Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
A(α x +α x )=α Ax +α Ax (5.28)
1 1 2 2 1 1 2 2
| i | i | i | i
for complex numbers α ,α and vectors x , x .
1 2 1 1
| i | i
Two special linear operators are the identity operator IV and zero operator
0, with properties
103
IV x = x (5.29)
| i | i
0x =0. (5.30)
| i
ThesubscriptVontheidentityoperatorisoftendroppedwhennoconfusion
can arise. The symbol 0 is somewhat overloaded, denoting ordinary complex
number 0, the zero vector, and the zero operator. This causes no confusion in
practice though. Note however,that 0 does not denote the zero vector!
| i
An operator A is invertible if there exists an inverse operator A 1 with the
−
property
A 1A=AA 1 =I. (5.31)
− −
Matrix representation
Linear operators on a vector space can be represented by matrices.
Let 1 , 2 ..., n be a basis (not necessarily orthonormal) for the vector
{| i | i | i}
space. Any vector x can be expanded as in (5.19),
| i
n
x = α i .
i
| i | i
Xi=1
Applying the linear operator A on both sides of the equation and using
linearity yields
n
Ax = α Ai .
i
| i | i
Xi=1
For each i, Ai is a vector in the vector space, and since 1 , 2 ..., n
| i {| i | i | i}
is a basis, there must exist complex numbers A for i,j =1,2,...,n such that
ji
Ai can be expanded
| i
n
Ai = A j . (5.32)
ji
| i | i
Xj=1
Inserting this into the previous equation gives
n n n n
Ax = α A j = ( A α )j .
i ji ji i
| i | i | i
Xi=1Xj=1 Xj=1 Xi=1
Representing vectors concretely as column vectors of expansion coefficients
as in equation (1.12) we see that the action of the linear operator A on compo-
nents of a vector x can be written as a transformation equation
| i
104
α A A A α
1 11 12 1n 1
···
α A A A α
 2  21 22 2n 2
. . . ··· . . . (5.33)
. . → . . . . . . . .
    
α  A A A α 
 n  n1 n2 ··· nn n
Already at this stage it is clear that the theory of transformations on vec-
tor spaces offers the potential for setting up a model of computation. Letting
the state of the computer be represented by the vector x , linear operators
| i
induces transitions between states of the computer. The details must of course
be elaborated, which is the subject of subsequent chapters.
If the vector space has an inner product and the basis is orthonormal there
is a convenient way to calculate the matrix elements A of the linear operator.
ji
Taking the inner product with a dual basis vector k on both sides of equation
h |
(5.32) and using orthonormality of the basis vectors gives
n n
k Ai = k A j = A k j
ji ji
h | | i h | | i h | i
Xj=1 Xj=1
n
= A δ =A ,
ji kj ki
Xj=1
that is
A = k Ai . (5.34)
ki
h | | i
This is a representation for the matrix elements that is very often used in
quantum mechanics.