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Dirac notation. Bra-ket notation (Read 4571 times)
Gerrit-Jan Linker
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Dirac notation. Bra-ket notation
13.01.08 at 21:41:21
 
Dirac notation. Bra-ket notation
 
N-dimensional complex vectorial spaces can be described using the Dirac notation as invented by Paul Dirac. The notation is also called the bra-ket notation.
 
Let V be the N-dimensional complex vector space. Let |a> be any vector that is element of V and let us call it ket. As any vector can be described by a lineair combination of a basis set of kets, {|i>}N as:
 
|a> = a1|1> + a2|2> + a3|3> + ... + aN|N>
 
ai will be the component of the vector |a> in the direction |i>.
Note the sum is for i=1 to N; there are N dimensions and the basis set has N basis vectors, directions.  
 
Since V is a complex vector space, the adjoint of any vector in the space can be defined named bra: bra = <a| = (|a>)adjoint.
Correspondingly there exist a bra basis set {<i|}N in which <i| = (|i>)adjoint.
The bra vector <a| can be developed as a lineair combination of the bra basis set:
 
<a| = a*1<1| + a*2<2| + a*3<3| + ... + a*N<N|
 
The representations chosen as vectors in the chosen bra basis set or ket basis set is arbitrary. An infinite number of sets of N vectors can be chosen as basis sets.
 
A compact representation of a ket vector in a given basis set is a column vector:
a = (a1, a2, a3, ..., aN)T
Note that we need to transpose the row vector to make it a column vector. This is purely due to the notation. A column vector is written vertically and takes much space.
 
A compact representation of a bra vector in a given basis set is a row vector:
aadjoint = (a*1, a*2, a*3, ..., a*N)
 
See also:
Bra-ket notation
http://en.wikipedia.org/wiki/Bra-ket_notation
The adjoint of a vector
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1200239019
Dirac Notation
http://people.seas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u1/dirac_notation/ dirac_notation.html
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« Last Edit: 13.01.08 at 22:23:55 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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