Fermi Golden Rule

3 Key excerpts on "Fermi Golden Rule"

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  eBook - ePub

  Basic Notions Of Condensed Matter Physics

  • Philip W. Anderson(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  This is the essential condition for the success of the Fermi-liquid theory. To demonstrate it, one uses perturbation theory, but only of the simplest kind, namely, the Golden Rule. The simplest decay process that can happen to a particle of momentum k is for it to excite a particle, either one out of the Fermi sea or one of the few excited particles, into an empty state, itself being scattered into a state k’ = k + q by the potential matrix element V(q) = ∫ d 3 r e iq ⋅ r V(r). (3.1) We give the diagrammatic description of this process in Fig. 3-2. Figure 3-2 “Self-energy part”: simplest case. Diagrams: As K. G. Wilson has remarked (1974a), it simply doesn’t matter which specific diagrammatic scheme one uses, so long as one works consistently within one’s own scheme. Diagrams will be used here simply as a shorthand for the physical processes I am describing. (A somewhat more complete introduction to diagrams will be given in this chapter.) Thus the diagram in Fig. 3-2 is shorthand for the following statement: I am to calculate the effect of the scattering process, described above, on the energy of particle k. I need to include two matrix elements (vertices) of the scattering potential V (which I stretch out into dotted lines for simplicity), since it is a second-order process. I draw forward-pointing lines for particles and backwards ones for holes (the propagation of the absence of a particle being equivalent to a particle moving backwards in time; or, to put it another way, the state, having emptied at t, must be refilled at some later t’. I have “borrowed” an electron. It’s just like selling short on the stock market)

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  eBook - ePub

  Atomic Physics

  • D.C.G Jones(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Appendix D .

  For an electron which starts in state 1, the approximate solution of interest is the probability of transition to state 2. This is given by the probability of finding the electron in state 2 at time t

  P 2 ( t ) =  | c 2 | 2 = (4 π 2 /h 2 ) | D 12 | 2 t 2  (sin ξ / ξ ) 2

  (6.15)

  where

  ξ = 2 π

   (ν 12

    -   ν

  f

  )  t /2

  and

  D 12

  =

  ∫

  − ∞

  ∞

  e x  

  u 2

  *  

  u 1

   dv

  The term D12 , usually known (for reasons which need not concern us) as the ‘quantum dipole moment matrix element’, defines the strength of the interaction, just as it did in the classical case.

  The physical meaning of (6.15) is not very transparent so must be discussed in some detail.

  6.5 FERMI’S ‘GOLDEN RULE’ FOR TRANSITIONS

  Equation (6.15) appears a little odd. Firstly, the probability of transition appears to vary as the square of time. Secondly, and more importantly, the bracket term (sinξ/ξ) becomes indeterminate at the ‘resonance frequency’ where ν12 = νf . These peculiarities are due to the fact that the calculation has dealt with the interaction of a perfectly monochromatic wave with an atom which possesses infinitely narrow energy levels. If either or both of these restrictions are removed, a physically comprehensible result is obtained.

  First, consider the atom interacting with an electromagnetic field which is not monochromatic but has a constant energy density ρ(ν) at frequencies near to ν12 . It is then necessary to rewrite (6.15) in terms of an integral over all frequencies (noting that only those near v12 will be important)

  P 2 ( t ) =  | c 2 | 2 = (4 π 2 /h 2 ) | D 12 | 2 ∫ t 2 ρ (ν) (sin ξ / ξ ) 2  d ρ

  (6.16)

  Changing variables and carrying out the integration leads to a more straightforward expression for P2 (t), and hence for the transition rate dP2 (t)/dt

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  Introduction to Elementary Particles

  • David Griffiths(Author)
  • 2020(Publication Date)
  • Wiley-VCH

    (Publisher)

  The phase space factor is purely kinematic ; it depends on the masses, energies, and momenta of the participants, and reflects the fact that a given process is more likely to occur the more ‘room to maneuver’ there is in the final state. For example, the decay of a heavy particle into light secondaries involves a large phase space factor, for there are many different ways to apportion the available energy. By contrast, the decay of the neutron, in which there is almost no extra mass to spare, is tightly constrained and the phase space factor is very small. * The ritual for calculating reaction rates was dubbed the Golden Rule by Enrico Fermi. In essence, Fermi’s Golden Rule says that a transition rate is given by the product of the phase space and the (absolute) square of the amplitude. You may have encountered the nonrelativistic version, in the context of time-dependent perturbation theory [ 2 ]. We need the relativistic version, which comes from quantum field theory [ 3 ]. I can’t derive it here; what I will do is state the Golden Rule and try to make it plausible. Actually, I’ll do it twice: once in a form appropriate to decays and again in a form suitable for scattering. 6.2.1 Golden Rule for Decays Suppose particle 1 (at rest) † decays into several other particles 2, 3, 4,.. ., n : 1 → 2 + 3 + 4 +... + n (6.14) The decay rate is given by the formula (6.15) where m i is the mass of the i th particle and p i is its four-momentum. S is a statistical factor that corrects for double-counting when there are identical particles in the final state: for each such group of s particles, S gets a factor of (1/ s !). For instance, if a → b + b + c + c + c, then S = (1 / 2!)(1 / 3!) = 1 / 12. If there are no identical particles in the final state (the most common circumstance), then S = 1. Remember: The dynamics of the process is contained in the amplitude, M (p 1, p 2,.

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