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Quantization of Electromagnetic Field: Classical to Decay in Hydrogen Atom, Exams of Quantum Mechanics

Syracuse UniversityQuantum Mechanics

An overview of the process of quantizing the electromagnetic field based on classical suggestions and its application to spontaneous decay in a hydrogen atom. It covers the replacement of classical fields with operator fields, the determination of the quantum lagrangian or hamiltonian, and the calculation of matrix elements for the interaction term.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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PHY662, Spring 2004, Apr. 29, 2004
29th April 2004
1 Miscellaneous
1. We conclude today with the quantization of the electromagnetic field and its
consequences.
2. The final exam is on Monday, May 3. It is scheduled for 5:30 PM, Room
202/204.
3. There will be an exam review session at 5:00 PM, Sunday, May 2, Room 202/204.
2 Quantizing the EM field
During most of this course, photons have been discussed only as parenthetical com-
ments. Starting with magnetic resonance, we have noticed that the transitions be-
tween states take place for electromagnetic fields that oscillate with frequency such
that ¯hω = ∆E. This is true for external classical potentials, as this is the type of
external potential that can lead to interference between states separated in energy by
∆E.
You of course know that light comes in waves with quantized energy steps of size ¯hω.
To realize this picture, you might impose local gauge invariance of the wave function
(ψ→e−iqΛ/¯hcψ) and maintain the Schrodinger equation with the introduction of the
vector potential ~
A. This gives the interaction between ~
Aand ψ. Our observations
of the electromagnetic field at macroscopic scales (the classical realm) suggests the
Hamiltonian for the field ~
A(and φ, but we will be assuming the Coulomb gauge where
φ= 0).
Today we will quickly sketch how to use the classical suggestion to create a correct
quantum theory and then apply this description to spontaneous decay of an excited
state in a hydrogen atom.
2.1 Classical considerations →a, a†
We will be replacing the classical vector field ~
A(~r)by an operator field ~
A(op)(~r). (Next
semester you might in addition replace the field ψ(~r)by the operator field ψ(op)(~r), so
1
pf3

Partial preview of the text

Download Quantization of Electromagnetic Field: Classical to Decay in Hydrogen Atom and more Exams Quantum Mechanics in PDF only on Docsity!

PHY662, Spring 2004, Apr. 29, 2004

29th April 2004

1 Miscellaneous

  1. We conclude today with the quantization of the electromagnetic field and its consequences.
  2. The final exam is on Monday, May 3. It is scheduled for 5:30 PM, Room 202/204.
  3. There will be an exam review session at 5:00 PM, Sunday, May 2, Room 202/204.

2 Quantizing the EM field

During most of this course, photons have been discussed only as parenthetical com- ments. Starting with magnetic resonance, we have noticed that the transitions be- tween states take place for electromagnetic fields that oscillate with frequency such that ¯hω = ∆E. This is true for external classical potentials, as this is the type of external potential that can lead to interference between states separated in energy by ∆E.

You of course know that light comes in waves with quantized energy steps of size ¯hω. To realize this picture, you might impose local gauge invariance of the wave function (ψ → e−iqΛ/hc¯ψ) and maintain the Schrodinger equation with the introduction of the vector potential A~. This gives the interaction between A~ and ψ. Our observations of the electromagnetic field at macroscopic scales (the classical realm) suggests the Hamiltonian for the field A~ (and φ, but we will be assuming the Coulomb gauge where φ = 0).

Today we will quickly sketch how to use the classical suggestion to create a correct quantum theory and then apply this description to spontaneous decay of an excited state in a hydrogen atom.

2.1 Classical considerations →a, a†

We will be replacing the classical vector field A~(~r) by an operator field A~(op)(~r). (Next semester you might in addition replace the field ψ(~r) by the operator field ψ(op)(~r), so

that you can consider creating and destroying fermions such as electrons.)

To summarize pp. 506-515 of Shankar:

  1. From macroscopic observation and symmetry, we have a good idea of what the Lagrangian L is for the EM field, expressed in terms of A~. One then wants to determine what quantum Lagrangian or Hamiltonian this classical theory ap- proximates.
  2. To reproduce the classical equations of motion, the quantum momenta conjugate to the coordinates should obey commutation relations like [q, p] = i¯h, in some form (but the “position” & “momenta” are functions of position, i.e., are fields). The classical momenta are found using Πi = (^) ∂∂ AL˙ i . For the EM field, Π = − 4 Eπc~.
  3. Shankar then discusses the constraints ∇ ·~ A~ = 0 and ∇ ·~ ~Π = 0, which come from the full gauge-invariant Lagrangian. These constraints are hard to directly incorporate into the canonical commutation relations.
  4. However , if you

(a) change coordinates to Fourier space, e.g., A~(~r) → A~(~k), and (b) express A~(~k) in bases ~(~k, j), j = 1, 2 , 3 , with ~(~k, 3) ‖ ~k, A~(~k) = ∑ j aj^ ( ~k)~(~k, j), the classical constraints simply become a 3 = 0 AND

H =

∑^2

j=

ω[a∗(~k, j)a(~k, j)] d^3 ~k.

  1. So we take a(~k, j = 1, 2) as normal coordinates for oscillators! We can carry out all the usual oscillator stuff, e.g., writing P ∝ a†^ + a, Q ∝ i(a†^ − a), so that H ∼

Q^2 + P 2 , with Q and P obeying the canonical commutation relations.

  1. This is just like oscillator world we wrote down last time (except for the interac- tions and counting the degrees of freedom). (The spectrum or ρ(ω) is also fixed for the EM field.) So, the quantum state of the field can be described by a set of states for quantum oscillators which can be described by the ladder number of each oscillator. The ladder number a†a is the occupation number or number of photons in that state. Adding or subtracting a photon corresponds to the raising or lowering of a quantum oscillator.

2.2 Spontaneous decay

Let us consider the very specific problem considered in the course text, that of a tran- sition | 2 lm; 0~k,j 〉 → |100; 1~k,j 〉 ,