Alpha Decay Probability

Classical Mechanics Level 4

γ = 1 ℏ ∫ r 0 r 1 2 m ( V − E ) d r \gamma = \dfrac{1}{\hbar} \int_{r_0}^{r_1} \sqrt{2m(V-E)}\, dr

Suppose that the charge of a nucleus and the energy E E of particles in the nucleus change in such a way that the equation above is shifted to γ − ln ⁡ 2 2 \gamma - \frac{\ln 2}{2} , where r 0 r_0 and r 1 r_1 are the points where the energy E E intersects the potential V V .

By what factor does the probability of alpha decay in a given amount of time (transmission coefficient) change?


The answer is 2.

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
May 10, 2016

Relevant wiki: Quantum Tunneling

The transmission probability is given by:

e − 2 γ . e^{-2\gamma}.

If γ \gamma is shifted by − 1 2 ln ⁡ 2 -\frac12 \ln 2 , then the transmission probability is therefore shifted by a factor of:

e − 2 ( − 1 2 ln ⁡ 2 ) = e ln ⁡ 2 = 2. e^{-2 (-\frac12 \ln 2)} = e^{\ln 2} = 2.

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