hw solutions 4 - Physics 519 Homework Set#4 Due in class...

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Physics 519 Homework Set #4 Spring 2013 Due in class 5/8/13 300 pts 1. (100 pts) Many particles. a. Suppose you have three particles, and three distinct one-particle states ( ψ a ( x ), ψ b ( x ), and ψ c ( x )) are available. How many different three-particle states can be constructed (i) if they are distinguishable particles, (ii) if they are identical bosons, (iii) if they are identical fermions? (iii) For distinguishable particles, there are 3 × 3 × 3 = 27 distiguishable states, ψ i ( x 1 ) ψ j ( x 2 ) ψ k ( x 3 ) where i, j, k = a, b, or c . (ii) For identical bosons, the wave function must be symmetric under the exchange of any pair of boson coordinates. There are 3 possi- ble states where all 3 bosons are in the same single particle state: ψ j ( x 1 ) ψ j ( x 2 ) ψ j ( x 2 ) where j = a, b, c . These states are clearly symmetric under x i x j . Next, we consider when the cases when 2 particles are in the same single particle state, such as ψ a ( x 1 ) ψ a ( x 2 ) ψ k ( x 3 ) where k = b, c . This state is already symmetric under the exchange of x 1 and x 2 . We must make it symmetric under the exchange of x 1 and x 3 , and x 2 and x 3 . We do so by adding the same state with the labels ex- changed: ψ a ( x 1 ) ψ a ( x 2 ) ψ k ( x 3 )+ ψ a ( x 3 ) ψ a ( x 2 ) ψ k ( x 1 )+ ψ a ( x 1 ) ψ a ( x 3 ) ψ k ( x 2 ) . There are 6 such states, the 2 above for k = b, c and 2 each when the 2 particles are in the states b and c . Finally, we consider the case when the particles are all in different states, such as ψ a ( x 1 ) ψ b ( x 2 ) ψ c ( x 3 ) . We make this symmetric by adding all permutations of the 3 labels: ψ a ( x 1 ) ψ b ( x 2 ) ψ c ( x 3 )+ ψ a ( x 1 ) ψ c ( x 2 ) ψ b ( x 3 )+ ψ b ( x 1 ) ψ a ( x 2 ) ψ c ( x 3 )+ ψ b ( x 1 ) ψ c ( x 2 ) ψ a ( x 3 )+ ψ c ( x 1 ) ψ a ( x 2 ) ψ b ( x 3 ) + ψ c ( x 1 ) ψ b ( x 2 ) ψ a ( x 3 ) . There is only one such state.
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