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# mid - Physics 115A Midterm Harry Nelson Monday Feb 10 2003...

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Physics 115A Midterm Harry Nelson Monday, Feb. 10, 2003 Closed Book; no calculators. For full credit, show your work and make your reasoning clear to the graders. The ‘boldface’ notation below is used for operators; thus, Ω is an abstract operator. In class we put a ‘twiddle’ under the Ω to denote that it was an operator. The symbol . = means ‘is represented by’. The quadratic formula for the roots to the equation ax 2 + bx + c = 0 is: x = - b ± b 2 - 4 ac 2 a . 1. (25 pts) Two kets have unit length: | V 1 ) , and | V 2 ) , so ( V 1 | V 1 ) = ( V 2 | V 2 ) = 1; these two kets are never equal, that is, | V 1 ) negationslash = | V 2 ) . The two projection operators are P 1 = | V 1 )( V 1 | and P 2 = | V 2 )( V 2 | . (a) Suppose | V 1 ) and V 2 ) are represented in an orthonormal basis by: | V 1 ) . = bracketleftBigg 1 0 bracketrightBigg , | V 2 ) . = - radicalBig 1 3 radicalBig 2 3 . i. Find the matrices that represent P 1 and P 2 . ii. Use the matrix representations to find the matrix that represents the commu- tator [ P 1 , P 2 ].

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mid - Physics 115A Midterm Harry Nelson Monday Feb 10 2003...

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