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(I) Simple quantum systems: potential barrier Consider a uniform potential barrier of height vo = 6 eV and width L = 900 pm. Assume that the wavefunction inside the barrier continuously decays and smoothly connects the region on the left and on the right as illustrated below. Incoming Outgoing amplitude amplitude The wavefunction inside the barrier is a solution of the time-independent Schrödinger equa- tion 2m da + vpW (x) = EW(x) where vo is the barrier height. Assume that the particle is travelling from the left to the right (from negative to positive on the z-axis) and inside the barrier it keeps traveling to the right. A) In the radioactive decay an a-particle (¿He²+) with de Broglie wavelength of A = 0.8 nm impacts the barrier. Assume that the barrier is very high and wide (from the perspective of the particle that is impacting the barrier from the left) and that the particle inside the barrier is described by a wavefunction V(2) = Ae-ar where a and A are constants. What is the probability the a-particle will tunnel through the barrier? B) If an electron with de Broglie A = 0.8 nm impacts the barrier. Provide quantitative reasoning for the questions below: You cannot assume that the that the barrier is very high and wide in this case, why? • Use the appropriate expression for the transmission probability and demonstrate whether the electron will have a higher probability of transmission than the a- particle?

(I) Simple quantum systems: potential barrier Consider a uniform potential barrier of height vo = 6 eV and width L = 900 pm. Assume that the wavefunction inside the barrier continuously decays and smoothly connects the region on the left and on the right as illustrated below. Incoming Outgoing amplitude amplitude The wavefunction inside the barrier is a solution of the time-independent Schrödinger equa- tion 2m da + vpW (x) = EW(x) where vo is the barrier height. Assume that the particle is travelling from the left to the right (from negative to positive on the z-axis) and inside the barrier it keeps traveling to the right. A) In the radioactive decay an a-particle (¿He²+) with de Broglie wavelength of A = 0.8 nm impacts the barrier. Assume that the barrier is very high and wide (from the perspective of the particle that is impacting the barrier from the left) and that the particle inside the barrier is described by a wavefunction V(2) = Ae-ar where a and A are constants. What is the probability the a-particle will tunnel through the barrier? B) If an electron with de Broglie A = 0.8 nm impacts the barrier. Provide quantitative reasoning for the questions below: You cannot assume that the that the barrier is very high and wide in this case, why? • Use the appropriate expression for the transmission probability and demonstrate whether the electron will have a higher probability of transmission than the a- particle?

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(I) Simple quantum systems: potential barrier Consider a uniform potential barrier of height to = 6 eV and width L = 900 pm. Assume that the wavefunction inside the barrier continuously decays and smoothly connects the region on the left and on the right as illustrated below. Incoming Outgoing amplitude amplitude The wavefunction inside the barrier is a solution of the time-independent Schrödinger equa- tion 2m dr2 + vV(x) = EV(r) where ep is the barrier height. Assume that the particle is travelling from the left to the right (from negative to positive on the z-axis) and inside the barrier it keeps traveling to the right. A) In the radioactive decay an a-particle (He2+) with de Broglie wavelength of A = 0.8 nm impacts the barrier. Assume that the barrier is very high and wide (from the perspective of the particle that is impacting the barrier from the left) and that the particle inside the barrier is described by a wavefunction V(2) = Ae-ar, where a and A are constants. What is the probability the a-particle will tunnel through the barrier? B) If an electron with de Broglie A = 0.8 nm impacts the barrier. Provide quantitative reasoning for the questions below: • You cannot assume that the that the barrier is very high and wide in this case, why? Use the appropriate expression for the transmission probability and demonstrate whether the electron will have a higher probability of transmission than the a- particle?
Transcribed Image Text:(I) Simple quantum systems: potential barrier Consider a uniform potential barrier of height to = 6 eV and width L = 900 pm. Assume that the wavefunction inside the barrier continuously decays and smoothly connects the region on the left and on the right as illustrated below. Incoming Outgoing amplitude amplitude The wavefunction inside the barrier is a solution of the time-independent Schrödinger equa- tion 2m dr2 + vV(x) = EV(r) where ep is the barrier height. Assume that the particle is travelling from the left to the right (from negative to positive on the z-axis) and inside the barrier it keeps traveling to the right. A) In the radioactive decay an a-particle (He2+) with de Broglie wavelength of A = 0.8 nm impacts the barrier. Assume that the barrier is very high and wide (from the perspective of the particle that is impacting the barrier from the left) and that the particle inside the barrier is described by a wavefunction V(2) = Ae-ar, where a and A are constants. What is the probability the a-particle will tunnel through the barrier? B) If an electron with de Broglie A = 0.8 nm impacts the barrier. Provide quantitative reasoning for the questions below: • You cannot assume that the that the barrier is very high and wide in this case, why? Use the appropriate expression for the transmission probability and demonstrate whether the electron will have a higher probability of transmission than the a- particle?
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