Questions tagged [fermis-golden-rule]

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How to deal with the appearance of delta functions in differential cross sections?

I've recently started learning quantum field theory, and I'm on chapter 3 of Hatfield's Quantum Field Theory of Point Particles and Strings. I was flipping through the book today, and I came across an ...
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Fermi's Golden Rule - Discrete state coupled to a continuum

I'm studying a discrete state coupled to a continuum of states. The Hamiltonian of this interaction, which i called Friedrichs Hamiltonian, is given by: $$H_{F}=E_{e}|e\rangle\langle e|+\int U|\chi_{U}...
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Quadrupole transition probability

I need to do some calculations for the transition probability for an E2 atomic transition. Basically I know the energy difference between the 2 levels, I know the laser power (and duration, as it is ...
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Relation between tunneling current and fermi golden rule (Bardeen model)

I am looking into the paper of Tersoff and Hamann - Theory of the scanning tunneling microscope. In this one there is the tunnel current written as $$I = \frac{2\pi e}{\hbar} \sum_{\mu ,\nu} f(E_\mu) [...
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How does one use Fermi's Golden Rule to calculate two-body rate of collision for a 1-D Fermi gas?

I am trying to verify a simulation I made of a bunch of non-interacting fermions (except elastic collisions) in a harmonic potential by comparing the rate of collisions from my simulation by using a ...
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Transitions in constant perturbations using time independent perturbation theory

The perturbed Hamiltonian is given as. $$H=\begin{cases} H^{(0)}&\text{for }t\leq 0 \\ H^{(0)}+V(x)&\text{for }t>0\end{cases}.$$ Here $V(x)$ does not depend explicitly on time but it can ...
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Calculation of the rate of transition of the Hydrogen electron from $|2lm\rangle$ to ground state

This is how it was calculated in my book: We take the electromagnetic field to be part of the system. The initial state of the system is the direct product $|2,l,m\rangle \times |0\rangle$. $|0 \...
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How is it possible to obtain delta function in the degenerate perturbation theory?

$$H = H_0 + V(t), V(0) = 0$$ Let $|i_0\rangle$ be the eigenstates of $H_0$, i.e. $H_0|i_0\rangle = E_{i_0}|i_0\rangle$, and $|i(t)\rangle$ is $|i_0\rangle$ after a time $t$ (with hamiltonian $H$). If $...
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Justification and interpretation of Fermi Golden Rule (second order) in Resonance Energy Transfer (RET)

I hope someone can give me some new insight to understand this. Fermi's golden rule is wildly used to calculate the rate for RET. I have some difficulties in understanding its justification and ...
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Photoelectric effect in hydrogen: evaluating the matrix element integral

I'm following page 503 of Shankar's Principle of Quantum Mechanics. The author is discussing the photoelectric effect and transition from the hydrogen ground state to a plane wave. Applying the ...
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Probabilities of transitions given by Fermi's golden rule

We were deriving Fermi's golden rule from first applying a weak time dependant (periodic) perturbation and then looking at first order changes to the Hamiltonian. We found that the probability of ...
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Calculation of electromagnetic transition probability

I'm reading From Nucleons to Nucleus by Suhonen, where (page 118) he introduced the electric or magnetic (with an index $\sigma$ such that $\sigma = E$ or $\sigma = M$) by the formula, \begin{align} ...
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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 ...
1 vote
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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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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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2 votes
1 answer
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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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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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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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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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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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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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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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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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$N$-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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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 ...
5 votes
3 answers
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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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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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5 votes
1 answer
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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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1 answer
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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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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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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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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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3 votes
1 answer
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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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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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1 vote
1 answer
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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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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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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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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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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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1 answer
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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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1 vote
1 answer
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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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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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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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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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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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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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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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1 answer
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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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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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