bluepilotg-2_07
- 9
- 0
- TL;DR Summary
- From line one to two in the image, the summations go from ##\int(\sum_{m=1}^\infty c_m\psi_m)* \, (\sum_{n=1}^\infty c_n\psi_n)dx## to ##\sum_{n=1}^\infty \sum_{m=1}^\infty c_m * c_n \int \psi_m * \psi_n dx##. Can someone explain why please.
I am self-studying quantum mechanics from Griffiths' textbook and some other sources. I have come across this derivation shown in the photo. I've taken all three major calculus courses for physics, linear algebra, ODE, PDE, Complex Analysis, etc.
However, I do still struggle with rules for summations. I do not understand why Griffiths goes from ##\int(\sum_{m=1}^\infty c_m\psi_m)* \, (\sum_{n=1}^\infty c_n\psi_n)dx## to ##\sum_{n=1}^\infty \sum_{m=1}^\infty c_m * c_n \int \psi_m * \psi_n dx##. I was under the impression that summations do not "multiply" in this way. I have not found another explanation online so far. Could someone kindly explain. Thanks.
However, I do still struggle with rules for summations. I do not understand why Griffiths goes from ##\int(\sum_{m=1}^\infty c_m\psi_m)* \, (\sum_{n=1}^\infty c_n\psi_n)dx## to ##\sum_{n=1}^\infty \sum_{m=1}^\infty c_m * c_n \int \psi_m * \psi_n dx##. I was under the impression that summations do not "multiply" in this way. I have not found another explanation online so far. Could someone kindly explain. Thanks.