solo - Physics 741 Graduate Quantum Mechanics 1 Homework...

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Physics 741 – Graduate Quantum Mechanics 1 Homework Set O Due Wednesday, October 15 1. [10] Two particles interacting via a spherically symmetric relative potential have Hamiltonian () 22 12 HV mm =++ PP QQ Define new operators ( ) ( ) ( ) 1 1 2 2 1 2 21 1 2 1 2 m m m m ≡+ + + ≡− + PPP Q Q Q pP P q Q Q (a) [4] Find the commutation relations for these operators with each other, i.e. , find ,, , , a n d , ij i j QP Qp qP qp ⎡⎤ ⎣⎦ This is pretty straightforward, we simply write down each of the commutators and work them out: 11 2 2 1 2 1 1 1 2 2 2 2 2 2 1 1 2 2 2 1 , , , , , ii j j i j i j ij ij j j i j i j ij ij i mQ mQ P P m Q P i i m Pm P m Q P m Q P i i Q δ ++ + + == = = + −− + + = = + =− [] 2 1 2 2 2 , 0 , ,0 j i j i j jj QP P i i mm QP mQ m P mP δδ += =−= +− = (b) [6] Write the Hamiltonian in terms of these new operators, and in terms of the constants and Mmm μ =+ = + The potential term is trivial. For the kinetic term, we first note that 1 1 2 2 1 1 2 2 2 1 2 2 2 1 1 2 1 2 2 2 m m m m m m M m m m m m m M = + + = + = + −+ = + + = P P P P P P P p Pp P P P P P P P p
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We now substitute these in for each of the kinetic terms.
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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solo - Physics 741 Graduate Quantum Mechanics 1 Homework...

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