Teaching Quantum Mechanics

First Quantum Mechanics tutorial.

I will never claim to be a good public speaker. In fact, I'm an awful public speaker, and when faced with twenty strange youngsters, everything in my brain promptly flew out. Fortunately, they knew just about nothing about the subject, so I could randomly bullshit them. Perhaps not randomly, but with lots of handwaving and giving inaccurate answers.

With some time in the morning, I refresh my memory a bit on QM -- I haven't touched that subject in 8 years -- and tried to put together some notes for them. What I find troubling is that the questions they asked were conceptual questions that are based on another subject, linear (vector) spaces. I don't really think I can teach a course on linear spaces because that is some maths they should know already. However, they don't, so in order for me to give them an accurate answer, I have to use concepts from maths they don't have.

What am I supposed to do? Go into the math, and start teaching them the math, so I can give a proper answer? Give a handwaving answer?

For example, they want to know what the difference is between a bra and a ket, and when to use which.

What does the text mean when it says
r|a> + s |b>   <-> <a|r* + <b|s*

Originally, my braindead self went:
"Arrow can mean a lot of things...I think they are saying:

r|a> + s |b>   = <a|r* + <b|s*"

Here I even messed up by using * for conjugate instead of + for adjoint. Still, even with the correct symbol, "=" is not correct. However, in such courses, people will use

|x>* = <x|

There is no way you can answer "how is a bra different from a ket" this way. The real reason the arrow is used is because they are different "objects" in two twin spaces. Bras are operators on the kets. If you just consider a matrix, they are both matrices, but they represent different things.

For example, why is |a><b| different from <a|b>? Because a is an operator operating on b in the second case. But a is an operatee in the second case with b as the operator. They are actually 4 different objects! But if you can't talk about linear spaces, you can only do some kind of vague handwaving and say "well, they're all matrices, and matrices don't commute", which is absolutely false.

However, if you explain linear spaces, then that is one entire course itself!

At any rate, sorry for the profusion of math. I feel like dumping my thoughts out. I think I'll just tell them to disregard whatever I say in the first tutorial section and force some math on them. :P