Modern Physics
Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Chapter 6, Problem 17E

(a)

To determine

The wave function of the reflected wave.

(b)
To determine

To verify: The ratio of reflected to incident probability density is 1

Formula Used:

The coefficient of reflected wave is

  2ik=(ikα)C

Calculation:

Now go back to the two conditions of a valid solution and eliminate the coefficient of the reflecred wave, B.

  2ik=(ikα)C

Solve the combined conditions for the coefficient of the decaying wave, C.

  C=2ik(ikα)   =2ik(ikα)(ikα)(ikα)   =2k22ikαk2+α2   

Use the coefficient of the decaying wave, C, the wave number of the incident wave, K, and the decay constant of the decaying wave, α , to complete the wave function, ψdee .

  ψdec=2k22ikαk2+α2eax

Substitute 2mE2for k2 and mE22for α2

  ψdec=(2mE2)2(mE2)i2mE2+mE22eax

Now go back to the two conditions of a valid solution and eliminate the coefficient of the reflected wave, B.

  2ik=(ikα)C

Solve the combined conditions for the coefficient of the decaying wave, C.

  C=2ik(ikα)  =2ik(ikα)(ikα)(ikα)  =2k22ikαk2+α2

Use the coefficient of the decaying wave, C, the wave number of the incident wave, K, and the decay constant of the decaying wave, α , to complete the wave function, ψdec .

  ψdec=2k22ikαk2+α2

Substitute 2mE2for k2 and mE22for α2

  ψdec=(2mE2)2(mE2)i2mE2+mE22eax

  =(4mE2)(2mE2)i5mE22eax=(8545i)eax

The ratio of the reflected probability density to the incident probability density can be found by diving the magnitude squared of the reflected wave, |ψref|2 by the magnitude squared of the incident wave, |ψinc|2 .

  |ψref|2|ψinc|2=[(35+45i)ei2me/ 2x][(3545i)ei2me/ 2x][ei2me/ 2x][ei2me/ 2x]          =(35+45i)(3545i)          =925+1625          =1

Conclusion:

The ratio of reflected probability density to the icidemt prpability density is 1

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