Need Help Understanding Closure Rule

[evidenso]

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

mads

 

[olgranpappy]

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

mads

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.

 

[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 can't picture it in my head.

 

[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|