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Chapter 6: Q52P (page 1356)

The penetration distance\(\eta \)in a finite potential well is the distance at which the wave function has decreased to 1/e of the wave function at the classical turning point:

蠄(x=L+畏)=1e蠄(L)

The penetration distance can be shown to be

畏=魔2m(U0-E)

The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find \(\eta \) for an electron having a kinetic energy of 13 eV in a potential well with U0 = 20 eV. (b) Find畏for a 20.0 MeV proton trapped in a 30.0-MeV-deep potential well.

Short Answer

Expert verified
  1. The value of 畏 for an electron having a kinetic energy of 13 eV in a potential well with U0 = 20 eV is 7.4脳10-11m
  2. The value of for a 20.0-MeV proton trapped in a 30.0-MeV-deep potential well is1.44脳10-15m

Step by step solution

01

(a) Determination of the value of \(\eta \) for an electron having a kinetic energy of 13 eV in a potential well with U0 = 20 eV.

Use the formula given in the question,

畏=魔2mU0-E

For electron,

m=9.11脳10-31kg

Substitute all the standard values and values listed in question in the above equation,

畏=1.055脳10-34J路s29.11脳10-31kg20eV-13eV1.602脳10-19J/eV=7.4脳10-11m

Thus, the value of for electron is 7.4脳10-11m.

02

(b) Determination of the value of \(\eta \) for a 20.0-MeV proton trapped in a 30.0-MeV-deep potential well.

Use the formula given in the question,

畏=魔2mU0-E

For proton,

m=1.67脳10-27kg

Substitute all the standard values and values listed in question in the above equation,

畏=1.055脳10-34J路s21.67脳10-27kg30eV-20eV1.602脳10-19J/eV=1.44脳10-15m

Thus, the value of 畏for electron is 1.44脳10-15m.

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