

web.mit.edu/6.763/www/Ft03/Lectures/lecture10.pdf                               accessed 24 Feb. 2016

    Interesting Fact

Gauge Fields

Electromagnetic field is a gauge field. The gauge field arises in a very simple way when we demand that the derivative of field have the same transformation property as the field when a local phase transformation is made. This requires the modification of the derivative to covariant derivative which brings a gauge field with it. However when we relate the gauge field to the electric and magnetic fields there are constraints which create problems when we try to quantize them.

As it happens , all the other fields, weak, strong and even the gravitational field are all gauge fields or fields with constraints. In the last 50 years we have learnt how to tackle the combined electro- weak and strong fields. Gravitation is proving even now very difficult to quantize.

The name of  Herman Weyl , a mathematician ,is associated with gauge formulation of e.m. fields.