Answer to Question #282970 in Quantum Mechanics for BIGIRIMANA Elie

Question #282970

If the real normalized functions f(x) and g(x) are not orthogonal, show that their sum f(x) +g(x) and their difference f(x)−g(x) are orthogonal. 


1
Expert's answer
2022-02-06T14:34:50-0500

If two functions are orthogonal, then their scalar product is equal to zero. Let's check:


"\\int_\\R(f(x)+g(x))(f(x)-g(x))dx = \\int_\\R f^2(x) -g^2(x)dx=\\\\\n=\\int_\\R f^2(x)dx -\\int_\\R g^2(x)dx"

As far as f(x) and g(x) are normalized,


"\\int_\\R f^2(x)dx=\\int_\\R g^2(x)dx = 1"

Thus, obtain:


"\\int_\\R(f(x)+g(x))(f(x)-g(x))dx = 1-1=0"

Hence, functions f(x) +g(x) and f(x)−g(x) are orthogonal.


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Comments

Alexandre
16.11.23, 20:13

Those work are so helpful !!!

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