# What is Fermi's golden rule: Definition and 22 Discussions

In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time (so long as the strength of the perturbation is independent of time) and is proportional to the strength of the coupling between the initial and final states of the system (described by the square of the matrix element of the perturbation) as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

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1. ### I Fermi's golden rule: why delta function instead of density states?

Sakurai, in ##\S## 5.7.3 Constant Perturbation mentions that the transition rate can be written in both ways: $$w_{i \to [n]} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \rho(E_n)$$ and $$w_{i \to n} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \delta(E_n - E_i)$$ where it must be understood that this expression is...
2. ### Solving QM Problem: Fermi's Golden Rule & Transitional Probability

Hello all, I would like some guidance on how to approach/solve the following QM problem. My thinking is that Fermi's Golden Rule would be used to find the transitional probability. I write down that the time-dependent wavefunction for the free particle is...
3. ### I Fermi's golden rule derivation a bit hazy

My thought is following: However this would give me w=E_m + E_n instead of E_m-E_n I guess something relating to hermiticity or adjointing something has gnoe wrong in my version. Can someone point me in the right direction please? :) Thank you in advance!
4. ### I Fermi's Golden Rule and the S-matrix

Hey there, This question was asked elsewhere, but I wasn't really satisfied with the answer. When I learned about Fermi's golden rule, ##{ \Gamma }_{ if }=2\pi { \left| \left< { f }|{ \delta V }|{ i } \right> \right| }^{ 2 }\rho \left( { E }_{ f } \right)##, it was derived from first order...
5. ### I QFT - Confusion about Fermi's Golden Rule & Cross-Sections

Hey there! I've recently been looking at calculating amplitudes, densities of states and scattering cross sections in QFT, but am having a little bit of trouble with the exact form of the cross section - particularly with factors of ##2E## for the energies of the incoming and outgoing particles...
6. ### I Fermi's golden rule derivation

I'm reading Modern Particle Physics by Mark Thomson and watching Susskind's lecture on QM. In Thompson's book, equation (2.41) the wavefunction is expressed in terms of complete set of states of the unperturbed Hamiltonian as \Psi(\textbf{x}, t) = \sum_{k} c_k(t)\phi_k(\textbf{x})e^{-iE_kt}...
7. ### Fermi's Golden Rule: Countable Quantum States & H-Theorem

Consider a system with countable quantum states. One can define Jij as the rate of transition of probability from i-th to j-th quantum state. In H-theorem, if one assumes both $$H:=\Sigma_{i} p_{i}log(p_{i})$$ $$J_{ij}=J_{ji}$$ then they can prove the H always decrease. The latter is Fermi's...
8. ### Density of states in Fermi's golden rule

Fermi's golden rule contains a term that is the density of the final states ##\rho(E_{final})##. For my problem we have no time depending potentials so that's the same as ##\rho(E_{initial})##. If I understand the definition of ##\rho## correctly, it's the number of states in an interval...
9. ### Are photons really particles or just a misconception?

Hello everyone, thanks for reading I'll explain my question. At first, light was described as electromagnetic waves, until Einstein proposed the photoelectric effect and thus creating the concept of photon, a particle of light with momentum and energy, but no mass. It could explain why the...
10. ### How Do You Calculate Cross Sections in Electron-Neutrino Scattering?

Homework Statement [/B] (a) Find the ratio of cross sections. (b) Find the cross section for electron-neutrino scattering by first writing down relevant factors. Homework EquationsThe Attempt at a Solution Part (a)[/B] These represent the neutral current scattering for the muon-neutrino and...
11. ### Quick question on Fermi Golden Rule

Adopted from my lecture notes, found it a little fishy: Shouldn't ##\frac{dp}{dE} = \frac{E}{p}## given that ##p = \sqrt{E^2 - m^2}##. Then the relation should be instead: \frac{dp}{dE} = \frac{E}{p} = \frac{E}{\sqrt{E^2 - m^2}}
12. ### Fermi Golden Rule - Beta Decay

We consider the following beta decay: ^A_ZX \rightarrow ^A_{Z+1} Y + e^{-} + \nu_e The Fermi golden rule is given by: \Gamma = \frac{2\pi}{\hbar} |A_{fi}|^2 \frac{dN}{dE_f} Reaction amplitude is given by ##A_{fi} = G_F M_{nucl} ## while density of states is given by ##dN = \frac{4 \pi...
13. ### Fermi's Golden Rule Density of States

In my particles course, it says we will use Fermi's golden rule to work out rates. FGR is: Γ=2π|Mfi|ρ For the case of non-relativistic phase space, my notes say the density of states can be found as follows (pretty much word for word): Apply boundary conditions Wave-function vanishing at box...
14. ### Photonic density of state in matter

I'm a beginner in quantum optics. I always become confusing when the material's refractive index is complex. This time is about the photonic density of states. We know that if the material is not absorbing or dissipative, meaning the refractive index is a real number, the local photonic density...
15. ### Fermi's Golden Rule: non-sinusoidal $t\to\infty$

From what I have seen, Fermi's Golden rule is applied to constant or sinusoidal time varying potentials. But what if the perturbation is of the form V_0\left( \mathbf{x}\right)f\left(t\right), where f(t) is not a constant or sinusoidal? I am not really familiar with the derivation of Fermi's...
16. ### Fermi's golden rule practice

This is sort of a reverse homework question...I'm hoping someone can give me a question, to which I'll try to provide an answer. I have a QM2 exam coming up that is pretty much guaranteed to include Fermi's golden rule. Trouble is, I'm having difficulty finding good practice problems. I'm...
17. ### Adiabatic perturbations and Fermi's golden rule

Hi all, I'm having some difficulty in my advanced QM class with adiabatic perturbations and Fermi's Golden Rule. I understand the physical reasons behind the conclusion of the adiabatic perturbation calculation (that a particle in the ground state will end up in the new Hamiltonian's ground...
18. ### Fermi's Golden Rule: Derivation with Discrete & Continuous Spectrum

When I read about the derivation of this formula from Cohen Tannoudji or similar books, I couldn't understand one thing. He starts off assuming that the unperturbed Hamiltonian has a non-degenerate discrete spectrum, writes down the formulas for the evolution of an initial state using the...
19. ### Fermi's Golden Rule - Inelastic collision probability

Fermi's Golden Rule for an inelastic transition states gives the probability of a transition, due to a perturbation (which is constant in a given time interval). For 2 energy states A and B, it is proportional, among others quantities, to what I'll call E. This E is the small range of energy...
20. ### Fermi's Golden Rule for E&M

With out having to use the Dirac equation for a photon, is there any formalism similar to Fermi's golden rule, except for the E&M wave equation derived from Maxwells' Equations? I have a simple system whose wave equation solutions can be nicely expressed in terms of an eigenfunction...
21. ### Fermi's Golden Rule (Decay Amplitude)

This question relates to Griffiths: Introduction to Elementary Particles, p. 196 The process in question is a neutral pion decay into two photons. It is stated that because the secondary particles are massless, the amplitude for this process is: M(p_2,p_3) where p_2 and p_3 are the momentum...
22. ### Fermi's Golden Rule and Density of States

So we all know, or can look up Fermi's golden rule to be something like: \sigma\left(E\right) \propto \rho\left(E\right) |\langle \psi_i | \mu | \psi_f\left(E\right) \rangle|^2 Sigma is the cross section (or can be the transition rate as well), rho is the density of states of the final state...