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help with just number 2 of this modern physics question

2. For the case of tunneling with \( E > U_0 \), starting with the boxed boundary conditions in the middle of part 203 of the book, derive the transmission and reflection probabilities in 6-12. Hint: Multiply the third boundary condition by \( k' \), then add and subtract the fourth condition to get \( C \) and \( D \) in terms of \( F \). Plug these into the first two and solve for \( B \) and \( F \) in terms of \( A \).
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Transcribed Image Text:2. For the case of tunneling with \( E > U_0 \), starting with the boxed boundary conditions in the middle of part 203 of the book, derive the transmission and reflection probabilities in 6-12. Hint: Multiply the third boundary condition by \( k' \), then add and subtract the fourth condition to get \( C \) and \( D \) in terms of \( F \). Plug these into the first two and solve for \( B \) and \( F \) in terms of \( A \).
**Quantum Tunneling and the Uncertainty Principle** 1. **Tunneling and Energy Conservation**: Tunneling is a phenomenon that exemplifies the uncertainty principle. It involves a particle traversing a potential barrier it seemingly lacks the energy to overcome. This process implies a temporary violation of energy conservation, allowing the particle to "jump over" the barrier. **Case Study**: - A particle with energy \( E = \frac{1}{2} U_0 \) approaches a barrier with potential \( U_0 \). - The barrier's width is represented by \(\delta\), which is deemed the penetration depth for this energy level. - Define \(\Delta E\) as the energy required for the particle to just surpass the potential barrier. - \(\Delta t\) is the duration a particle of this energy takes to traverse the distance \(\delta\). - **Objective**: Demonstrate that \(\Delta E \Delta t = \frac{1}{2} \hbar\). *Note*: This derivation is not rigorous; however, the result aligns with expectations, ensuring accuracy in scale. **Diagram Explanation**: - The diagram shows a potential barrier of height \( U_0 \) and width \(\delta\). - The energy level \( E = \frac{1}{2} U_0 \) is noted, illustrating that the particle's energy is half the potential barrier's height. - The setup visualizes how particles with insufficient apparent energy can still achieve penetration through the barrier within the constraints of quantum mechanics.
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Transcribed Image Text:**Quantum Tunneling and the Uncertainty Principle** 1. **Tunneling and Energy Conservation**: Tunneling is a phenomenon that exemplifies the uncertainty principle. It involves a particle traversing a potential barrier it seemingly lacks the energy to overcome. This process implies a temporary violation of energy conservation, allowing the particle to "jump over" the barrier. **Case Study**: - A particle with energy \( E = \frac{1}{2} U_0 \) approaches a barrier with potential \( U_0 \). - The barrier's width is represented by \(\delta\), which is deemed the penetration depth for this energy level. - Define \(\Delta E\) as the energy required for the particle to just surpass the potential barrier. - \(\Delta t\) is the duration a particle of this energy takes to traverse the distance \(\delta\). - **Objective**: Demonstrate that \(\Delta E \Delta t = \frac{1}{2} \hbar\). *Note*: This derivation is not rigorous; however, the result aligns with expectations, ensuring accuracy in scale. **Diagram Explanation**: - The diagram shows a potential barrier of height \( U_0 \) and width \(\delta\). - The energy level \( E = \frac{1}{2} U_0 \) is noted, illustrating that the particle's energy is half the potential barrier's height. - The setup visualizes how particles with insufficient apparent energy can still achieve penetration through the barrier within the constraints of quantum mechanics.
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