2.6. HERMITIAN OPERATORS13Answer: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.2.6Hermitian Operators2.6.1Solution herm-aQuestion:A matrixAis defined to convert any vectorvectorr=xˆı+yˆintovectorr2= 2xˆı+4yˆ. Verifythat ˆıand ˆare orthonormal eigenvectors of this matrix, with eigenvalues 2, respectively 4.Answer:Takex= 1,y= 0 to get thatvectorr= ˆıtransforms intovectorr2= 2ˆı.Therefore ˆıis aneigenvector, and the eigenvalue is 2. The same way, takex= 0,y= 1 to get that ˆtransformsinto 4ˆ, so ˆis an eigenvector with eigenvalue 4. The vectors ˆıand ˆare also orthogonal andof length 1, so they are orthonormal.In linear algebra, you would write the relationshipvectorr2=Avectorrout as:parenleftBiggx2y2parenrightBigg=parenleftBigg2004parenrightBiggparenleftBiggxyparenrightBigg=parenleftBigg2x4yparenrightBiggIn short, vectors are represented by columns of numbers and matrices by square tables of
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