Bra & Ket notation in quantum physics

[blueink]

Homework Statement

If |A> = x |B> + y |C> where |B> and |C> are orthonormal, then what happens when <A|A> ?

The Attempt at a Solution

Would <A| = x* <B| + y* <C|?

I'm not really sure where to go from there

 

[tiny-tim]

hi blueink! welcome to pf!

now expand (x* <B| + y* <C|)(x |B> + y |C>)

[and what is <B||B>? what is <B||C>?]

 

[mfb]

I'm not really sure where to go from there

Write down <A|A> with your <A|, and use the distributive property of the scalar product:
<X| (|Y>+|Z>) = <X|Y> + <X|Z>
Afterwards, simplify.

 

[blueink]

Expanding (x* <B| + y* <C|)(x |B> + y |C>)
=(x* <B|)(x |B>)+(x* <B|)(y |C>)+(y* <C|)(x |B>)+(y* <C|)(y |C>)
=x*x <B|B> + x*y <B|C> + y*x <C|B> + y*y<C|C>

Where <B|B> and <C|C> = 1 and <B|C> and <C|B> = 0?

 

[tiny-tim]

yup!

that's what's so beautiful about orthonormal systems!

(erm … you did mean orthonormal, and not just orthogonal? )

 

[blueink]

yup!

that's what's so beautiful about orthonormal systems!

(erm … you did mean orthonormal, and not just orthogonal? )

I did mean orthonormal, ahh confusion! thanks :)