In class we deliberately avoided the general case... (Which left us under the impression we won't be required to go into such 'impressive mathematics')
It's not in the course book as well. Purcell only mentions something about some symmetry that's somehow responsible for this phenomenon without going into details.
סעיפים 7.1 -7.2. בהרצאה עשינו זאת בדרך של 7.2.