Forest FermiGoldenRule Notes - Revision history

Oborn: /* Fermi's Golden Rule */

2009-04-26T21:14:47Z

Fermi's Golden Rule

2009-02-09T17:31:33Z

2009-02-09T17:31:07Z

2009-02-09T17:18:50Z

2 Body scattering in CM frame

2009-02-09T17:17:21Z

Example: Pi-zero (\pi^0) decay

2009-02-09T17:16:11Z

Example: Two -Body decay (fission fragments)

2009-02-04T17:21:41Z

Transition Probability

2008-10-28T18:33:36Z

Transition Probability

2008-10-28T18:32:36Z

Transition Probability

2008-10-28T18:30:41Z

Transition Probability

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 Fermi's Golden rule says that the transition rate <math>W</math> is given by a transition matrix element (or "Amplitude") <math>M_{i,f}</math> weighted by the phase space and Plank's constant <math>\hbar</math> such that
-: <math>W = \frac{2 \pi}{\hbar} |M_{i,f}|^2 \times </math>(Phase Space)
+: <math>W = \frac{2 \pi}{\hbar} |M_{i,f}|^2 \times </math>(Phase Space) = \frac{0.693}{t_{1/2}}
 ==Transition Amplitude==
 :<math>| M_{i,f}| ^2 \equiv \int \psi_f^{*}(\vec{r}) H_{int} \psi_i(\vec{r}) dr^3</math>

← Older revision		Revision as of 17:31, 9 February 2009
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	: Notice that the number of integrals is equal to the number of particles in the interaction. The units of <math>\|M\|^2</math> change with the number of integrals <math>(n)</math> such that <math>[\|M\|^2] = \left (\frac{MeV}{c}\right)^{4-n}</math>		: Notice that the number of integrals is equal to the number of particles in the interaction. The units of <math>\|M\|^2</math> change with the number of integrals <math>(n)</math> such that <math>[\|M\|^2] = \left (\frac{MeV}{c}\right)^{4-n}</math>

−	[http://wiki.iac.isu.edu/index.php/Classes Back to Classes]	+	[http://wiki.iac.isu.edu/index.php/Classes Back to Classes] [[Classes]]

← Older revision		Revision as of 17:31, 9 February 2009
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	: Notice that the number of integrals is equal to the number of particles in the interaction. The units of <math>\|M\|^2</math> change with the number of integrals <math>(n)</math> such that <math>[\|M\|^2] = \left (\frac{MeV}{c}\right)^{4-n}</math>		: Notice that the number of integrals is equal to the number of particles in the interaction. The units of <math>\|M\|^2</math> change with the number of integrals <math>(n)</math> such that <math>[\|M\|^2] = \left (\frac{MeV}{c}\right)^{4-n}</math>

−	[http://iac.isu.edu~~/mediawiki~~/index.php/Classes Back to Classes]	+	[http://wiki.iac.isu.edu/index.php/Classes Back to Classes]

← Older revision		Revision as of 17:18, 9 February 2009
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−	The general expression for the cross section via Fermi's golden rule is given as	+	The general expression for the cross section via Fermi's golden rule is similar to the above pion decay example if you consider m_1 and m_2 forming an intermediate state and is given as

	: <math>d^2 \sigma = \left( \frac{\hbar }{8 \pi}\right )^2 \frac{S \|M\|^2}{\sqrt{ \left [ \left ( p_1 \right )_{\mu} \left ( p_2\right )^{\mu})\right ]^2 - (m_1 m_2)^2}} \left [ \left ( \frac{d^3p_3}{E_3}\right) \left ( \frac{d^3p_4}{E_4}\right) \right ] \delta^4(p_1^{\mu}+p_2^{\mu}-p_3^{\mu}-p_4^{\mu})</math>		: <math>d^2 \sigma = \left( \frac{\hbar }{8 \pi}\right )^2 \frac{S \|M\|^2}{\sqrt{ \left [ \left ( p_1 \right )_{\mu} \left ( p_2\right )^{\mu})\right ]^2 - (m_1 m_2)^2}} \left [ \left ( \frac{d^3p_3}{E_3}\right) \left ( \frac{d^3p_4}{E_4}\right) \right ] \delta^4(p_1^{\mu}+p_2^{\mu}-p_3^{\mu}-p_4^{\mu})</math>

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 ; Caveat
-: Sometime <math>|M|</math> will depend on the momentum vector directions in which case the integral must be done after evaluating the matrix element amplitude.  An example of this is when the transition is spin dependent <math>(\vec{p} \cdot \vec{s})</math> as in the hyperfine interaction or polarization based transitions.
+: Sometimes <math>|M|</math> will depend on the momentum vector directions in which case the integral must be done after evaluating the matrix element amplitude.  An example of this is when the transition is spin dependent <math>(\vec{p} \cdot \vec{s})</math> as in the hyperfine interaction or polarization based transitions.
 === Example: Two -Body decay (fission fragments) ===

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 === Example: Two -Body decay (fission fragments) ===
-Now consider a more general case in whigh the decay daughters have mass:
+Now consider a more general case in which the decay daughters have mass:
 :<math>d \Gamma = \hbar W = | M |^2 \frac{S}{2 \hbar m_1}  \left [  \left ( \frac{d^3\vec{p}_2}{(2 \pi)^3 2E_2} \right )\left ( \frac{d^3\vec{p}_3}{(2 \pi)^3 2E_3} \right ) \right ] (2 \pi)^4 \delta^4(p_1^\mu-p_2^\mu-p_3^\mu)</math>

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 : <math>dN = \frac{dN}{dE} dE = \frac{dN}{dE} \frac{2 \hbar}{T} dx</math>
-:<math>P(T) = 4 |< \beta | H_{int} | \alpha>|^2 \frac{dN}{dE} \frac{T}{2 \hbar} \int_{\infty}^{\infty}  \frac{\sin^2 (x)}{(x)^2} dx</math>
+:<math>P(T) = 4 |< \beta | H_{int} | \alpha>|^2 \frac{dN}{dE} \frac{T}{2 \hbar} \int_{-\infty}^{\infty}  \frac{\sin^2 (x)}{(x)^2} dx</math>
-:<math>int_{\infty}^{\infty}  \frac{\sin^2 (x)}{(x)^2} dx = \pi</math>
+:<math>\int_{-\infty}^{\infty}  \frac{\sin^2 (x)}{(x)^2} dx = \pi</math>

@@ Line 184: / Line 184: @@
-:<math>P|< \beta | H_{int} | \alpha>|^2  </math> = M_
+:<math>P|< \beta | H_{int} | \alpha>|^2  </math> =:<math>| M_{i,f}| ^2 \equiv \int \psi_f^{*}(\vec{r}) H_{int} \psi_i(\vec{r}) dr^3</math>
-:<math>\frac{dN}{dE}  </math>
 ==Single Particle decay==

← Older revision		Revision as of 18:32, 28 October 2008
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	:<math>P(T) = 4 \|< \beta \| H_{int} \| \alpha>\|^2 \frac{dN}{dE} \frac{T}{2 \hbar} \int_{\infty}^{\infty} \frac{\sin^2 (x)}{(x)^2} dx</math>		:<math>P(T) = 4 \|< \beta \| H_{int} \| \alpha>\|^2 \frac{dN}{dE} \frac{T}{2 \hbar} \int_{\infty}^{\infty} \frac{\sin^2 (x)}{(x)^2} dx</math>
		+
		+	:<math>int_{\infty}^{\infty} \frac{\sin^2 (x)}{(x)^2} dx = \pi</math>
		+
		+
		+	:<math>P(T) = \frac{2 \pi T}{ \hbar}\|< \beta \| H_{int} \| \alpha>\|^2 \frac{dN}{dE} </math>
		+
		+
		+	:<math>P\|< \beta \| H_{int} \| \alpha>\|^2 </math> = M_
		+	:<math>\frac{dN}{dE} </math>

	==Single Particle decay==		==Single Particle decay==