Questions tagged [fermis-golden-rule]

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Fermi's Golden rule: Accounting for Decoherence

On the Wikipedia page for Fermi's golden rule, there is a vague statement that is given in passing: ... if there is some decoherence in the process, like relaxation or collision of the atoms, or like ...
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40 views

Why does a "constant" perturbation favour the transition at $\omega_{fi}=0$?

For a constant perturbation of the form $$\hat{H'}(t)=\hat{V}\theta(t)$$ to a time-independent Hamiltonian $\hat{H}_0$, the transition probability at time $t$ from an eigenstate $|i\rangle$ of $\hat{H}...
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Fermi's golden rule notation

I am currently reading through Sakurai (1st ed) and he states that Fermi's golden rule can sometimes be written in terms of a Dirac delta function, with the assumption that $$ \rho (E_n) \equiv \delta ...
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245 views

Energy conservation for finite times in Fermi’s Golden Rule

In the derivation of Fermi’s Golden rule for the application of a sudden constant perturbation, we get the following formula for the rate: $$ P_{f \leftarrow i}(t) = |\langle f|V|i\rangle|^2 \frac{4\...
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1answer
93 views

Transition rate derivation in non-relativistic quantum scattering

I am reading Principles of Quantum Mechanics by Shankar, here's a derivation I am puzzled. To evaluate probability of particle entering detector in some solid angle, using $S$-matrix and Fermi's ...
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2answers
100 views

Why do we need large time assumption for energy conservation in electron transitions?

For electron absorption calculations (with an electric field perturbation $\Delta H = eE_0x \cos(\omega t)$) we end up with an integral like: $$c_2(t) \propto \int \rho(\omega) \left( \frac{\sin(\...
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5answers
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Why is Schrödinger's cat in a superposition and not a mixture if you model decay with Fermi's golden rule?

I am teaching quantum information for undergraduate math students and as a perspective I thought it would be cool for them to discuss Schrödinger's cat a bit. More precisely I'd like to come up with ...
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19 views

Why are electromagnetic field modes considered a continuum of states (e.g. in the Fermi Golden Rule calculation)?

When we consider a state transition e.g. from 2p to 1s in the hydrogen atom, the energy gets emitted in the form of a photon. As an assumption underlying the Golden Rule application, we expect an ...
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51 views

Getting $\frac{2\pi}{\hbar}$ factor in Fermi's Golden Rule without using limit of no time dependence

Fermi's golden rule is: $$R = \frac{2\pi}{\hbar} |\langle f| \Delta H_0 |i \rangle|^2 \rho(E)$$ and it looks like all derivations of this at some point use the limit of a time dependent perturbation ...
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35 views

Does time-dependent perturbation theory work for time-independent perturbations?

Are the results from time-dependent perturbation theory for time dependent Hamiltonians of the form $H = H_0 + \Delta H(t)$ (such as the result below) equally valid for time independent Hamiltonians ...
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46 views

Fermi's golden rule, continuuous spectrum

Usually when the Fermi's golden rule is derived using time-dependent perturbation theory, the notation suggests that the system under consideration has discrete spectrum (quantum harmonic oscillator, ...
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1answer
59 views

Two-Body Decay Conservation of Energy

I was trying to derive transition rate for a two-body decay process. In one of the reference I'm following, it consider $a\rightarrow1+2$ decay, and said the daughter particles in center-of-mass ...
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1answer
41 views

$N$-body phase space for Fermi golden rule

I was following along Mark Thomson's Modern Particle Physics, and stumble upe the derivation of d$n$ of Fermi golden rule on page 62: "... For the decay of a particle to a final state consisting ...
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2answers
166 views

Dirac-delta function in Derivation of Fermi's Golden Rule

I was following along Mark Thomson's Modern Particle Physics, and got stuck on this book's derivation of Fermi's Golden Rule (On page 53): "... If there are d$n$ accessible final states in the ...
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155 views

Can inelastic scattering still give rise to diffraction?

I am having a conceptual difficulty reconciling inelastic events and diffraction, particularly whether or not you can have inelastic diffraction. Here is my thought experiment that I am working ...
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1answer
75 views

Electric quadrupole allowed transitions mathematical proof

Consider the electric quadruple moment operators as follows: $Q_{20} = \frac{e}{2}(x^2+y^2-2z^2) $ $Q_{2 \pm1} = \frac{e\sqrt{6}}{2}z(x\pm iy) $ $Q_{2 \pm2} = - \frac{e\sqrt{6}}{4}(x\pm iy)^2 $ I ...
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1answer
77 views

Is there any simple way to predict beta decay half lives?

Question For nuclides that decay by alpha emission, the Geiger-Nuttall law gives a simple and reasonably accurate estimate of the half-life. Essentially, one can model the alpha particle as a ...
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1answer
232 views

How does dirac's delta function appear in transition rate in fermi's golden rule?

In the context of time dependent perturbation theory as in 8.06, video's code L 11.2 from mit ocw, I can't see any Dirac delta function appear anywhere. When I read about "Fermi's Golden Rule&...
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1answer
90 views

Question about the radial hydrogen eigenfunctions

When calculating the selection rules for electronic transition in the hydrogen atom in dipole approximation, we always focus on the angular integrals. But why the integral $$ \int_{0}^{\infty}[rR_{nl}(...
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1answer
70 views

Mistake in Walter Greiner's "Quantum Mechanics" Special chapters

I am going through section 2.4 and 2.5 of Walter Greiner's book "Quantum Mechanics: Special Chapters". In section 2.4, there is a detailed analysis of the elastic scattering of a free ...
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49 views

Width decay and Fermi's golden rule [closed]

The width decay $\Gamma$ is the probability per time of a decay and the more accessible states there are in a decay, the more $\Gamma$ grows. Are these accessible states the decay's channels, or the ...
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1answer
53 views

Two-body decay: Heavier particles live longer than light particles

From Fermi's Golden Rule one can derive that the decay rate for a two-particle decay ($A\to B+C$) is given by $$\Gamma = \frac{p^*}{32\pi^2m_A^2} \int |{\cal M}|^2 d\Omega,$$ where the absolute value ...
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47 views

Fermi's golden rule, Hamiltonian

I have 1 question about the Fermi's golden rule. The question is: In the introduction of this theory, for explaining the $\beta$ decay, we suppose that the Hamiltonian is of the form: $H=H_0+H_I$ ...
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1answer
107 views

Does a photon interact with the spin of an electron?

In optical transitions which involve collisions between photons (from light) and electrons present in a solid, say, the transition rate is typically given by Fermi's golden rule. But the equation ...
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2k views

What is the difference between the joint density of states and the density of state?

I think I understood the density of states, but I didn't understand the joint DOS. What is the main difference? What is the exact definition of the joint DOS? When do we use the joint DOS and when do ...
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53 views

Delta function in Fermi's Golden rule [duplicate]

I am currently trying to understand the Fermi's golden rule. We consider a system with Hamiltonian: $$\hat H = \hat H_0 + \hat Ue^{i \omega t},$$ where the expectation value of $\hat U$ i much ...
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1answer
179 views

Components of Electric Quadrupole Oscillator Strength

Fermi's Golden Rule states that the rate of a transition of an electron going from the ground state $0$ into some state $n$, is directly proportional to the square of the first order perturbation $\...
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1answer
115 views

Transition probability in the case of "strong" perturbation

We know that Fermi's Golden rule is true only for weak and short perturbation, when the transition probability $P_{fi}\ll 1$. But what if perturbation is relatively strong, so we can't use this ...
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1answer
224 views

Limit of the $\sin^2$ function in the derivation of Fermi's golden rule

In the derivation of Fermi's golden rule one typically arrives at an expression of the form $$ \frac{\sin^2(\omega t)}{\omega^2} $$ which is then converted to $$ \pi t\delta(\omega). $$ I cannot ...
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1answer
104 views

Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
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58 views

2 in the Fermi’s Golden Rule

In the derivation of the Fermi's golden rule many authors expand periodic perturbation in this form $$\hat{V}=\hat{F} e^{-i \omega t}+\hat G e^{i \omega t}$$ However I do not understand the reason. ...
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1answer
535 views

Fermi golden rule: occupation factor

Fermi's golden rule for transitions between single-particle states $a$ and $b$ is $$ \Gamma_{ a \to b} = \frac{2\pi}{\hbar}\vert M_{ab} \vert^2\delta(\epsilon_a - \epsilon_b) \, .\tag{1} $$ Here $\...
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1answer
226 views

Fourier transform of a scatering potential

The Fermi golden rules states $$ \Gamma(\vec{k},\vec{k}') = \frac{2\pi}{\hbar} \left| \left \langle \vec{k}|V|\vec{k}' \right \rangle \right|^2 \delta(E(\vec{k})-E(\vec{k}')) \, .$$ Many places (for ...
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1answer
334 views

Why is the Fermi Golden rule called so?

I was studying time dependent perturbation theory and this rule came under the case of constant (weak) perturbations. I understood the rule and the derivation but what I cannot understand is that is ...
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1answer
63 views

Feynman, Hibbs Perturbations and Energy

I am currently self-studying from Feynman & Hibbs’ Quantum Mecahnics and Path Integrals, but having an issue understanding a step in their development of first-order perturbations. They define $$...
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400 views

How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?

In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
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1answer
1k views

What is the matrix element?

Can someone give me an Eli5 description of what the matrix element is, particularly in regards to Fermi's Golden Rule? Fermi's golden rule describes the likelihood of a transition per unit time. ...
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2answers
328 views

Probability density in Fermi golden rule

Consider Fermi golden rule $$\Gamma _{{i\rightarrow f}}={\frac {2\pi }{\hbar }}\left|\langle f|H'|i\rangle \right|^{{2}}\rho $$ I don't understand why $\left|\langle f|H'|i\rangle \right|^{{2}}$ is ...
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149 views

Fermi's golden rule (transition rate) for two widely separated states

My problem has to do with quantum (e.g. electronic) transitions of a single particle between two orthogonal states. I know, for example, that light can couple two orthogonal states in a Hydrogen atom ...
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47 views

Wigner-Ekhart theorem for Fermi Golden Rule in semiconductors

I have seen many authors mentioning that there is a way to work in total angular momentum basis and calculate the matrix elements using the Wigner-Ekhart theorem. Here the author even says that ...
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1answer
82 views

Why is it intuitively unreasonable for this transition probability to grow quadratically in $t$?

In Sakurai's "Modern Quantum Mechanics" section 5.6, there is a seemingly simple statement made that I do not understand the logic of. The author is considering a physical situation in which we "turn-...
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108 views

Identifying diagrams for optical processes

I was reading some papers on the study of the optical properties of some metals and came upon these conference proceedings by Hopfield from 1972. They are on the study of the infrared properties of ...
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1answer
283 views

Fermi golden rule and decay width

Consider QM perturbation theory. For the hamiltonian $\hat{H} = \hat{H}_{0} + \hat{V}$, the set of eigenstates $\{|n\rangle\}$ of $\hat{H}_{0}$ and assuming time independence of $\hat{V}$, one has ...
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133 views

What is norm of matrix element in Fermi Golden Rule

Fermi Golden Rule says: $\Gamma \propto |M_{ij}|^2$ I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet ...
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600 views

How to explain the long lifetime of Rydberg atoms with Fermi golden rule?

How to explain the long lifetime of Rydberg atoms with Fermi's golden rule? Wikipedia says it is partly due to tiny wavefunction overlap with inner orbitals, but what about the outer ones?
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1answer
340 views

Deriving the 2s to 1s transition rate

The hydrogen $2s$ to $1s$ is forbidden so it has a long mean-life (0.125 s vs 1.6 ns). Fermis golden rule can be used to derive the $2p\to1s$, but it predicts a zero rate (I think) when applied to the ...
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51 views

Density of States and Quantum Jumps

The specific question that I'm working on is "If I have a particle in the bound state of a 1-D delta function potential at $t = - \infty$, and I apply a harmonic perturbation $V(x,t) = V_0xcos(\omega ...
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1answer
126 views

What is the physical significance of fourier transforming a potential?

Fermi's golden rule essentially states that the transfer rate between two plane waves is proportional to the Fourier transform of the potential (with respect to the difference in momenta). What ...
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1answer
152 views

Why is the relativistic transition matrix $\mathcal {M}$ Lorentz invariant?

I am currently studying particle physics and recently reached the part of particle decay. Here we converted the Fermi's Golden Rule: $$\Gamma_{fi} = \frac{2\pi}{\hbar}|T_{fi}|^2 \rho(E_i) $$ to its ...
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3answers
231 views

X-ray Lasers and Forbidden Transitions

My notes from an introductory course about lasers say that There does not exist a laser emitting in the X-ray because the spontaneous decay lifetime is too short to have stimulated emission. In ...