Dirac notation Schwarz Inequality Proof

[interdinghy]

Homework Statement

This isn't really a problem so much as me not being able to see how a proof has proceeded. I've only just today learned about Dirac notation so I'm not too good at actually working with it. The proof in the book is:

|Z> = |V> - <W|V>/|W|^2|W>

<Z|Z> = <V - ( <W|V>/|W|^2 ) W| V - ( <W|V>/|W|^2 ) W> =

<V|V> - ( <W|V><V|W>/|W|^2 ) - ( <W|V>*<W|V>/|W|^2 ) + ( <W|V>*<W|V><W|W>/|W|^4| ) >= 0

This last step has me completely lost, and if someone could explain it I'd really appreciate it. It looks like he's pulled out the <V| and |V> and combined them to get <V|V>, but I don't get where the minus sign comes from, or how he gets any of the rest of the equation, so I'm assuming I'm wrong about where the <V|V> comes from.

I'm deeply sorry about the notation here, I typed this on a phone or I would've used LaTeX.

Homework Equations

<Z|Z> >= 0

The Attempt at a Solution

Mentioned in the first section.

 

[kreil]

So you know that

<br /> |Z\rangle = |V\rangle - \frac{\langle W|V\rangle}{|W|^2} |W\rangle<br />

So what do you think \langle Z| equals?

 

[interdinghy]

So you know that

<br /> |Z\rangle = |V\rangle - \frac{\langle W|V\rangle}{|W|^2} |W\rangle<br />

So what do you think \langle Z| equals?

\langle Z| should just equal <br /> \langle V| - \frac{\langle W|V\rangle}{|W|^2} \langle W|<br />

right? So if I look at \langle Z|Z \rangle, I have to distribute, but I can't see how he's getting the terms he's getting, namely because I don't think I'm too comfortable with Dirac notation yet.

1: \langle V | V \rangle = \langle V | V \rangle obviously,

2: \langle V | - \frac{\langle W|V\rangle}{|W|^2} W \rangle seems like it should equal - \frac{\langle W|V\rangle \langle V|W \rangle}{|W|^2}

3: Probably the most confusing term for me:

\langle - \frac{\langle V|W \rangle}{|W|^2}W|V \rangle

This seems to me like it should just be - \frac{\langle V|W \rangle \langle W|V \rangle}{|W|^2} but this term doesn't appear anywhere in his final formulation really. I mean I guess \langle V|W \rangle is equivalent to his \langle W|V \rangle ^*, but why does he get that conjugate instead of the \langle V|W \rangle that I got?

4: then \langle - \frac{\langle W|V\rangle}{|W|^2} W|- \frac{\langle W|V\rangle}{|W|^2} W \rangle is the first term where I think I'm getting confused. For some reason the term he gets from this has a complex conjugate in it, which I assume means he's moving | - \frac{\langle W | V \rangle}{|W|^2} \rangle to the bra side, which I assume makes the top term a complex conjugate, leaving me with \langle - \frac{\langle W|V\rangle}{|W|^2}\frac{\langle W|V\rangle ^*}{|W|^2} W| W \rangle = \frac{ \langle W|V \rangle \langle W|V \rangle ^* \langle W|W \rangle}{|W|^4}

At the end, my final answer would look like: \langle V | V \rangle - \frac{\langle W|V\rangle \langle V|W \rangle}{|W|^2} - \frac{\langle V|W \rangle \langle W|V \rangle}{|W|^2} + \frac{ \langle W|V \rangle \langle W|V \rangle ^* \langle W|W \rangle}{|W|^4}

This seems like it may at least be equivalent to his formulation based on what I said in 3, but even this took kind of a mental leap for me in part 4 with moving a term over to the bra side. Is that a correct step? If so, why?

Thanks so much for any help. I'm losing sleep over this.

 

[kreil]

No problem, anyone that gets a science degree loses some sleep over a problem or two, so I know what boat you're in!

Does it help if I say that,

<br /> |Z\rangle = \langle Z|^{\dagger}<br />
<br /> \langle Z| = |Z\rangle ^{\dagger}<br />

and that,

<br /> \langle W |V \rangle ^{\dagger} = \langle W | V \rangle ^* = \langle V|W\rangle ...?<br />

(In other words, the inner product of a bra and a ket is a complex number, and the Hermitian conjugate of a complex number is simply its complex conjugate.)

This all means that,

<br /> \langle Z| = |Z\rangle^{\dagger} = \left( |V\rangle - \frac{\langle W|V\rangle}{|W|^2} |W\rangle \right) ^{\dagger} =\left( \langle V | - \frac{\langle W|V \rangle ^*}{|W|^2} \langle W| \right)<br />

So,

<br /> \langle Z | Z \rangle = \left( \langle V | - \frac{\langle W|V \rangle ^*}{|W|^2} \langle W| \right) \left( |V\rangle - \frac{\langle W|V\rangle}{|W|^2} |W\rangle \right)<br />
See also,
http://en.wikipedia.org/wiki/Bra–ket_notation#Inner_products_and_bras

 

[interdinghy]

Yes, that clears it up completely, thank you.

I can now actually see the part in the book where the asterisk denoting that the initial setup involved a complex conjugate has been worn away due to shoddy printing, so I feel a lot better about this whole thing in general now.

Thanks so much!

 

[fricke]

So you know that

<br /> |Z\rangle = |V\rangle - \frac{\langle W|V\rangle}{|W|^2} |W\rangle<br />

So what do you think \langle Z| equals?

Hmm.. I don't get why the proof starts with that |Z>
Mind to explain it, please? thanks :)

 

[Shephine Shaji]

Towards the end they consider, <W|V><V|W> = |<V|W>|, which I cannot understand. Please explain someone.