Calculating Transition Probability and Width in a System with Perturbation

[helpcometk]

Homework Statement

Consider a system with states |i> (with energy Ei) and |f> with energy eF) AND A PPERTURBATION OF THE FORM w(t)=wcosωt

i) WHAT IS THE PROBABILITY P(t) FOR A TRANSITION BETWEEN |i> AND |f> ?
ii)FOR WHICH VALUE OF ω IS THE PROBABILITY MAXIMAL ?GIVE THE PROBABILITY IN THIS CASE.
iii)WHAT IS THE WIDTH OF P(t) AT THE MAXIMUM?(VIEWED AS A FUNCTION OF ω).EXPLAIN HOW IT IS RELATED TO THE TIME-ENERGY UNCERTAINTY RELATION.

Homework Equations

i have searched in chris foot and in some other textbooks but i found no clues .

The Attempt at a Solution

in the attachement igive what chris foot gives about this subject. do you think Ω is the frequency i need on ii) ?

 

[mark726]

Thank you for bringing up this interesting topic. I am happy to share my insights on this problem.

i) The probability P(t) for a transition between |i> and |f> can be calculated using Fermi's Golden Rule, which states that P(t) = |<f|H'|i>|^2 * (sin((Ef-Ei)t/2))^2, where H' is the perturbation operator and Ef and Ei are the energies of states |f> and |i>, respectively. In this case, the perturbation operator is wcosωt, so P(t) = |<f|wcosωt|i>|^2 * (sin((Ef-Ei)t/2))^2. The probability depends on the magnitude of the perturbation w, the frequency ω, and the time t.

ii) The probability will be maximal when the term |<f|wcosωt|i>|^2 is maximal. This occurs when the cosine function is at its maximum value, which happens when ω = 0. In this case, P(t) = |<f|w|i>|^2 * (sin((Ef-Ei)t/2))^2. The probability is directly proportional to the square of the magnitude of the perturbation w.

iii) The width of P(t) at the maximum is related to the time-energy uncertainty relation, which states that the uncertainty in energy (ΔE) and the uncertainty in time (Δt) are inversely proportional to each other, i.e. ΔE * Δt ~ hbar. In this case, the width of P(t) at the maximum is related to the uncertainty in energy, which is given by ΔE = |Ef-Ei|. As ω increases, the width of P(t) decreases, meaning that the uncertainty in energy decreases. This is because higher frequencies correspond to shorter periods of time, and thus, a more precise measurement of energy can be made.

I hope this helps in your understanding of this problem. Please let me know if you have any further questions.