Working with Bra and Ket vector notation

[rwooduk]

I have 2 ket vectors that i need to prove are orthogonal and normalised

I know <Up|Down> = <Down|Up> = 0 <---- Orthogonal Condition

I know <Up|Up> = <Down|Down> = 1 <---- Normalisation Condition

My problem is I have 2 ket vectors, say |A> and |B>, containing |Up> and |Down> terms.

How do I put the two ket vectors together? Do i simply put |A>|B> = And multiply the terms?

 

[CompuChip]

No, you put <A|B> (for which you first need to find |B> from <B|). Then use the fact that the inner product <.|.> is bilinear to split everything up in the basis combinations you know equal 0 or 1.

 

[rwooduk]

Thanks that helps a lot!

edit how do you turn a ket into a bra? I've been through all my lecture notes, the workshop questions, the suggested book and there's nothing. she does this everytime! sets questions we haven't covered and i spent half the day trying to find something vaguely relevant on the internet. found this

"To turn a ket into a bra for purposes of taking inner products, write the complex conjugates of its components as a row."

how am i supposed to take the complex conjugate of a spin up vector?

i'll just write the full thing, please don't give a direct answer, i just need an example of the method or a youtube video of the method to use please

|A> = 1/SQRT2 (|UP>+|DOWN>)

 

[rwooduk]

OK, i tried it this way:

|A> = 1/SQRT2 |UP> + 1/SQRT2 |DOWN> = (1 0) <----- A 2X1 MATRIX i.e. stood upright

<A| = (<A|^T)* <---- THE TRANSPOSE OF THE COMPLEX CONJUGATE = (1 0)

THERFORE

<A|A> = (1 0)(1 0) = 1 <---- WHERE (1 0) IS A 2X1 MATRIX i.e. stood upright

Therefore normalised

<A|B> = (1 0)(0 1) = 0 <---- WHERE (0 1) IS A 2X1 MATRIX i.e. stood upright

Therefore orthogonal

that look right? don't know what happens to the 1/SQRT2 though??!?

 

[retro10x]

when you take the complex conjugate of the Ket |A>, you take the conjugate of the 1/sqrt2 and flip the kets into bras. So |A>=(1/sqrt2)(|up>+|down>)
<A|=|A>*

<A|=(1/sqrt2)*(|up>+|down>)*=(1/sqrt2)*(<up|+<down|)=(1/sqrt2)(<up|+<down|)

where * denotes the complex conjugate, which doesn't affect real numbers