PC Chapter 07

# PC Chapter 07 - 7 Tunneling Phenomena 7-1(a The reflection...

This preview shows pages 1–4. Sign up to view the full content.

103 7 Tunneling Phenomena 7-1 (a) The reflection coefficient is the ratio of the reflected intensity to the incident wave intensity, or ( 29 (29 - = + 2 2 1 21 1 i R i . But (29(29 -=- - =- +=+= 22 1 1 1 * 1 112 i ii i , so that = 1 R in this case. (b) To the left of the step the particle is free. The solutions to Schrödinger’s equation are ± ikx e with wavenumber = h 12 2 2 mE k . To the right of the step (29 = U xU and the equation is ( 29 (29 ψ ψ =- h 2 2 dm U Ex dx . With ψ - = kx xe , we find ψ ψ = 2 2 2 d kx dx , so that -  =   h 2 2 mUE k . Substituting = h 2 2 mE k shows that = - 1 E UE or = 1 2 E U . (c) For 10 MeV protons, = 10 MeV E and = 2 938.28 MeV m c . Using ( 29 - == h 15 197.3 MeV fm 1 fm 10 m c , we find ( 29 δ = = h 2 197.3 MeV fm 1 1.44 fm 2 2 938.28 MeV 10 MeV c k mE c . 7-2 (a) To the left of the step the particle is free with kinetic energy E and corresponding wavenumber = h 1 2 2 mE k : ψ - =+ 11 ikx x Ae Be 0 x To the right of the step the kinetic energy is reduced to - EU and the wavenumber is now - = h 2 2 2 mEU k ψ - i x Ce De 0 x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
104 CHAPTER 7 TUNNELING PHENOMENA with = 0 D for waves incident on the step from the left. At = 0 x both ψ and ψ d dx must be continuous: (29 ψ =+= 0 ABC ψ = -= 12 0 d ik A B ik C dx . (b) Eliminating C gives + =- 1 2 k AB k or  - =+   11 22 kk . Thus, ( 29 ( 29 ( 29 ( ( - - = == + + =-= + 2 2 2 2 1 1 4 1 B R A TR (c) As EU , 2 0 k , and 1 R , 0 T (no transmission), in agreement with the result for any energy < . For →∞ E , and 0 R , 1 T (perfect transmission) suggesting correctly that very energetic particles do not see the step and so are unaffected by it. 7-3 With = 25 MeV E and = 20 MeV U , the ratio of wavenumber is ( 29 = = -- 1 2 25 5 2.236 25 20 kE k . Then from Problem 7-2 - + 2 2 51 0.146 R and 1 0.854 . Thus, 14.6% of the incoming particles would be reflected and 85.4% would be transmitted. For electrons with the same energy, the transparency and reflectivity of the step are unchanged. 7-4 The reflection coefficient for this case is given in Problem 7-2 as ( ( ( ( - - = + + 2 2 2 1 1 B R A . The wavenumbers are those for electrons with kinetic energies = 54.0 eV E and - 54.0 eV 10.0 eV 64.0 eV : = - 1 2 54 eV 0.918 6 64 eV k . Then, ( ( - - = + 2 3 2 0.918 6 1 1.80 10 0.918 6 1 R is the fraction of the incident beam that is reflected at the boundary.
MODERN PHYSICS 105 7-5 (a) The transmission probability according to Equation 7.9 is (29 α  =+  -  2 2 1 1 sinh 4 U L TE EUE with [ ] α - = h 12 2 mUE . For << EU , we find ( 29 α h 2 2 2 2 1 mUL L by hypothesis. Thus, we may write α α≈ 1 sinh 2 L Le . Also -≈ UEU , giving αα +≈ 22 1 1 16 16 LL UU ee T E EE and a probability for transmission α - == 2 16 L E P T Ee U . (b) Numerical Estimates: ( 29 - h 34 1.055 10 Js 1) For - 31 9.11 10 kg m , - - 21 1.60 10 J UE , - = 10 10 m L ; [ ] α - - = h 81 2 5.12 10 m and α - = 2 0.90 L e 2) For - 31 9.11 10 kg m , - - 19 1.60 10 J , - = 10 10 m L ; α - 91 5.12 10 m and α - = 2 0.36 L e 3) For - 27 6.7 10 kg m , - - 13 1.60 10 J , - = 15 10 m L ; α - 1 41 4.4 10 m and α - = 2 0.41 L e 4) For = 8 kg m , -= 1 J , = 0.02 m L ; α - 3 3.8 10 m and α - =≈ 33 2 1.5 10 0 L 7-6 Equation 7.9 gives for the transmission probability α - 2 2 1 1 sinh 4 U L . For 0.1% transmission (29 = 0.001 , and the resulting equation must be solved for E using = 1 nm L and = 5 eV U . We adopt = E x U as the unknown and write α - = =- hh 2 2 2 2 1 m LUE mUL Lx and = -- 2 1 1 U E U E xx . For this case ( 29 ( × h 32 2 2 2 511 10 eV 5.00 eV 1 nm 2 11.46 197.3 eV nm

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 11/05/2009 for the course PHSX 313 taught by Professor Staff during the Fall '09 term at Kansas.

### Page1 / 18

PC Chapter 07 - 7 Tunneling Phenomena 7-1(a The reflection...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online