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31 lines
877 B
Groff
31 lines
877 B
Groff
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3.7
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Suppose f and g are eigenfunctions of Q, with eigenvalue q. Show any linear combination of f and g are eigenfunctions of Q with eigenvalue q.
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|Qf> = q|f>
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|Qg> = w|g>
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A and B are (possibly complex) constants.
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|Q(A*f+B*g)> = A|Qf> + B|Qg> = Aq|f> + Bq|g>
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= q(A|f> + B|g>).
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QED
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Check that f(x)=exp(x) and g(x)=exp(-x) are eigenfunctions of the operator d^2/dx^2, with the same eigenvalue. construct two linear combiations of f and g that are orthogonal eigenfunctions on the interval [-1,1]
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if Q = d^2/dx^2 then we have the differential equation
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f'' = qf
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(e^x)'' = e^x, so this is an eigenfunction with q=1.
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(e^-x)'' = e^-x, so this is an eigenfunction with q=1.
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e^x + e^-x is also an eigenfunction of q, per earlier proof, and so is e^x - e^-x. These functions are just 2*sinh and 2*cosh, which are orthogonal functions because sinh is odd and cosh is even.
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