For the case of alpha decay, [...] the simple underlying mechanism is quantum mechanical tunneling through a potential barrier. You will find a simple explanation in any elementary quantum mechanics textbook; for example, Ohanion's Principles of Quantum Mechanics has a nice example of alpha decay on page 89. The fact that the process is probabilistic, and the exponential dependence on time, are straightforward consequences of quantum mechanics. (The time dependence is a case of "Fermi's golden rule" --- see, for example, page 292 of Ohanion.)
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