Fermi's Golden Rule, Study Guides, Projects, Research of Quantum Mechanics

Download Fermi's Golden Rule and more Study Guides, Projects, Research Quantum Mechanics in PDF only on Docsity! Fermi’s Golden Rule Emmanuel N. Koukaras A.M.: 198 Abstract We present a proof of Fermi’s Golden rule from an educational perspective without compromising formalism. We insert this relation in (4) and project the result on |ϕn〉 : H0 ∑ k ck(t)|ϕk〉e−iEkt/h̄ + W (t) ∑ k ck(t)|ϕk〉e−iEkt/h̄ = ih̄ ∂ ∂t ∑ k ck(t)|ϕk〉e−iEkt/h̄ ⇒ ∑ k ck(t)Ek|ϕk〉e−iEkt/h̄ + ∑ k ck(t)W (t)|ϕk〉e−iEkt/h̄ = ih̄ ∑ k ∂ck(t) ∂t |ϕk〉e−iEkt/h̄ + ih̄ ∑ k ck(t)|ϕk〉 ( − iEk h̄ ) e−iEkt/h̄ ⇒ Encn(t)e−iEnt/h̄ + ∑ k ck(t)Wnk(t)e−iEkt/h̄ = ih̄ ∂cn(t) ∂t e−iEnt/h̄ + Encn(t)e−iEnt/h̄ ⇒ ∑ k ck(t)Wnk(t)e−iEkt/h̄ = ih̄ ∂cn(t) ∂t e−iEnt/h̄ ⇒ ∂cn(t) ∂t = 1 ih̄ ∑ k ck(t)Wnk(t)eiωnkt (6) where ,Wnk = 〈ϕn|W (t)|ϕk〉 , the perturbation matrix element ,ωnk = (En −Ek)/h̄ Up to this point we have made no approximation. The difficulty in solving (6) arises from the fact that the coefficients are expressed in terms of themselves. In order to evaluate the coefficients from (6) we make two assumptions: 1. The system is initially in state |i〉, thus, all of the coefficients at t=0 are equal to zero, except for ci : cj(t = 0) = δij 2. The perturbation is very weak and applied for a short period of time, such that all of the coefficients remain nearly unchanged. With these in mind, (6) gives us: ∂cn(t) ∂t = 1 ih̄ ci(t)Wni(t)eiωnit (7) so for any final state the coefficient will be (cf (t) ≈ cf (0) = 0): 4 cf (t) = 1 ih̄ ∫ t 0 Wfi(t′)eiωfit ′ dt′ (8) remarks In order to derive (8) we forced all of the coefficients of the states to remain virtually unchanged, at time t, from the values they initially had (t=0). For that we must pay a price. Equation (8) holds only for perturba- tions that last a very short period of time, i.e. that don’t have enough time to significantly alter the state of the system. Also, by zeroing out the coefficients of states in (6) we deprived the system of any capability of reaching the final state by alternate routes, i.e. only direct transitions from state |i〉 to |f〉 are possible. We have extensively used frases that refer to transitions between eigen- states of the unperturbed hamiltonian H0. Such frases are commonplace in Quantum Mechanics literature1 but are also a point of much controversy and discussions. The controversy has to do mainly with the interpretation one gives to the mathematical results of Quantum Mechanics. As an ex- ample of the situation a quotation is given from the very well known and accepted book Quantum Mechanics, by L. E. Ballentine[5, p. 351] (who supports[6] an interpretation for the wavefunction other than the orthodox interpretation): “When problems of this sort are discussed formally, it is com- mon to speak of the perturbation as causing transitions between the eigenstates H0. If this means only that the system has ab- sorbed from the perturbing field (or emitted to it) the energy difference h̄ωfi = εf − εi, and so has changed its energy, there is no harm in such language. But if the statement is interpreted to mean that the state has changed from its initial value of |Ψ(0)〉 = |i〉 to a final value of |Ψ(T )〉 = |f〉, then it is incor- rect. . . . If the state vector |Ψ〉 is of the form (5) it is correct to say that the probability of the energy being εf is |cf |2. In the for- mal notation this becomes Prob(E = εf |Ψ) = |αf |2, which is a correct formula of quantum theory. But it is nonsense to speak of the probability of the state being |f〉 when in fact the state is |Ψ〉.” The application of the perturbation changes the state of the system from the initial state |ϕi〉 to a final state |ϕf 〉, both of which are eigenstates of 1This language is used by Cohen-Tannoudji, Atkins, Merzbacher [1, 2, 3, 4] and many more. 5 the unperturbed hamiltonian H0. The probability of finding the system in the eigenstate |ϕf 〉 is: Pif (t) = |〈ϕf |ψ(t)〉|2 (9) Using (8) we have: Pif (t) = 1 h̄2 ∣∣∣∣ ∫ t 0 eiωfit ′ Wfi(t′)dt′ ∣∣∣∣ 2 (10) 3. High frequency harmonic perturbation The case of an oscillating (i.e. harmonic) perturbation is a most important one. Once the effects of an oscillating perturbation are known then the gen- eral case can be evaluated since an arbitrary perturbation can be expressed as a superposition of harmonic functions. An example of an oscillating per- turbation is an electromagnetic wave such as a laser pulse. We define the oscillating perturbation having the form: W (t) = 2Wcos(ωt) = W ( eiωt + e−iωt ) (11) Inserting this in (8) we obtain: cf (t) = Wfi ih̄ ∫ t 0 ( eiωt′ + e−iωt′ ) eiωfit ′ dt′ = Wfi ih̄ { ei(ωfi+ω)t − 1 i(ωfi + ω) + ei(ωfi−ω)t − 1 i(ωfi − ω) } (12) Thus, equation (10) becomes: Pif (t) = W 2 fi h̄2 ∣∣∣∣∣ ei(ωfi+ω)t − 1 i(ωfi + ω) + ei(ωfi−ω)t − 1 i(ωfi − ω) ∣∣∣∣∣ 2 (13) At this point we make an approximation assuming that the oscillating angular frequency of the perturbation has a value near the Bohr angular frequency of |ϕi〉 and |ϕf 〉, ωfi: ω ' ωfi which can also be written: |ω − ωfi| ¿ |ωfi| With this approximation, the first term in equation (13) becomes negli- gible compared to the second one. This is made obvious from the fact that 6 be less than 1, which we have when: t ¿ h̄ |Wfi| (22) and using, (20): 1 |ωfi| ¿ h̄ |Wfi| (23) which we can read as: the matrix element of the perturbation must be much smaller than the energy separation between the initial and final states. 4. Quantum jumps to the continuum When the final state is part of a continuum of states (i.e. when the energy belongs to a continuous part of the spectrum of H0) we must account for all the states to which the system can jump to. This is done by integrating the probability as given by equation (17) with the density of states ρ(E) as weights: P(t) = ∫ {Eacc} Pif (t)ρ(E)dE (24) ,where {Eacc} denotes all the states that the system can jump to under the influence of the perturbation. (What we have actually done is create a probability density from the prob- ability equation). The probability function is sharply peaked at ω = ωfi and as a result acts as a delta function in the integral and thus selects the value for the density function at ω = ωfi. By substituting equation (17) in (24) we have: P(t) = ∫ {Eacc} W 2 fi h̄2 F (t, ω − ωfi)ρ(E)dE = ∫ {Eacc} W 2 fi h̄2 { sin[(ωfi − ω)t/2] (ωfi − ω)/2 }2 ρ(E)dE as shown in the figure (2) the range of energies is very narrow and as a result the matrix element Wfi and the density of states ρ(E) can be considered as constant: 9 Figure 2: The function F (t, ω) acts as a delta function. P(t) = W 2 fi h̄2 ρ(Efi) ∫ {Eacc} { sin[(ωfi − ω)t/2] (ωfi − ω)/2 }2 dE = W 2 fi h̄2 ρ(Efi) ∫ {Eacc} { sin[(ωfi − ω)t/2] (ωfi − ω)/2 }2 h̄dω = W 2 fi h̄2 ρ(Efi)h̄ ( 2 t ) t2 ∫ +∞ −∞ sin2x x2 dx where we substituted E = h̄ω, x = (ωfi − ω)t/2 and used the fact that for frequencies far from ωfi the function sin2x/x2 is negligible so we can extend the limits to infinity. The value for the integral is well known and equal to π, thus we obtain: 10 P(t) = 2π h̄ W 2 fiρ(Efi)t (25) The result is significant in that the probability has a linear dependence in time. The transition rate is: W = dP(t) dt = 2π h̄ W 2 fiρ(Efi) (26) This result is known as Fermi’s Golden Rule. This equation states that in order to calculate the transition rate all we have to do is multiply the square modulus of the perturbation matrix element between the two states, by the the density of states at the transition frequency. 11