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Chapter 7: Problem 72

A 12.0 -eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is \(2.5 \%,\) find its width.

Short Answer

Expert verified
To find the width of the barrier, the given energies and the probability should be substituted into the rearranged quantum tunneling formula and compute the final calculation.

Step by step solution

01

Know the Formula

Firstly, understand the formula to calculate the transmission coefficient, i.e., the probability of quantum tunneling in terms of energy and barrier width. It is given as: \(T = e^{(-2 \sqrt{2m(V-E)} \lambda/h)}\) where T is the transmission coefficient (or probability), V is the barrier height, E is the electron energy, m is the mass of the electron, λ is the width of the barrier, and h is Planck's constant.
02

Arrange for Width

We aim to calculate the barrier width λ. So, rearrange the formula to solve for λ, which gives: \(\lambda = -\frac{h \ln(T)}{2 \sqrt{2m(V-E))}}\). Considering the eV energy units, the converted Planck's constant \( h = 4.135667696 \times 10^{-15} eV.s \) and the electron mass \( m = 9.10938356 \times 10^{-31}kg \) should be used.
03

Substitute and Solve

Now, substitute V = 15.0 eV, E = 12.0 eV, T= 0.025 (Converting 2.5% to decimal) into the above formula to calculate λ.
04

Final Calculation

After putting the values into the formula, perform the mathematical computation to calculate the width of the barrier λ, which is the answer to this problem.

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