ch13 - 1 CHAPTER 13 1(a electron-proton system mr me(1 5.45...

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CHAPTER 13 1. (a) electron-proton system m r = m e 1 + m e / M p = (1 - 5 .45 × 10 - 4 ) m e (b) electron-deuteron system m r = m e 1 + m e / M d = (1 - 2 .722 × 10 - 4 ) m e (c) For two identical particles of mass m , we have m r = m 2 2. One way to see that P 12 is hermitian, is to note that the eigenvalues ±1 are both real. Another way is to consider i , j dx 1 dx 2 ψ ij * ( x 1 , x 2 ) P 12 ψ ij ( x 1 , x 2 ) = i , j dx 1 dx 2 ψ ij * ( x 1 , x 2 ) ψ ji ( x 2 , x 1 ) = j , i dy 1 dy 2 ψ ji * ( y 2 , y 1 ) ψ ij ( y 1 , y 2 ) = dy 1 dy 2 ( P 12 ψ ij ( y 1 , y 2 ))* ψ ij ( y 1 , y 2 ) j , i 3. If the two electrons are in the same spin state, then the spatial wave function must be antisymmetric. One of the electrons can be in the ground state, corresponding to n = 1, but the other must be in the next lowest energy state, corresponding to n = 2. The wave function will be ψ ground ( x 1 , x 2 ) = 1 2 u 1 ( x 1 ) u 2 ( x 2 ) - u 2 ( x 1 ) u 1 ( x 2 ) ( 29 4. The energy for the n -th level is E n = h 2 π 2 2 ma 2 n 2 ε n 2 Only two electrons can go into a particular level, so that with N electrons, the lowest N /2 levels must be filled. The energy thus is E to t = 2 ε n 2 n = 1 N /2 2 ε 1 3 N 2 3 = ε N 3 12 If N is odd, then the above is uncertain by a factor of ε N 2 which differs from the above by (12/ N ) ε , a small number if N is very large. 5. The problem is one of two electrons interacting with each other. The form of the interaction is a square well potential. The reduction of the two-body problem to a one-particle system is straightforward. With the notation 1
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x = x 1 - x 2 ; X = x 1 + x 2 2 ; P = p 1 + p 2 , the wave function has the form ψ ( x 1 , x 2 ) = e iPX u ( x ) , where u ( x ) is a solution of - h 2 m d 2 u ( x ) dx 2 + V ( x ) u ( x ) = Eu ( x ) Note that we have taken into account the fact that the reduced mass is m /2. The spatial interchange of the two electrons corresponds to the exchange x - x .Let us denote the lowest bound state wave function by u 0 ( x ) and the next lowest one by u 1 ( x ). We know that
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