What is this vector notation?

[egal]

I saw this |V> notation in the following text,


I don't understand these equations, what do I need to search to learn about this topic?

 

[topsquark]

I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.

So if we have a linear operator $$\Omega$$ and a scalar $$\alpha$$ then $$\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>$$ and the rest follows.

I should mention that in Quantum Mechanics $$\alpha$$ and $$\beta$$ could very well be complex numbers, in which case
$$( <V| \alpha ) \Omega = \alpha ^* <V| \Omega$$ so the second line might not be correct.

-Dan

Addendum: Sorry. Most of the LaTeX should be written in a single sentence. I don't know why it did this or how to fix it. But it's readable.

 

[Ted]

@topsquark Use imath instead of math for inline math. Or, using the f(x) icon in the toolbar, make sure to select inline.

 

[topsquark]

This is LaTeX test. No content has been changed.

I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.

So if we have a linear operator $\Omega$ and a scalar $\alpha$ then $\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>$ and the rest follows.

I should mention that in Quantum Mechanics $\alpha$ and $\beta$ could very well be complex numbers, in which case
$( <V| \alpha ) \Omega = \alpha ^* <V| \Omega$ so the second line might not be correct.

It works! Yay!

-Dan

 

[egal]

This is LaTeX test. No content has been changed.

I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.

So if we have a linear operator $\Omega$ and a scalar $\alpha$ then $\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>$ and the rest follows.

I should mention that in Quantum Mechanics $\alpha$ and $\beta$ could very well be complex numbers, in which case
$( <V| \alpha ) \Omega = \alpha ^* <V| \Omega$ so the second line might not be correct.

It works! Yay!

-Dan

I guess my math is not good enough to understand these, now I'm learning linear algebra then I'll look into these equations, thanks for your time

 

[topsquark]

The definition of a linear operator is pretty basic to Linear Algebra. If you haven't seen that in your course yet then just be patient... you'll get there. On the other hand once you've seen it, if you are still having problems please make sure you ask lots of questions because it shows up everywhere.

-Dan