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(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator 2, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q, with eigenvalue q.

(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator 2, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q, with eigenvalue q.

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Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d?/dx², with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).
Transcribed Image Text:Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d?/dx², with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).
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