Questions tagged [fermis-golden-rule]

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5
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3answers
139 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 ...
0
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1answer
40 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 ...
4
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1answer
41 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 ...
0
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1answer
62 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&...
1
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1answer
82 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
57 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 ...
1
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0answers
42 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 ...
0
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0answers
33 views

Time-Depedent Pertubation Theory - Ionization Rate of a Hydrogen Atom

So, i was studying Time-Depedent Pertubation Theory, using the book "Lectures on Quantum Mechanics" by Steven Weinberg when i ran across this problem: "Calculate the rate of ionization ...
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0answers
20 views

Rate of Photon Absorption in a Semiconductor - Fermi's Golden Rule

In this paper, the rate of photon absorption in a semiconductor is given by Fermi's golden rule: $$ w_{1,2} = \frac{2\pi}{\hbar}\frac{e^{2}}{m^{2}c^{2}}\frac{2V}{(2\pi)^{3}}\int_{B.Z.}|\mathbf{A}\cdot\...
2
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1answer
48 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 ...
0
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0answers
36 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: $H=H_0+H_I$ ...
1
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1answer
50 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 ...
1
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0answers
358 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 ...
1
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0answers
42 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 ...
2
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1answer
129 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 $\...
0
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0answers
55 views

Fermi’s golden rule integral over energy states, time constraints

In perturbation theory, we can, to first-order, arrive at an expression for transition rates that looks like $$ \Gamma = \frac{2}{\hbar} |M_{if}|^2 \frac{\sin{\frac{E_f-E_i}{\hbar} t}}{E_f-E_i}. $$ ...
0
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1answer
106 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 ...
2
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1answer
173 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 ...
1
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1answer
100 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
55 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. ...
0
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1answer
361 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 $\...
2
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1answer
171 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 ...
2
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1answer
271 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 ...
1
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1answer
58 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 $$...
1
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0answers
321 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 ...
0
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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. ...
1
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2answers
232 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
128 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 ...
0
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0answers
46 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 ...
1
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1answer
72 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-...
3
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0answers
92 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 ...
1
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1answer
254 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 ...
-1
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1answer
130 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 ...
6
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2answers
467 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?
1
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1answer
264 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 ...
1
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0answers
49 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
84 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 ...
1
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1answer
108 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 ...
4
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3answers
180 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 ...
0
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0answers
78 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
414 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|...
0
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1answer
444 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 ...
1
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1answer
80 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. ...
11
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1answer
811 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 ...
2
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1answer
699 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?
3
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3answers
337 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. ...
4
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1answer
386 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
votes
1answer
207 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 ...
1
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0answers
846 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 ...
10
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1answer
677 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\...