Fermis golden rule and transition rates

  • Context: Graduate 
  • Thread starter Thread starter mupsi
  • Start date Start date
  • Tags Tags
    Transition
Click For Summary

Discussion Overview

The discussion revolves around the application of Fermi's golden rule in calculating transition rates for electrons in solids, particularly in the context of electron-phonon interactions. Participants explore the necessity of incorporating distribution functions when determining transition probabilities and the implications of many-particle versus one-particle state transitions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the need for an additional probability for unoccupied states when using Fermi's golden rule, suggesting that the transition rate already encapsulates this probability.
  • Another participant argues that the transition probability is proportional to the density of final states, which may include degenerate states and a continuum of states, indicating that the density of states must be considered in the transition probability expression.
  • A different viewpoint suggests that for one-particle transitions, only the golden rule is necessary, while many-particle systems require distribution functions, raising questions about the treatment of these transitions as fundamentally different.
  • One participant emphasizes the need to analyze the different terms in the transition probability expression as per the golden rule, suggesting it is a workable approximation.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of distribution functions in transition probability calculations, indicating that there is no consensus on whether the treatment of one-particle and many-particle transitions should be considered fundamentally different.

Contextual Notes

Participants highlight the complexity of transition probabilities in many-particle systems and the potential redundancy in introducing probabilities for unoccupied states, but do not resolve the underlying assumptions or mathematical steps involved.

mupsi
Messages
32
Reaction score
1
When calculating the mean lifetime of an electron in a solid under the influence of a perturbation (for example electron-phonon interaction) we often apply Fermi's golden rule but the rate always has to be weighted by appropriate distribution functions (for example fermi functions). These take into account wether a given state is occupied of unoccupied and hence a candidate for the transition. Why do we need an "additional" probability for an unoccupied state? I mean, in the context of the Hilbert space the transition of a many-particle state to another isn't different from transitions between one particle states fundamentally. Isn't the probability already contained in the transition rate? In other words: The golden rule states: Given that the electron was in the state A (probability p(A)=1) what is the probability to find it in a continuum of states close to B at a time t? The rate is the time derivative of that probability. Isn't it redundant then to introduce a probability (1-P(B))? I hope I expressed myself clearly.
 
Physics news on Phys.org
mupsi said:
Given that the electron was in the state A (probability p(A)=1) what is the probability to find it in a continuum of states close to B at a time t? The rate is the time derivative of that probability. Isn't it redundant then to introduce a probability (1-P(B))? I hope I expressed myself clearly.

well it looks very simple that now an electron has to transit between states A and B ... so the probability of the combined states A and B should be 1 . rather than probability of finding the electron in state A is to be 1.

The transition probability is proportional to the density of final states . It is reasonably common for the final state to be composed of several states with the same energy - such states are said to be "degenerate" states. In many cases there will be a continuum of final states, so that this density of final states is expressed as a function of energy.
i think one has to analyze the different terms appearing in the transition probability expression (as in the golden rule) which is a workable approximation.
 
If you'd calculate the transitions of one particle you would only use the golden rule. The probability to find the electron in the continuum with energy close to Ea would then be an integral of the rate -no distribution function needed. When you use many particle systems you have to use distribution functions additionally. But fundamentally, a transition from a particular one-particle state to another is not different from the transition of a many particle state to another. The underlying math is the same. But why are they treated as if they're different?
 
*integral over time
 

Similar threads

Undergrad Fermi's golden rule: why delta function instead of density states?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Fermi's Golden Rule and the S-matrix
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Time Dependent Sinusoidal Perturbation Energy Conservation
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Fermi's Golden Rule: Countable Quantum States & H-Theorem
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Orbitals and selection rules in a synchrotron
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad QFT - Confusion about Fermi's Golden Rule & Cross-Sections
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Why the photoelectric absorption section finite at threshold
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Fermi's Golden Rule: non-sinusoidal [itex]t\to\infty[/itex]
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Determining the Number of Points in the n-Space
  • · Replies 16 ·
Replies
16
Views
2K
Graduate Time dependence in quantum theory
  • · Replies 1 ·
Replies
1
Views
1K