Questions tagged [fermis-golden-rule]

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1answer
104 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
87 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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0answers
50 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
68 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
80 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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0answers
41 views

Absorption probability derivation in Feynman QED

In Feynmann QED (page 11), the tansition probability $$a_{lk} = \frac{4\sin^2(\Delta T/(2\hbar))}{\Delta^2}|u_{lk}|^2\,, \quad \Delta = E_l - E_k - \hbar\omega.$$ In the same page, there is an ...
2
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1answer
162 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
52 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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0answers
210 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
643 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
150 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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0answers
89 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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0answers
43 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
56 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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0answers
75 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
211 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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1answer
108 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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2answers
258 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 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
205 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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0answers
48 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 ...
2
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1answer
68 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
86 views

Why is the relativistic transition matrix 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
153 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 ...
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0answers
67 views

Fermi's Golden Rule; what frame of reference?

Fermi's Golden Rule is given by: $$ \newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]...
3
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1answer
341 views

Fermi golden rule and density matrix

Fermi Golden Rule expresses up to the first order the rate of departure from a state $|\psi_i>$ under the influence of a perturbation $V$ $$ W=\frac{2\pi}{\hbar} \int dk_f \mathcal{D}(k_f) \, \left|...
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1answer
327 views

Correlation Amplitude in QM

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79: Question: Why does it state that the oscillations ...
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1answer
65 views

Conductivity in Semi Conductor With band structure

I am trying to figure out how to compute the conductivity(or gain) in a semi conductor (excited by light at optical frequency $E=\hbar\omega$) using its band structure and matrix momentum element. ...
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1answer
543 views

Arbitrary normalisation of a free particle wave function

$\newcommand{\vec}[1]{\mathbf{#1}} \newcommand{\dd}{\mathrm{d}}$I'm reading Landau and Lifshitz' book on non-relativistic quantum mechanics and I have some doubts about a passage in the chapter about ...
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1answer
554 views

When does the Fermi golden rule break down?

It is widely used in all fields of physics. However, how accurate is it? It seems that many people use it without estimating its accuracy. When does it break down? Any example?
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3answers
244 views

Measurement and transition rates

We can use time dependant perturbation theory (specifically Fermi's Golden Rule) to calculate the transition rate (probability of transition per unit time) from one energy eigenstate to another. ...
3
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1answer
289 views

Fermi's golden rule and Fock states

I am having trouble understanding the derivation of the rate of spontaneous and stimulated emission given in this link. We have a perturbation that takes the form: $$ \hat H=\sum_{\vec k}f(\vec r,\...
2
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1answer
144 views

How to prove that the spectral line-width is given by the imaginary part the self energy?

I am trying to understand the computational methods to calculate the spectral line-width as done in this paper, http://www.nature.com/articles/ncomms11755 Here, they say that the line-width is ...
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0answers
721 views

Relation between differential cross section and Fermi's Golden rule

It is stated and it seems also reasonable that from Fermi's Golden rule one should be able to obtain the differential cross-section (in first Born approximation). Does anyone know how to derive their ...
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1answer
427 views

Fermi's golden rule and S-matrix

As I understand, Fermi's golden rule is a result from first order perturbation, which says that the transition rate of an initial state $|i\rangle$ to a final state $|f\rangle$ is $$ \Gamma_{i\...
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0answers
179 views

Normalize plane wave on an infinite domain.

I need to make an exercise related to quantum mechanics. (Specifically I need to apply Fermi's golden rule where the initial and final states are both plane waves). The system is 1 dimensional, ...
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0answers
343 views

How to calculate the resonance peak of a NV center during a ESR/ODMR measurement?

I am thinking of making an magnetometer with ODMR measurement of a NV center in nanodiamond. But before I do the experiment, I want to estimate the sensitivity of my experiment beforehand ...
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1answer
235 views

Question on doing the integral for Fermi golden rule

Today in the lecture, my professor did something which confused me As an example, we consider the photoelectric effect, in which an electron bound in a Coulomb potential is ionized after ...
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0answers
177 views

Fermi's golden rule and the DoS of scattering states

Can the Fermi's golden rule $$\Gamma_{fi} ~=~ \rho(E_f) \frac{2\pi}{\hbar} |M_{fi}|^2$$ be applied for transitions of discrete states to scattering states? If yes, then what should the density of ...
5
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1answer
451 views

Fermi's golden rule and infinite probablity?

I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: ...
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5answers
408 views

Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)$?

How to prove $$\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)~?$$ I have encountered this limit while learning time dependent perturbation and transition ...
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0answers
74 views

Is the Fermi golden rule really accurate for calculating the life time of an atomic level?

In my impression, Fermi golden rule is routinely used in calculating the life time of an excited atomic level. But it is based on the first order perturbation theory, so it is not expected to be ...
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1answer
832 views

Fermi's Golden Rule

Consider a system with countable quantum states. One can define $J_{ij}$ as the rate of transition of probability from i-th to j-th quantum state. In H-theorem, if one assumes both $$ H:=\sum_{i} p_{i}...
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1answer
307 views

Rate in Fermi's golden rule

There is a very clear derivation of Fermi's golden rule (actually Dirac's) here. Everything runs smoothly until, somehow, the equivalence $$ \Gamma_{a \rightarrow b} = \frac{P_{a \rightarrow b}}{t} $$...
5
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1answer
388 views

Fermi Golden Rule derivation of quasi-electron lifetime

I wonder if there is a detailed derivation of the quasi-electron lifetime: \begin{equation} \frac{1}{\tau_k}=\frac{2\pi}{\hbar}\frac{1}{V^2}\sum_{k', q}\sum_{\sigma}|V_q|^2f_{k'}(1-f_{k-q})(1-f_{k'+q}...
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2answers
123 views

Hyperfine lifetime calculation: what is the spin eigenfunctions?

I'm trying to calculate the lifetime of the 21 cm line in hydrogen and have the following expression: $$\frac{1}{\tau} = \frac{4\alpha}{3}\omega_{if}^3|\langle a_f|\vec{x}|a_i\rangle|^2.$$ The ...
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0answers
63 views

Evaluation of a probability from Fermi Golden rule

In Marc Bee's book, he has described the principle of spectroscopy with reservoir (the material) and the probe as interacting systems with their own hamiltonians $H_R$ and $H_p$ respectively and $H_c$ ...
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0answers
235 views

Relativistic Fermi Golden Rule?

On online slide notes, it is mentioned that: Fermi Golden Rule: $$P_{if}=\frac{2\pi}{\hbar}|M_{if}|^2\rho_f$$ where $\rho_f$ is density of final sates --number of quantum states per unit volume - ...
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1answer
547 views

Fermi Golden Rule

First order time dependent perturbation theory tells us that under the influence of a perturbation $Ve^{i\omega t}$, a system that started in the state $|n\rangle$ at time $t=0$ has probability $$P_k(...
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1answer
102 views

What is the sum over the transition rates?

I was looking at the solution to an exercise, and I came over this expression: $$P_{i\to f} = \sum \limits_{f} {2 \pi \over \hbar }\; |\langle f |\hat V | i \rangle |^2 \delta(E_{fi}-E),$$ where ...
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0answers
176 views

Gauge invariance of Fermi's golden rule

I am having some issues with gauge invariance of Fermi's golden rule. Say we have a system Hamiltonian for a particle in an electric field and some additional potential $V$ with \begin{equation}H=(p-A(...