Final answer:
Orthonormality of basis functions is demonstrated by verifying orthogonality, which means their overlap integral is zero, and normalization, which means the integral of each squared function over all space equals one. The example wave functions with a phase shift π are not orthogonal, and normalization is shown with the sine wave function and calculating the normalization constant Bn.
Step-by-step explanation:
To demonstrate that two basis functions are orthonormal, we use the concepts of orthogonality and normalization. Orthogonality means the integral of the product of the two functions over all space is zero, and normalization means that each function has an integral over all space that equals one. Let's consider the wave functions y1(x, t) = A cos (kx - wt) and y2(x, t) = A cos (kx - wt + π). Using trigonometric identities, we can combine these two to form a resulting wave function.
The first step is to examine the overlap integral of the two wave functions, which would be zero for orthogonal functions. However, in this case, due to the phase shift of π, y1 and y2 are not orthogonal, as the phase shift of π does not change the integral to zero. While these two waves share the same amplitude and frequency and differ only by a constant phase shift, their superposition does not result in zero (which one would expect from orthogonal functions) but rather in a phase-shifted wave.
For an example of normalization, consider the wave function Yn(x) = Bn sin(πnx/L), where L is the length of the region and n is a quantum number. To find the normalization constant Bn, we impose the normalization condition which, when evaluated, yields the correct value for Bn such that the integral of the square of Yn over the interval is one, fulfilling the normalization requirement.
Therefore, to show two basis functions are orthonormal, you need to verify both the orthogonality and the normalization conditions using integration over the appropriate limits. This involves evaluations of integrals for the product of the basis functions as well as their individual squared norms.