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Unformatted text preview: Midterm Examination February 11, 2008 Physics 115B, Fall 2008 E. Abers Midterm Examination with solutions and comments Question 1 [20 Points] A quantum mechanical system has just three linearly independent states, and the three vectors  ψ 1 ) ,  ψ 2 ) , and  ψ 3 ) are an orthonormal basis. A linear operator B is defined as follows: B  ψ 1 ) = i  ψ 2 ) B  ψ 2 ) = i  ψ 1 ) B  ψ 3 ) = 0 (0.0.1) Part a) What is the matrix representation for B in this basis? Part b) What are the eigenvalues of B ? SOLUTION Part a) The matrix representation is defined by B  ψ i ) = summationdisplay j B ji  ψ j ) or equivalently B ij = ( ψ i  B  ψ j ) so B 12 = ( ψ 1  B  ψ 2 ) = i ( ψ 1  ψ 1 ) = i etc. The matrix representation is  i i Note: Too many of you interchanged the rows and columns. And some of you found only two of the three eigenvalues. (A three by three Hermitean matrix always has three independent eigenvectors.) Part b) 0 = det( B λ ) = λ 3 λ = ( λ 1)( λ + 1) λ The eigenvalues are 1, 1, and 0. 2 Midterm Examination Question 2 [20 Points] A particle in three dimensions is in a state  ψ ) , whose wave function, written in spherical coordinates, is ψ ( r, θ, φ...
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 Winter '08
 Abers
 Physics, Linear Algebra, Energy, Midterm Examination, Hilbert space, possible values

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