Fermi's Golden Rule & Transition Probability

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Discussion Overview

The discussion revolves around Fermi's Golden Rule and its application to transition probabilities in quantum mechanics, particularly in the context of particle collisions and the implications of constant versus time-dependent potentials. Participants explore the conditions under which the Dirac delta function appears in transition probabilities and the significance of interaction and transition times.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that for the Dirac delta function to appear, the product of transition time and transition frequency must be much larger than one, raising questions about the implications for particle collisions.
  • Another participant argues that after a sufficiently long time post-transition, the final state's energy will approximate the initial state's energy, due to the mixing of energy eigenstates influenced by the perturbation in the Hamiltonian.
  • A different viewpoint emphasizes that the derivation of transition probability under constant potential indicates that the measurement time pertains only to the interaction period, not the time after the potential is switched off.
  • Concerns are raised regarding the application of Fermi's Golden Rule to elastic particle collisions, questioning how the assumption of equal initial and final energies aligns with the requirement that interaction time cannot be infinite.
  • One participant requests clarification on the derivation mentioned, noting a potential confusion between constant and time-dependent potentials.

Areas of Agreement / Disagreement

Participants express differing interpretations of the implications of Fermi's Golden Rule, particularly regarding the conditions for applying it to elastic collisions. There is no consensus on the justification for the assumptions made about interaction and transition times.

Contextual Notes

Participants highlight the importance of distinguishing between interaction time and transition time, as well as the role of measurement timing in determining energy states. There is an acknowledgment of the complexity involved in applying Fermi's Golden Rule to specific scenarios, such as particle collisions.

Who May Find This Useful

This discussion may be of interest to those studying quantum mechanics, particularly in the areas of transition probabilities, particle physics, and the implications of different potential types on quantum states.

plasmon
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As i studied Fermi golden rule. It involves transition probability of an initial state transiting to a final state in case of constant potential. As i understand the product of time of transition and the frequency of transition should be very much larger than one in order that the dirac delta function appears.

Does this mean that the time of particle collision(for e.g two particles come from past and collide with each other) should be infinite in order that the energy of final and initial states are same.
 
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It means that after waiting a reasonably large time after the transition the final state has the same energy as the initial state.
Due to the perturbation term in your Hamiltonian your final state is a mixture of many energy eigenstates with energy peaked around the value of the initial state. In time, the coefficients belonging to the other states tend to zero.

So the earlier after the transition you measure the final state's energy, the more uncertainty you will have in prediciting your result.
 
Take a look into the derivation of transition probability in the case of the constant potential. The derivation shows that the time of the measurement is actually the time during which the interaction potential was switched on. It does not include the time after the potential was switched off.

The inequality that results in Dirac delta function is.

(Interaction Time)(Transition Frequency)>>1

Transition frequency= Difference in energy of final and initial state divided by Planck's constant.

There are two time scales involved here.

(i) Interaction time (Strength of interaction).

(ii) Transition time.

Now Since the interaction time cannot be infinite. What is the justification of applying Fermi Golden rule on elastic particle collisions in colliders, where we assume that the initial and final energy of particle is the same.

I have an idea the inequality actually means that

Interaction time>>>>Transition time (Interaction Time not is not infinite)

So in the end, only those states survives having energy same as the initial states.
 
Last edited:
Could you provide a link to said derivation? You keep talking about a constant potential, but what you describe is in fact a time-dependend potential.
 
Here, is one of the many links.

http://moleng.physics.upatras.gr/personnel/Koukaras/download/FermiGR.pdf
 
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