hw2add - PHYS 507 Answers to Homework II 1. (a) T ( a ) | p...

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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.

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hw2add - PHYS 507 Answers to Homework II 1. (a) T ( a ) | p...

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