**Quantum Mechanics Problem: Tunneling Probability Across a Potential Barrier**1. **Problem Description:** - **System Setup:** Consider a system consisting of two conducting wires separated by a small distance \( L \). An electron has the potential to tunnel across this separation, also known as the air gap, by overcoming a potential difference. - **Given Conditions:** - The potential difference between the wires is denoted by \( U_0 - E = 0.25 \) eV. - The probability of an electron successfully tunneling across this potential barrier is 0.02%. - **Questions to Address:** - What is the length of the gap, \( L \)? - If the potential difference is increased to 0.75 eV, by how much must the gap be reduced to maintain the same tunneling probability?**Conceptual Framework:**- **Quantum Tunneling:** This phenomenon allows particles, such as electrons, to pass through a potential barrier that would be insurmountable according to classical physics. The probability \( P \) of tunneling is closely related to the height and width of the barrier.- **Probability Calculation:** - The tunneling probability \( P \) is often expressed in terms of the barrier properties and can be represented as: \[ P \propto e^{-2 \kappa L} \] where \( \kappa \) is the decay constant given by: \[ \kappa = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}} \] Here, \( m \) is the mass of the electron and \( \hbar \) is the reduced Planck's constant.- **Adjusting the Barrier:** - If the potential difference changes, the decay constant \( \kappa \) and the required gap \( L \) must be adjusted accordingly to keep the tunneling probability constant.**Graph and Diagram Explanation:**- No graphs or diagrams are available in the provided text. However, if a graph were included, it might illustrate the relationship between the potential barrier height (\( U_0 - E \)), the gap distance (\( L \)), and the tunneling probability (\( P \)). A typical representation might show an exponential decay of tunneling probability with increasing barrier width and height.This problem effectively integrates principles of quantum mechanics, particularly the

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1. In a system of two conducting wires separated by a small distance L, an electron can potentially tunnel across a potential difference (air gap) between the wires. Assuming the difference Uo - E = 0.25 eV, and the probability of an electron tunneling across the potential barrier is .02%, what is the length of the gap? If the difference is 0.75 eV, how much smaller will the gap have to retain the same tunneling probability?

1. In a system of two conducting wires separated by a small distance L, an electron can potentially tunnel across a potential difference (air gap) between the wires. Assuming the difference Uo - E = 0.25 eV, and the probability of an electron tunneling across the potential barrier is .02%, what is the length of the gap? If the difference is 0.75 eV, how much smaller will the gap have to retain the same tunneling probability?