mp-chapt-3-sol - Chapter 3. Problem Solutions 1. A photon...

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Inha University Department of Physics Chapter 3. Problem Solutions 1. A photon and a particle have the same wavelength. Can anything be said about how their linear momenta compare? About how the photon's energy compares with the particle's total energy? About how the photon’s energy compares with the particle's kinetic energy? Sol From Equation (3.1), any particle’s wavelength is determined by its momentum, and hence particles with the same wavelength have the same momenta. With a common momentum p , the photon’s energy is pc , and the particle’s energy is , which is necessarily greater than pc for a massive particle. The particle’s kinetic energy is 2 2 2 ) ( ) ( mc pc + ( 29 ( 29 2 2 2 2 2 mc mc pc mc E K - + = - = For low values of p ( p << mc for a nonrelativistic massive particle), the kinetic energy is K p 2 /2 m , which is necessarily less than pc. For a relativistic massive particle, K pc mc 2 , and K is less than the photon energy. The kinetic energy of a massive particle will always be less than pc, as can be seen by using E = ( pc ) 2 + ( mc 2 ) 2 to obtain . ) ( 2 2 2 2 Kmc K pc = -
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Inha University Department of Physics Chapter 3. Problem Solutions 3. Find the de Broglie wavelength of a 1.0-mg grain of sand blown by the wind at a speed of 20 m/s. Sol For this nonrelativistic case, m; 10 3 3 m/s) kg)(20 10 0 1 s J 10 63 6 29 6 34 - - - × = × × = = . . ( . mv h l quantum effects certainly would not be noticed for such an object. 5. By what percentage will a nonrelativistle calculation of the de Broglie wavelength of a 100-keV electron be in error? Sol Because the de Broglie wavelength depends only on the electron's momentum, the percentage error in the wavelength will be the same as the percentage error in the reciprocal of the momentum, with the nonrelativistic calculation giving the higher wavelength due to a lower calculated momentum. The nonrelativistic momentum is s, m kg 10 71 1 J/eV) 10 eV)(1.6 10 kg)(100 10 1 9 2 2 22 19 - 3 31 / . . ( × = × × × = = - - mK p nr and the relativistic momentum is ( 29 ( 29 m/s, kg 10 79 1 MeV 511 0 100 0 1 22 2 2 2 2 2 × = + = - + = - . / ) . ( . ( c mc mc K c p r
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Inha University Department of Physics Chapter 3. Problem Solutions 7. The atomic spacing in rock salt, NaCl, is 0.282 nm. Find the kinetic energy (in eV) of a neutron with a de Broglie wavelength of 0.282 nm. Is a relativistic calculation needed? Such neutrons can be used to study crystal structure. Sol A nonrelativistic calculation gives keeping extra figures in the intermediate calculations. The percentage error in the computed de Broglie wavelength is then %. . . . . / ) / ( ) / ( 8 4 71 1 71 1 79 1 = - = - = - nr nr r r r nr p p p p h p h p h ( 29 eV 10 03 1 m) 10 eV)(0.282 10 6 939 2 m) eV 10 24 1 2 2 2 3 2 9 - 6 2 6 2 2 2 2 2 2 - - × = × × × = = = = . . ( . ( ) ( / l
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mp-chapt-3-sol - Chapter 3. Problem Solutions 1. A photon...

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