When a particle such as an electron in an STM tunnels through a barrier, the Heisenberg uncertainty relation can be applied to get an approximate upper limit on the tunneling distance. Suppose an electron with energy E₁ attempts to tunnel through the barrier sketched in Figure P28.57. For an STM, the width of the barrier is the distance between the STM tip and the object being studied, which is typically 0.5 nm. 

(a) Take this value for the barrier width as Δx and use the Heisenberg uncertainty relation to fi nd the uncertainty Δp in the momentum of the electron. 

(b) Assuming  Δp is the electron’s total momentum, compute the speed v of the electron. 

(c) How long does it take the electron to travel through the barrier? 

(d) Use the time found in part (c) in the Heisenberg uncertainty relation (Eq. 28.15) to find the uncertainty  ΔE in the energy of the electron. If  ΔE is the total energy of the electron, compare it with the kinetic energy calculated with the value for the speed found in part (b). The tunneling probability will be significant if the energy uncertainty ΔE is comparable to or larger than the barrier “height” (measured in terms of the electron’s potential energy).