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Unformatted text preview: 2.6. HERMITIAN OPERATORS 13 Answer: By definition of Inv, and then using [1, p. 43]: Inv sin( kx ) = sin( − kx ) = − sin( kx ) Inv cos( kx ) = cos( − kx ) = cos( kx ) So by definition, both are eigenfunctions, and with eigenvalues − 1 and 1, respectively. 2.6 Hermitian Operators 2.6.1 Solution herm-a Question: A matrix A is defined to convert any vector vectorr = x ˆ ı + y ˆ into vectorr 2 = 2 x ˆ ı +4 y ˆ . Verify that ˆ ı and ˆ are orthonormal eigenvectors of this matrix, with eigenvalues 2, respectively 4. Answer: Take x = 1 , y = 0 to get that vectorr = ˆ ı transforms into vectorr 2 = 2ˆ ı . Therefore ˆ ı is an eigenvector, and the eigenvalue is 2. The same way, take x = 0 , y = 1 to get that ˆ transforms into 4ˆ , so ˆ is an eigenvector with eigenvalue 4. The vectors ˆ ı and ˆ are also orthogonal and of length 1, so they are orthonormal. In linear algebra, you would write the relationship vectorr 2 = Avectorr out as: parenleftBigg x 2 y 2 parenrightBigg = parenleftBigg 2 0 0 4 parenrightBiggparenleftBigg x y parenrightBigg = parenleftBigg 2 x 4 y parenrightBigg In short, vectors are represented by columns of numbers and matrices by square tables of numbers. 2.6.2 Solution herm-b Question: A matrix A is defined to convert any vector vectorr = ( x, y ) into the vector vectorr 2 = ( x + y, x + y ). Verify that (cos 45 ◦ , sin 45 ◦ ) and (cos 45 ◦ , − sin 45 ◦ ) are orthonormal eigenvectors of this matrix, with eigenvalues 2 respectively 0. Note: cos 45of this matrix, with eigenvalues 2 respectively 0....
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This note was uploaded on 11/13/2011 for the course PHY 4458 taught by Professor Garvin during the Fall '11 term at University of Florida.
- Fall '11