Introduction to Quantum Mechanics · Problem 9.19

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · quantum-tunneling-beer-can

Problem

About how long would it take for a (full) can of beer at room temperature to topple over spontaneously, as a result of quantum tunneling? *Hint:* Treat it as a uniform cylinder of mass mm, radius RR, and height hh. As the can tips, let xx be the height of the center above its equilibrium position (h/2h/2). The potential energy is mgxmgx, and it topples when xx reaches the critical value x0=R2+(h/2)2−h/2x_0 = \sqrt{R^2 + (h/2)^2} - h/2. Calculate the tunneling probability (Equation 9.23), for E=0E = 0. Use Equation 9.29, with the thermal energy ((1/2)mv2=(1/2)kBT)\left((1/2)mv^2 = (1/2)k_B T\right) to estimate the velocity. Put in reasonable numbers, and give your final answer in years.

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