Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ộ, with the same eigenvalue q. Show that any linear combination of fand g is itself an eigenfunction of ô, 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 combinations of fand g that are orthogonal eigenfunctions on the interval (-1, 1).
Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ộ, with the same eigenvalue q. Show that any linear combination of fand g is itself an eigenfunction of ô, 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 combinations of fand g that are orthogonal eigenfunctions on the interval (-1, 1).
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Transcribed Image Text:Problem 3.7
(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ộ,
with the same eigenvalue q. Show that any linear combination of fand g is
itself an eigenfunction of ô, 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
combinations of fand g that are orthogonal eigenfunctions on the interval
(-1, 1).
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