- #1
- 716
- 162
Consider two momentum eigenstates ##\phi_1## and ##\phi_2## representing momenta ##p_1## and ##p_2##. For the sake of easy numbers, ##p_1=1*\hbar## (with ##k=1##) and ##p_2=2*\hbar## (with ##k=2##). Thus, ##\phi_1=e^{ix}## and ##\phi_2=e^{2ix}##. Orthogonality states that
##\int \phi_1^*\phi_2dx=\int e^{-ix}e^{2ix}dx=0##
Why is this?
I understand how orthogonality would work with dirac deltas (i.e. I know why position eigenfunctions are orthogonal in position space and why momentum eigenfunctions are orthogonal in momentum space, etc.) but I am unclear of how it works with plane waves.
Also, I am specifically asking why ##\int e^{-ix}e^{2ix}dx=0##, not why orthogonality works in general (I understand its derivation using the definition of hermitian operators and the inner product).
##\int \phi_1^*\phi_2dx=\int e^{-ix}e^{2ix}dx=0##
Why is this?
I understand how orthogonality would work with dirac deltas (i.e. I know why position eigenfunctions are orthogonal in position space and why momentum eigenfunctions are orthogonal in momentum space, etc.) but I am unclear of how it works with plane waves.
Also, I am specifically asking why ##\int e^{-ix}e^{2ix}dx=0##, not why orthogonality works in general (I understand its derivation using the definition of hermitian operators and the inner product).