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(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 d²/dx?, with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1). %3D

(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 d²/dx?, with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1). %3D

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(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ĝ, 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 d2/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:(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ĝ, 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 d2/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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