Feynman ammonia molecule states

[harpf]

I'm trying to follow Feynman's explanation on page 9-3 of Volume 3 of The Feynman Lectures on Physics. I've attached a copy of the section in question.

To normalize CII he notes that
< II | II > = < II | 1 >< 1 | II > + < II | 2 >< 2 | II > = 1
I am not clear how he derives the conclusion
CII = 1/√2 (C1 + C2)

I tried to solve the first equation unsuccessfully like this-
< II | 1 >< 1 | II > + < II | 2 >< 2 | II > = 1
CII C1 + CII C2 = 1
[1/√2 (C1 + C2)] C1 + {1/√2 (C1 + C2)} C2 = 1
[1/√2 (C1 + C2)] (C1 + C2) = 1
1/√2 (C1 + C2) (C1 + C2) = 1
which isn’t working for me.

Thank you for clarifying.

 

[M Quack]

The step from the first to the second line of your attempt is wrong. You've lost half of the matrix elements.

 

[harpf]

Thank you for your response, M Quack, which is clear and helpful.

If CII = < II | Φ >, I’m assuming I can substitute it for both < II | 1 > and < II | 2 >. You can see I’ve taken the same approach with C1 and C2. I suspect to go further I may have to use C1 = C2 and possibly C1*, which I am frankly weak on, and < 2 | 1 > = < 1 | 2 > = 1 / 2.

The source of my confusion may be a basic misunderstanding of state transitions and the meaning of the coefficients, C1, C2, and CII. I am really struggling to move forward. Thank you.

 

[M Quack]

<ii | 1> <1 | ii> = (cii^* c1) (c1^* cii)

= (1/√2 (c1+c2)^* c1) (c1^* 1/√2 (c1+c2)
= 1/2 (c1^* c1 + c2^*c1) (c1^*c1+c1^*c2)
= 1/2 (1 + 0) (1 + 0)

Do you see what I mean when I say you dropped half the matrix elements?

(Please don't ask me why preview insists on making everything lowercase)

 

[harpf]

That’s really helpful. The fog begins to lift. Thanks again.

I now realize the difference between < II | Φ >, which is CII, and < II | 1 >, which is the product of | 1 > and < II |.
Also, < II | = | II >* = CII*.

No, I can’t say I see what you mean about the matrix elements (although implying I was able to identify half of them seems generous).

 

[M Quack]

<II|i> is nothing else than a matrix element between the old and new wave functions. Maybe that is speaking a bit loosely as usually a matrix element is <x|M|x> where |x> is the wave function or state, and M some operator or matrix, depending on the notation.

 

[harpf]

I appreciate your help.