Physics
Fermi Golden Rule
The Fermi Golden Rule is a fundamental principle in quantum mechanics that describes the rate at which transitions occur between quantum states. It states that the transition rate is proportional to the square of the matrix element of the perturbation that causes the transition, and to the density of final states available to the system.
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8 Key excerpts on "Fermi Golden Rule"
eBook - PDFElementary Particle Physics
An Intuitive Introduction
- Andrew J. Larkoski(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
4 Fermi’s Golden Rule and Feynman Diagrams Golden Rule #1: Do unto others as you would have them do unto you. The foundation of civilization. Golden Rule #2: The transition rate from an initial to a final state in quantum mechanics is the matrix element squared of the Hamiltonian times the density of states. The foundation of predictions in particle physics. 1 Special relativity and applications of group theory aren’t sufficient alone to study particle physics. We need to couch them within the framework of quantum mechanics, especially as applied to experiments like the LHC. Our primary experimental tool in particle physics is colliding particles and observing the detritus that comes from it. Quantum mechanically, we cannot say for certain what the outcome of any given particle collision will be, but we can determine the probability of a particular collision. Just as in non-relativistic quantum mechanics, the probability that a given pair of particles interacts and produces a collection of particles from the collision is controlled by the overlap of the initial state with the final- state wavefunctions. So, we need to develop a method for calculating these wavefunction overlaps and extracting corresponding probabilities. Probability is a dimensionless quantity; it is just a number between 0 and 1. However, this makes it extremely subtle because it is absolute. To calculate the probability of the outcome of any given particle collision means that we need to know all possible outcomes a priori. In quantum mechanics, this is essentially the statement that the eigenstate basis for a given potential is complete: it can describe any potential outcome of an experiment. Unfortunately, this is effectively impossible in particle physics because the number of final states is uncountably infinite and (in most cases) the set of final states is not even fully known.
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 1997(Publication Date)
- WSPC(Publisher)
The remaining integral reduces to the form -foo r-sin ax dx = 7ra, (6.33) producing the result ,(D (t) 2nt (nVm) g(EM). (6.34) As a final step, we can write the expression for W±H, which is the transition probability Fermi's Golden Rule 171 l 2 t 2 /fi -6 -4 -2 0 2 4 6 co (units of Tl/t) Figure 6.1 Probability of making the transition from state m) to state n) after a time t as a function of u nrn = f E n — Bm ) /H. per unit time (to first order). It is given by the important expression v v nm known as Fermi's golden rule.* 2TT (nVm) g(E<®) (6.35) 6.3.2 Extension to Scattering Problems Here, we develop a more general form of Fermi's golden rule applicable to scattering calculations. In a typical scattering situation one considers a stream of quanta (i.e., neutrons, photons, or charged particles) entering, and then leaving, the field set up by some material medium. The quantum system is composed of the scattered particles + scattering medium. Both long before and long after the scattering interaction takes place, the system is in an energy eigenstate of H 0 , the Hamiltonian for particles and scatterer far apart, where there is no mutual interaction between the two. While quanta approach and recede from the scatterer, an interaction potential V exists that induces transitions into various scattered states. The perturbation is turned on gradually starting at t 0 = —oo, and then turned off gradually until t = +oo. This slow turn-on and turn-off of the interaction potential is sometimes referred to as adiabatic switching. The probability that scattering takes place between some initial system state m) to some final state n) is given by Eq. 6.20 in conjunction with the *More specifically, this result was coined golden rule no. 2 by Fermi [28]. 172 Time-Dependent Perturbation Theory, Transition Probabilities, and Scattering perturbation expansion for Ui(t).
eBook - PDF- James Yardley(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
7 and 39. Academic Press, New York, 1965. 1.5 FERMI'S G O L D E N R U L E 11 Note that the sums in Eq. (1.5-4) range from j = — oo to j = + GO, while those in Eqs. (1.5-5) and (1.5-6) range from k = 1 to k = oo. Thus for 0 < eth~ l < 2n we have Ρ — 2V-et 1 et (1.5-7) Furthermore, for et/h « 1, i.e., for times which are sufficiently short, P aiJi ^(2nV 2 /he)t. (1.5-8) A quantity known as density of states, p, may be defined as ρ = € _ 1 . If e is the energy spacing between the states {φ 3 ), then ρ corresponds to the number of states per unit energy in the continuum or near continuum represented by {φ 3 ). Often we are not so much interested in P a{j} as we are in the transition probability per unit time, k a{j} , which is, from Eq. (1.5-8), k aU) = (2n/h)V 2 p. (1.5-9) This is a commonly used form of the Fermi Golden Rule. When this expression is valid we may write ^ = -f c e U 1 [ N e ] , (1-5-10) where [N f l ] represents the concentration of molecules in state φ α . Thus, k a{j} is the rate constant for decay of state a. A more general form for the Fermi Golden Rule may be derived as follows. Suppose the system is initially in state φ α . At t = 0 it is coupled to a set of states {/}. Then from Eq. (1.5-2), we have Kf^y( 2(1 T a/) )-( 1 . 5 -H ) The function in the brackets is plotted in Fig. 1-3 as a function of oe aj , the a-j transition frequency. It may be seen that this function appears to be a sharply peaked function centered about ω α] = 0. In fact, we may write this function as 2nt ô(oe aj ) where δ(ω) is the Kroneker δ function for ω space. This function has the property that j <5(co) = 1 and that j /(co) δ(ω) άω = /(0) where the integral is over all frequencies. When carrying out dimensional analysis, it is worth noting that δ(ω) has units of ω 1 . This shows that transitions occur primarily to states φ ] with energies Ej ~ E a . The transition probability
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 2007(Publication Date)
- WSPC(Publisher)
Chapter 6 TIME-DEPENDENT PERTURBATION THEORY, TRANSITION PROBABILITIES, AND SCATTERING The central tool for calculating transition rates in quantum mechanics is known as Fermi’sgoldenrule ;itisaresultthatcomesdirectlyfrom time-dependentperturbation theory . Todeveloptherelevantformalism,itisconvenienttointroducethe interaction picture of quantum mechanics, an alternative to the Schrödinger and Heisenberg pictures first presented in Chapter 3. A major application of the golden rule is to the calculation of scattering cross-sections. As an example, in this chapter we will derive the so-called double-differential cross-section for thermal neutron scattering. It will be seen that the probability that a neutron transfers a particular momentum and energy to the scattering medium depends on a knowledge of dynamic structure factors ; these are functions that appear in light and x-ray scattering calculations as well. 6.1 The Interaction Picture in Quantum Mechanics In the Schrödinger picture of quantum mechanics, the time-evolution of a system’s state vector | ψ s ( t ) angbracketright is dictated by the Schrödinger equation, i.e., Eq. 3.120. In Section 3.3 it was shown that given the state vector at t = 0 , i.e., | ψ s (0) angbracketright , the solution for the state vector at time t can be expressed as | ψ s ( t ) angbracketright = ˆ U ( t ) | ψ s (0) angbracketright , where ˆ U ( t ) is a unitary time-evolution operator satisfying the differential equation i planckover2pi1 d dt ˆ U ( t )= ˆ H ˆ U ( t ) . (6.1) In cases when the Hamiltonian ˆ H is independent of time, Eq. 6.1 can be integrated and the solution is simply ˆ U ( t ) = exp parenleftBig -i ˆ Ht/ planckover2pi1 parenrightBig . However, certain situations, e.g., many problems encountered in quantum electrodynamics, share the common feature that the Hamiltonian can be split into two pieces, namely, a time-independent Hamil-tonian ˆ H 0 and a time-dependent potential ˆ V ( t ) , i.e., ˆ H = ˆ H 0 + ˆ V ( t ) .
- Hiroyuki Yokoyama, Kikuo Ujihara, Yokoyama Hiroyuki, Kikuo Ujihara(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
This result, which is an example of Fermi 's golden rule, predicts that for times large enough that energy conservation is established, yet short enough for first-order perturbation theory to remain va/id, the excited atomic state Spontaneous Emission in Optical Cavities: A Tutorial Review 9 decays at the (time-independent) rate Y¡ of Equation 23. More precisely, the prediction of this calculation is that for this range of times, the evolution of the upper state population is given by dP (t) _e -=-y =-y P(t) dt f f e ' (24) the second approximate equality relying on the fact that P.(t) = 1 in the regime of validity of first-order perturbation theory. Remarkably, it turns out that this result can be readily extended to the form dP_(t) __ P() - Y¡ t, dt e (25) that is, the spontaneous emission rate Y¡ is valid for all times and any degree of depletion of the upper state population - except of course for very short times, where a quadratic dependence of P .(t) on time is to be expected! (There are also very long time departures from the exponential decay law, but we won't discuss them in this review.) Clearly, frrst-order perturbation theory is not in a position to predict the result (25). The derivation of this equation is the subject that we now turn to. c. WEISSKOPF-WIGNER THEORY A better way to approximately solve Equations 16 and 17 is given by the Weisskopf-Wigner theory of spontaneous emission. 13 Here, one proceeds by formally integrating Equation 17 and inserting the result in Equation 16 to get da(t) _ ~~ 12i 1 d, -i(rok-ro0 )(t-t') ( ') --- -k.,¡ gk t e a t . dt k o (26) As was the case in the Fermi Golden Rule discussion, we replace the sum over k by an integral.
eBook - PDFQuantum Dynamics
Applications in Biological and Materials Systems
- Eric R. Bittner(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
To do this we note E 2 o = 2 I c ε o (4.241) Quantum Dynamics (and Other Un-American Activities) 119 and replace I with ( dI / d ω ) d ω . P f i ( t ) = ∞ 0 d ω P f i ( t ,ω ) (4.242) = 2 c ε o dI d ω ω o | f | μ · ε | i | 2 ∞ 0 sin 2 ((¯ h ω o − ¯ h ω )( t / (2¯ h ))) (¯ h ω o − ¯ h ω ) 2 d ω (4.243) To get this we assume that dI / d ω and the matrix element of the coupling vary slowly with frequency as compared to the sin 2 ( x ) / x 2 term. Thus, as far as doing integrals are concerned, they are both constants. With ω o so fixed, we can do the integral over d w and get π t / (2¯ h 2 ), and we obtain the golden rule transition rate: k f i = π c ε o ¯ h 2 | f | μ · ε | i | 2 dI d ω ω o (4.244) Notice also that this equation predicts that the rate for excitation is identical to the rate for de-excitation. This is because the radiation field contains both + ω and − ω terms (unless the field is circularly polarized), and the transition rate from a state of lower energy to a higher energy is the same as that of the transition from a higher energy state to a lower energy state. However, we know that systems can emit spontaneously in which a state of higher energy can go to a state of lower energy in the absence of an external field. This is difficult to explain in the present framework since we have assumed that | i is stationary. Let us assume that we have an ensemble of atoms in a cavity containing electromagnetic radiation and the system is in thermodynamic equilibrium. (Thought you could escape thermodynamics, eh?) Let E 1 and E 2 be the energies of two states of the atom with E 2 > E 1 . When equilibrium has been established, the number of atoms in the two states is determined by the Boltzmann equation: N 2 N 1 = Ne − E 2 β Ne − E 1 β = e − β ( E 2 − E 1 ) (4.245) where β = 1 / kT .
eBook - PDF- David Griffiths(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
The phase space factor contains only kinematical information; it depends on the masses, energies, and momenta of the participants, and reflects the fact that a given process is more likely to occur * This is related to the fact that the Coulomb potential has infinite range (see footnote on P. 17). t In this discussion I have assumed that the target itself is stationary, and that the incident particle is simply deflected as it passes through the scattering potential. My purpose was to introduce the essential ideas in the simplest possible context. But in Section 6.2 the formalism is completely general; it includes the recoil of the target, and allows for a change in the identity of the participants during the scattering process (in the reaction ir~ + p + * K + + 2~, for example, dû might represent the solid angle into which the K + scatters). i The amplitude is also called the matrix element; the phase space is sometimes called the density of final states. 6.2 THE GOLDEN RULE 195 the more "room to maneuver" there is in the final state. For example, the decay of a heavy particle into many 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 (n > p + e + p e ), in which there is almost no extra mass to spare, is tightly constrained, and the phase space factor is very small.* The transition rate for a given process is determined by the amplitude and the phase space according to Fermi's "Golden Rule": transition rate = \M\ 2 X (phase space) (6.13) h A derivation of the Golden Rule in the nonrelativistic context will be found in any quantum mechanics text 2 ; for the relativistic version one must consult a book on quantum field theory. 3 We shall not go into that here; for our purposes it will suffice to quote the quantitative formulation of the Golden Rule in the two cases of interest: Golden Rule for Decays.
eBook - PDF- Ying Fu, Min Qiu(Authors)
- 2011(Publication Date)
- Jenny Stanford Publishing(Publisher)
The damping of the wave is associated with the absorption of the electromagnetic energy. The absorption coefficient α is defined by the energy Generalized Golden Rule 43 intensity ( ∝ | E | 2 ) decrease by a factor of 2.718282 so that α = 2 βω c (2.68) 2.3 GENERALIZED GOLDEN RULE We now discuss the transition processes of electron states involving electromagnetic field. The processes include both the absorption and emission of electromagnetic energy by the electron to and from its surrounding environment. We consider the perturbation of an electromagnetic field in the form of V ( t ) = 2 V cos ( ωt ) (2.69) where V and ω are time-independent. Rather importantly, V and ω are physical quantities so that they are real, and thus V ( t ) is also real. Assuming that the elec-tron occupies initially an eigenstate | k with energy E k and starting from Eq. 2.18, the probability for the system to be found in a different eigenstate of | q with energy E q is q | ˆ T ( t, 0) | k = q | V | k e i ( E q − E k + ω ) t/ − 1 E q − E k + ω + e i ( E q − E k − ω ) t/ − 1 E q − E k − ω (2.70) P q ← k ( t ) = 2 2 q | V | k 2 1 − cos ( ω qk − t ) ω 2 qk − + 1 − cos ( ω qk + t ) ω 2 qk + + 2 cos ( ωt ) cos ( ωt ) − cos ( ω qk t ) ω 2 qk − ω 2 (2.71) where ω qk = E q − E k , ω qk ± = ω qk ± ω . If ω is finite and E q > E k , P q ← k , − ( t ) = 2 2 q | V | k 2 1 − cos ( ω qk − t ) ω 2 qk − (2.72) by neglecting the rapidly oscillating term containing e iω qk + t in Eq. 2.70. We then go through similar analysis from Eq. 2.23 to Eq. 2.28 to derive the generalized Golden rule: w q ← k , − = 2 π q | V | k 2 N DOS ( E k + ω ) (2.73) and the transition rate becomes p q ← k = q | V | k 2 Γ q ← k , − ( E q − E k − ω − Δ E k ) 2 + Γ 2 q ← k , − (2.74) where Δ E k is given by Eq. 2.36 and Γ q ← k , − = w q ← k , − / 2.
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