Introduction to Quantum Physics and Information Processing

48 Introduction to Quantum Physics and Information Processing

We can now use the basis transformation equations 3.10 and write

ˆ

S

x

=

~

2



(|0i + |1i)

√

2

(h0| + h1|)

√

2

+

(|0i − |1i)

√

2

(h0| − h1|)

√

2



=

~

2

{|0ih1| + |1ih0|}

=

~

2

"

0 1

1 0

#

.

To specify an operator, one has to give its action on the computational

basis states. This will give the matrix representation of the operator in the

natural basis.

Exercise 3.5. Show that in the computational basis,

ˆ

S

y

=

~

2

"

0 −i

i 0

#

.

Exercise 3.6. Show by diagonalizing these representations of

ˆ

S

x

and

ˆ

S

y

, you get

the eigenvectors of Equations 3.10.

Exercise 3.7. Calculate the effect of

ˆ

S

x

and

ˆ

S

y

on the states |0iand |1i.

3.2.5 Functions of ...