QUANTUM PHYSICS FUNDAMENTALS - Chapter 7, Exercise 1 Solution
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Proton Tunneling Through the Same Barrier as an Electron

PROBLEM
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A proton (mass 1.673 x 10^-27 kg) approaches the same barrier as the
chapter's worked example - 1 eV (1.602 x 10^-19 J) high, 1 nm wide.
Calculate kappa and estimate the transmission probability. Compare
it to the electron's own result.

SOLUTION
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Step 1: Calculate kappa.

    kappa = sqrt(2 m (V0 - E)) / hbar
    kappa = sqrt(2 x 1.673 x 10^-27 x 1.602 x 10^-19) / 1.0546 x 10^-34
    kappa = sqrt(5.360 x 10^-46) / 1.0546 x 10^-34
    kappa = 2.315 x 10^-23 / 1.0546 x 10^-34
    kappa = 2.196 x 10^11 m^-1

Step 2: Calculate the transmission probability, with L = 1 x 10^-9 m.

    T = e^(-2 kappa L)
    T = e^(-2 x 2.196 x 10^11 x 1 x 10^-9)
    T = e^(-439.2)
    T = approximately 10^-191

ANSWER: The proton's transmission probability through the identical
barrier is approximately 10^-191 - effectively zero, compared to the
electron's own real, nonzero result of approximately 3.5 x 10^-5
(roughly a 1-in-30,000 chance).

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WHY THIS WORKS AS AN ANSWER

This is a direct, striking real illustration of the exponential mass
dependence in the tunneling formula. The proton is only about 1,836
times more massive than the electron, but because kappa depends on
the SQUARE ROOT of mass, and the transmission probability depends
exponentially on kappa, that modest mass ratio (sqrt(1836) is only
about 43) produces an almost incomprehensibly larger drop in tunneling
probability once it's inside an exponential. This is exactly why
quantum tunneling is a routinely observed, practically significant
effect for electrons (in the scanning tunneling microscope and tunnel
diodes covered in this chapter) but is utterly negligible for a
proton crossing an identically-sized barrier under ordinary
conditions - and it is also why, in the chapter's own real stellar-
fusion example, the enormous NUMBER of protons colliding in the Sun's
core, not a large individual tunneling probability, is what makes the
overall fusion rate real and observable.