Physics
Physics
3rd Edition
ISBN: 9780073512150
Author: Alan Giambattista, Betty Richardson, Robert C. Richardson Dr.
Publisher: McGraw-Hill Education
Question
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Chapter 28, Problem 50P

(a)

To determine

The particle with greater tunnelling probability.

(a)

Expert Solution
Check Mark

Answer to Problem 50P

The particle with greater tunnelling probability is proton.

Explanation of Solution

Write the expression for tunnelling probability.

P=e2κa                                                            (I)

Here, P is the tunnelling probability, a is the barrier width and κ is a measure of the barrier height.

Write the expression for κ barrier width.

κ=8π2mh2(VE)                                                            (II)

Here, m is the mass of the particle, h is the Planck’s constant, V is the barrier height and E is the total energy.

κ is directly proportional to m and P is inversely proportional κ. Thus, an increased mass results in increased κ which in turn decreases P.

Since the mass of deuteron is greater than that of the proton, the proton has greater tunnelling probability.

(b)

To determine

The ratio of the tunnelling probabilities.

(b)

Expert Solution
Check Mark

Answer to Problem 50P

The ratio of the tunnelling probabilities is 121.

Explanation of Solution

Write the expression for tunnelling probability of proton and deuteron.

PpPd=e2κpae2κda                                                           (III)

Here, Pp is the tunnelling probability of the proton and Pd is the tunnelling probability of the deuteron.

Write the expression for κp and κd.

κp=8π2mph2(VE)                                                           (IV)

Here, mp is the mass of the proton.

κd=8π2mdh2(VE)                                                            (V)

Here, md is the mass of the deuteron.

Subtracting (IV) form (V)

κdκp=8π2h2(VE)(mdmp)                                                     (VI)

The mass of the deuteron is two times the mass of the proton i.e. md=2mp

Rearranging (I)

PpPd=e2(κdκp)a                                                          (VII)

Substituting  (VI) in (VII) and using md=2mp

PpPd=e2a8π2mph2(VE)(21)                                                          (VIII)

Conclusion

Substitute 10.0fm for a, 3.0 MeV for E, 10.0 MeV for V, 1.673×1034 Js for mp and 6.626×1034 Js for h in (VIII) to find Pp/Pd.

PpPd=exp(2(10.0 fm)(21)8π2×1.673×1027 kg(6.626×1034 Js)2(10.0 MeV3.0 MeV))=exp(2(10.0×1015 m)(21)8π2×1.673×1027 kg(6.626×1034 Js)2(10.03.0)×106 eV×1.602×1019Js1 eV)=exp(4.8)=121

Thus, the ratio of tunnelling probability is 121.

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Chapter 28 Solutions

Physics

Ch. 28 - Prob. 2CQCh. 28 - Prob. 3CQCh. 28 - Prob. 4CQCh. 28 - Prob. 5CQCh. 28 - Prob. 6CQCh. 28 - Prob. 7CQCh. 28 - Prob. 8CQCh. 28 - Prob. 9CQCh. 28 - Prob. 10CQCh. 28 - Prob. 11CQCh. 28 - Prob. 12CQCh. 28 - Prob. 13CQCh. 28 - Prob. 14CQCh. 28 - Prob. 15CQCh. 28 - Prob. 16CQCh. 28 - Prob. 17CQCh. 28 - Prob. 18CQCh. 28 - Prob. 1MCQCh. 28 - Prob. 2MCQCh. 28 - Prob. 3MCQCh. 28 - Prob. 4MCQCh. 28 - Prob. 5MCQCh. 28 - Prob. 6MCQCh. 28 - Prob. 7MCQCh. 28 - Prob. 8MCQCh. 28 - Prob. 9MCQCh. 28 - Prob. 10MCQCh. 28 - Prob. 1PCh. 28 - Prob. 2PCh. 28 - Prob. 3PCh. 28 - Prob. 4PCh. 28 - Prob. 5PCh. 28 - Prob. 6PCh. 28 - Prob. 7PCh. 28 - Prob. 8PCh. 28 - Prob. 9PCh. 28 - Prob. 10PCh. 28 - Prob. 11PCh. 28 - Prob. 12PCh. 28 - Prob. 13PCh. 28 - Prob. 14PCh. 28 - Prob. 15PCh. 28 - Prob. 16PCh. 28 - Prob. 17PCh. 28 - Prob. 18PCh. 28 - Prob. 19PCh. 28 - Prob. 20PCh. 28 - Prob. 21PCh. 28 - Prob. 22PCh. 28 - Prob. 23PCh. 28 - Prob. 24PCh. 28 - Prob. 25PCh. 28 - Prob. 26PCh. 28 - Prob. 27PCh. 28 - Prob. 28PCh. 28 - Prob. 29PCh. 28 - Prob. 30PCh. 28 - Prob. 31PCh. 28 - Prob. 32PCh. 28 - Prob. 33PCh. 28 - Prob. 34PCh. 28 - Prob. 35PCh. 28 - Prob. 36PCh. 28 - Prob. 37PCh. 28 - Prob. 38PCh. 28 - Prob. 39PCh. 28 - Prob. 40PCh. 28 - Prob. 41PCh. 28 - Prob. 42PCh. 28 - Prob. 43PCh. 28 - Prob. 44PCh. 28 - Prob. 45PCh. 28 - Prob. 46PCh. 28 - Prob. 47PCh. 28 - Prob. 48PCh. 28 - Prob. 49PCh. 28 - Prob. 50PCh. 28 - Prob. 51PCh. 28 - Prob. 52PCh. 28 - Prob. 53PCh. 28 - Prob. 54PCh. 28 - Prob. 55PCh. 28 - Prob. 56PCh. 28 - Prob. 57PCh. 28 - Prob. 58PCh. 28 - Prob. 59PCh. 28 - Prob. 60PCh. 28 - Prob. 61PCh. 28 - Prob. 62PCh. 28 - Prob. 63PCh. 28 - Prob. 64PCh. 28 - Prob. 65PCh. 28 - Prob. 66PCh. 28 - Prob. 67PCh. 28 - Prob. 68PCh. 28 - Prob. 69PCh. 28 - Prob. 70PCh. 28 - Prob. 71PCh. 28 - Prob. 72PCh. 28 - Prob. 73PCh. 28 - Prob. 74PCh. 28 - Prob. 75PCh. 28 - Prob. 76PCh. 28 - Prob. 77PCh. 28 - Prob. 78PCh. 28 - Prob. 79PCh. 28 - Prob. 80PCh. 28 - Prob. 81PCh. 28 - Prob. 82PCh. 28 - Prob. 83PCh. 28 - Prob. 84PCh. 28 - Prob. 85PCh. 28 - Prob. 86PCh. 28 - Prob. 87PCh. 28 - Prob. 88P
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