Question

An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of (a) 10%, (b) 1.0%, and (c) 0.10%?

Accepted Answer

Understand the problem: This is a quantum mechanics problem involving quantum tunneling. The tunneling probability is determined by the height and width of the potential-energy barrier, as well as the energy of the electron. The formula for tunneling probability is derived from the Schrödinger equation and is given approximately by: $$ T \approx e^{-2 \kappa L} $$, where $$ \kappa = \sqrt{\frac{2m(U - E)}{\hbar^2}} $$, $$ L  is $$the width of the barrier, $$ U  is $$the barrier height, $$ E  is $$the energy of the electron, $$ m  is $$the mass of the electron, and $$ \hbar  is $$the reduced Planck's constant. Identify the given values: The width of the barrier $$ L = 1.0 \, 	ext{nm} = 1.0 	imes 10^{-9} \; 	ext{m} $$, the height of the barrier $$ U = 5.0 \; 	ext{eV} $$, and the tunneling probabilities $$ T $$ are given for three cases: 10% (0.1), 1.0% (0.01), and 0.10% (0.001). The mass of the electron $$ m = 9.11 	imes 10^{-31} \; 	ext{kg} $$, and $$ \hbar = 1.055 	imes 10^{-34} \; 	ext{J·s} . $$Rearrange the tunneling probability formula to solve for $$ E $$: Start with $$ T \approx e^{-2 \kappa L} . $$Taking the natural logarithm of both sides gives $$ \ln(T) = -2 \kappa L . $$Solve for $$ \kappa $$: $$ \kappa = -\frac{\ln(T)}{2L} . $$Substitute $$ \kappa $$ into its definition $$ \kappa = \sqrt{\frac{2m(U - E)}{\hbar^2}}  to $$solve for $$ E $$: $$ E = U - \frac{\hbar^2 \kappa^2}{2m} . $$Substitute the known values into the formula for $$ E $$: For each tunneling probability $$ T $$, calculate $$ \kappa $$ using $$ \kappa = -\frac{\ln(T)}{2L} . $$Then, substitute $$ \kappa $$ into $$ E = U - \frac{\hbar^2 \kappa^2}{2m}  to $$find the energy $$ E  of $$the electron. Perform this calculation for $$ T = 0.1 $$, $$ T = 0.01 $$, and $$ T = 0.001 . $$Interpret the results: The calculated energies $$ E $$ for each tunneling probability represent the energy levels at which the electron has a 10%, 1.0%, and 0.10% chance of tunneling through the barrier. These values demonstrate the relationship between the energy of the electron and its tunneling probability, highlighting the quantum mechanical nature of the problem.

An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of (a) 10%, (b) 1.0%, and (c) 0.10%?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Quantum Tunneling

Potential Energy Barrier

Tunneling Probability