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Hello guys!

I've been learning how to estimate half life using Schrodinger's time-independent wave equation. In class, we divided the energy barrier into five smaller segments just like this webpage http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/alpdet.html#c1

I was wondering if we could divide the barrier into a large number of segments n to arrive at better approximations using calculus. In class, we used the following equation for the probability of tunneling

My work is shown below

$$ψ = e^{-βx}

\\β = \frac {\sqrt {2m_α(U - E_α)}} {ħ}

\\U = \frac {kq_1q_α} {x}

\\ψ^2 = e^{-2βx}

\\ψ_{total} = e^{-2∫β(x)dx}$$

I solved the integral and put a few numbers but didn't really get anything meaningful, so any help would be greatly appreciated

I've been learning how to estimate half life using Schrodinger's time-independent wave equation. In class, we divided the energy barrier into five smaller segments just like this webpage http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/alpdet.html#c1

I was wondering if we could divide the barrier into a large number of segments n to arrive at better approximations using calculus. In class, we used the following equation for the probability of tunneling

My work is shown below

$$ψ = e^{-βx}

\\β = \frac {\sqrt {2m_α(U - E_α)}} {ħ}

\\U = \frac {kq_1q_α} {x}

\\ψ^2 = e^{-2βx}

\\ψ_{total} = e^{-2∫β(x)dx}$$

I solved the integral and put a few numbers but didn't really get anything meaningful, so any help would be greatly appreciated

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