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Unformatted text preview: Solutions to Problem Set 1 Physics 341 by: Callum Quigley 1 Commuting Hermitian Operators To begin, we know that A is a Hermitian operator and so there exists a complete basis for V such that A is diagonal in this basis. Call this basis A = { a i n i , n N , i = 1 ,...,g n } , where g n labels the degeneracy of the nth eigenvalue. Then we have A  a i n i = a n  a i n i ,  a i n i A . (1.1) Before going further, lets establish some notation. Well write V n for the g ndimensional subspace of V with common eigenvalue a n , and denote the basis for this subspace as A n . Clearly, V = V V 1 ... with V i V j =  i for i 6 = j (1.2) A = A A 1 ... with A i A j = {} for i 6 = j. (1.3) Now fix a value of n . For any  a i n i A n , consider the vector B  a i n i . Because [ A,B ] = 0 it follows that A ( B  a i n i ) = BA  a i n i = a n ( B  a i n i ) (1.4) and so B  a i n i V n . Since this is true for all n , weve shown that B is blockdiagonal in the basis A , namely BV n V n . If g n = 1, then B is clearly diagonal in that 1 1 block. For g n > 1, B will, in general, not be diagonal within V n . However, we know that B is also Hermitian, so it too can be diagonalized. The key point is that within the subspace V n , A acts like the identity operator, A  V n = a n 1 , for any basis of V n and so it will always be diagonal. In particular, we can chose a new basis A n for each V n (while not disrupting the decomposition V = V V 1 ... ) such that B is diagonal in each subspace. In this way, we have constructed a basis for V such that both A and B are diagonal. 2 Unitary 2 2 Matrices Consider the general 2 2 matrix U with complex entries: U = a b c d , with a,b,c,d C . Unitarity requires that U 1 = U , and so this gives four constraints among the entries of U . Defining u = det U = ad bc , these constraints are a = u d * b = u c * (2.1) c = u b * d = u a * . (2.2) 1 If we combine the first and last of these (or equivalently the second and third), we see that  u  2 = 1 and so it is a pure phase. Lets write this as u = exp(2 i ). For the moment, lets consider the case where...
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 Spring '10
 Sghy
 Physics, mechanics

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