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Quantum Mechanics Primer Book Questions

I've recently been struggling on a couple of questions, mainly due to understanding rather than lack of mathematical knowledge.

https://isaacphysics.org/questions/qmp_ch4_q32?board=qmp_ch4_29_38_r1&stage=university

For this question, I'm pretty sure that the wavefunction for the incident particles is psi(x) = B_1 * e^(i * k * x), and the wavefunction for the transmitted particles is psi(x) = B_2 * e^(i * K * x).
As current, j = - h / (2 * pi * m) * Im(psi * dpsi*/dx), j_2 / j_1 = (B_2 / B_1)^2 * K / k, which is (4 * K * k) / (K + k)^2, and is apparently wrong, but I have no idea why.

https://isaacphysics.org/questions/qmp_ch4_q29?board=qmp_ch4_29_38_r1&stage=university
Part C)
I used the result of Ex 4.27 to find the flux of transmitted particles, and divided this by the flux of incoming particles to find the probability per collision with the nuclear edge for the alpha particle to escape the nucleus by tunnelling, as required, and got roughly 1.0*10^-38, a factor of 10 from the given value (but which I believe is fine as they only say its ~10^-39).
I then multiplied this value by the answer of part A, the number of times per second the alpha particle hits the edge of the nucleus every second, to find the probability of the alpha particle escaping, and so the uranium decaying in a second, to get 8.3*10^-18, which is apparently also wrong.

Any help with these 2 questions would be greatly appreciated.
Original post by CelestialMaster7
I've recently been struggling on a couple of questions, mainly due to understanding rather than lack of mathematical knowledge.

https://isaacphysics.org/questions/qmp_ch4_q32?board=qmp_ch4_29_38_r1&stage=university

For this question, I'm pretty sure that the wavefunction for the incident particles is psi(x) = B_1 * e^(i * k * x), and the wavefunction for the transmitted particles is psi(x) = B_2 * e^(i * K * x).
As current, j = - h / (2 * pi * m) * Im(psi * dpsi*/dx), j_2 / j_1 = (B_2 / B_1)^2 * K / k, which is (4 * K * k) / (K + k)^2, and is apparently wrong, but I have no idea why.

https://isaacphysics.org/questions/qmp_ch4_q29?board=qmp_ch4_29_38_r1&stage=university
Part C)
I used the result of Ex 4.27 to find the flux of transmitted particles, and divided this by the flux of incoming particles to find the probability per collision with the nuclear edge for the alpha particle to escape the nucleus by tunnelling, as required, and got roughly 1.0*10^-38, a factor of 10 from the given value (but which I believe is fine as they only say its ~10^-39).
I then multiplied this value by the answer of part A, the number of times per second the alpha particle hits the edge of the nucleus every second, to find the probability of the alpha particle escaping, and so the uranium decaying in a second, to get 8.3*10^-18, which is apparently also wrong.

Any help with these 2 questions would be greatly appreciated.






For this question, I'm pretty sure that the wavefunction for the incident particles is psi(x) = B_1 * e^(i * k * x), and the wavefunction for the transmitted particles is psi(x) = B_2 * e^(i * K * x).



For this question, I have a very different opinion on your pretty sure wavefunctions. You are solving a different scattering problem: these wavefunctions are more like the scattering wavefunctions for step potential.

https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/06%3A_Scattering_from_Potential_Steps_and_Square_Barriers/6.02%3A_Potential_step

The problem that you quote from Isaac Physics is more involving than what you think.



As for this question part (c), it seems to depend on the answer for part (b). You may want to check with Isaac Physics.
Original post by Eimmanuel
For this question, I have a very different opinion on your pretty sure wavefunctions. You are solving a different scattering problem: these wavefunctions are more like the scattering wavefunctions for step potential.
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/06%3A_Scattering_from_Potential_Steps_and_Square_Barriers/6.02%3A_Potential_step
The problem that you quote from Isaac Physics is more involving than what you think.

As for this question part (c), it seems to depend on the answer for part (b). You may want to check with Isaac Physics.

Thank you - this helped me find my mistake and I got the correct answer!

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