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Unformatted text preview: PHYS 507 Answers to Homework II 1. (a) T ( a )  p i = exp i ¯ h pa ¶  p i = exp i ¯ h p a ¶  p i . (b) There are different ways of doing this. This is a longer way. Start with ˜ φ ( p ) = h p  φ i = h p  T ( a )  ψ i . As a result, we first need to calculate h p  T ( a ). The dual complement of this expression is T ( a ) †  p i = T ( a )  p i = exp + i ¯ h p a ¶  p i , as a result h p  T ( a ) = exp i ¯ h p a ¶ h p  . This leads to ˜ φ ( p ) = exp i ¯ h p a ¶ h p  ψ i = exp i ¯ h p a ¶ ˜ ψ ( p ) . A shorter way: First expand the kets in terms of momentum eigenkes using the momentum space wavefunction.  ψ i = Z  p ih p  ψ i dp = Z ˜ ψ ( p )  p i dp ⇒  φ i = T ( a )  ψ i = Z ˜ ψ ( p ) T ( a )  p i dp = Z ˜ ψ ( p )exp i ¯ h p a ¶  p i dp ⇒ ˜ φ ( p ) = exp i ¯ h p a ¶ ˜ ψ ( p ) In any case, we see that the effect of translation in momentum space is a phase change where the amount of the phase depends on the particular value of mo mentum. An immediate conclusion is that the distribution of momentum does not change by translation, i.e.,  ˜ φ ( p )  2 =  ˜ ψ ( p )  2 . 2. (a) S ( μ 1 ) S ( μ 2 ) = S ( μ 1 + μ 2 ) and S ( μ ) † = S ( μ ). Since S ( μ ) S ( μ ) † = S ( μ ) S ( μ ) = S (0) = 1, S ( μ ) is a unitary operator. (b) These commutators will be useful for part (c). i ¯ h [ A,x ] = i 2¯ h [ px + xp,x ] = i 2¯ h ([ p,x ] x + x [ p,x ]) = i 2¯ h (2 ¯ h i x ) = x . i ¯ h [ A,p ] = i 2¯ h [ px + xp,p ] = i 2¯ h ( p [ x,p ] x + [ x,p ] p ) = i 2¯ h ( 2 ¯ h i p ) = p . 1 (c) Using shorthand notation B = iA/ ¯ h , we have S ( μ ) † = exp( μB ) and S ( μ ) † xS ( μ ) = x + μ [ B,x ] + μ 2 2! [ B, [ B,x ]] + ··· + μ n n ! [ B, [ B,... [ B,x ]]] + ··· From part (b) we can easily see that [ B, [ B,... [ B,x ]]] = x . Therefore S ( μ ) † xS ( μ ) = 1 + μ + μ 2 2! + ··· + μ n n ! + ··· ¶ x = e μ x . Similarly, S ( μ ) † pS ( μ ) = p + μ [ B,p ] + μ 2 2! [ B, [ B,p ]] + ··· + μ n n !...
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
 Spring '11
 starg

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