Closure rule

1. Oct 11, 2008

evidenso

hey
Im having problem about closure rule
can anyone explain the closure rule?
why does it gives one

2. Oct 12, 2008

olgranpappy

you might want to elaborate on your question a little more... but an equally vague answer would be that the closure rule gives one because the states form a complete set.

3. Oct 12, 2008

evidenso

well it's stated as this
$$\sum{|r><r|}=I$$
I do understand a lot of QM but why is it gived as a summed product. how does bracket notation work in the sense?. what is the difference to $$\sum{<r|r>}$$. I cant picture it in my head.

4. Oct 12, 2008

olgranpappy

sorry. I tried to write a more complete post using TeX... but the forums are not letting me post it.

So... briefly:

<a|b> is an inner product in Dirac's notation. A number.

|a><b| is an "outer product". This is an operator (called a "dyadic"). It acts on states. For example,
the action on a state |c> is to produce a ket proportional to |a>, namely |a><b|c>.

To prove the expression for a complete set write an arbitraty ket |psi> in terms of a sum over the complete set {|r>}. The coefficient of each term in the sum can be rewritten in terms of the inner product of |psi> with |r>. Rearranging and noting that psi is arbitrary gives I=sum_r |r><r|