A proton and a deuteron (which has the same charge as the...

A proton and a deuteron (which has the same charge as the proton but $2.0$ times the mass) are incident on a barrier of thickness $10.0 \mathrm{fm}$ and "height" $10.0 \mathrm{MeV}$. Each particle has a kinetic energy of $3.0 \mathrm{MeV}$. (a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.

A proton and a deuteron (which has the same charge as the proton but $2.0$ times the mass) are incident on a barrier of thickness $10.0 \mathrm{fm}$ and "height" $10.0 \mathrm{MeV}$. Each particle has a kinetic energy of $3.0 \mathrm{MeV}$.
(a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.

Key Concepts

Quantum Tunneling

Quantum tunneling is the phenomenon by which a particle has a finite probability of crossing a potential energy barrier, even when its energy is below the height of that barrier. This counterintuitive behavior, arising from the wave-like nature of particles in quantum mechanics, contrasts with classical predictions where the particle would be completely reflected.

Barrier Penetration Probability

The probability of a particle tunneling through a barrier is quantified by the transmission coefficient, which depends exponentially on both the width and the height of the barrier as well as on the particle’s energy. This relationship is derived from solving the Schrödinger equation for a particle encountering a barrier, and it illustrates how even small changes in these parameters can lead to significant differences in tunneling probabilities.

Mass Dependence in Tunneling

In quantum tunneling, the mass of the particle plays a crucial role because the exponential decay of the wavefunction inside the barrier is proportional to the square root of the mass. A heavier particle will have a faster-decaying wavefunction within the barrier, resulting in a smaller probability of tunneling compared to a lighter particle with otherwise identical energy and barrier conditions.

Transcript

00:01 In this exercise, we have two particles, one proton and one delteron, that are incident on a barrier that has a height of 10 mega -electrum volts and a thickness of 10 femtometers.

00:16 Both the delteron and the proton have a kinetic energy of 3 mega -electron volts.

00:23 Our goal in question a is to find what particle has the greatest probability of tunneling.

00:32 Here in red, i have written some information that is going to be useful for us.

00:37 So first, we have the probability of tunneling as proportional to the exponential of minus 2 kappa times a.

00:47 A is the width of the barrier.

00:52 That's the size of the barrier.

00:54 And kappa is equal to the square root of two times the mass of the particle divided by h bar square.

01:02 Times u0 u0 is the height of the barrier minus the kinetic energy of the particle so notice that since the probability is proportional to the exponential of minus 2 kappa a then whichever particle has the smallest kappa has the greatest probability and notice that kappa for the delteron is the square root of 2 times the mass of the delteron, which in the mass of the delteron itself is 2 times the mass of the proton, divided by h -bar -squared times u0 minus the kinetic energy.

01:53 Notice that the kinetic energy is the same for both the electron and the proton.

01:57 Kappa of the proton is two times the mass of the proton divided by h squared u0 minus e.

02:08 So from here we can see that kp, that kpa p, which is the kappa for the proton, is smaller than kappa for the delta.

02:20 And with this, we have that the probability of tunneling for the proton is greater than the probability of tunneling for the delteron.

02:36 So the proton has the greatest probability.

02:40 In question b, we have to find the ratio between the probability of tunneling for the proton and the probability of tunneling for the delterent.

02:50 So this is equal to the exponential of 2, of minus 2, kappa p a divided by the exponential of minus 2 kappa d a, which is equal to the exponential...

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