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Complex Conjugation

Here is a simple trick for computing the complex conjugate of complex expressions involving real variables using a replacement rule.

[Graphics:../Images/index_gr_1.gif]
[Graphics:../Images/index_gr_2.gif]

A few special superscripts are not interpreted as powers (see sections 2.8.15 and 3.10.2 of The Mathematica Book), such as

[Graphics:../Images/index_gr_3.gif]

[Graphics:../Images/index_gr_4.gif]

We can program SuperStar to compute the conjugate.

[Graphics:../Images/index_gr_5.gif]

Now conjugates can be computed using standard notation, as follows.

[Graphics:../Images/index_gr_6.gif]
[Graphics:../Images/index_gr_7.gif]

Since [Graphics:../Images/index_gr_8.gif] we can quickly and efficiently compute the absolute value of complex expressions involving real parameters.

[Graphics:../Images/index_gr_9.gif]
[Graphics:../Images/index_gr_10.gif]
[Graphics:../Images/index_gr_11.gif]

Similarly, here is a rule for computing the Hermitian (matrix) adjoint using SuperDagger.

[Graphics:../Images/index_gr_12.gif]

After introducing

[Graphics:../Images/index_gr_13.gif]

it is easy to show that the following general [Graphics:../Images/index_gr_14.gif] matrix,

[Graphics:../Images/index_gr_15.gif]
[Graphics:../Images/index_gr_16.gif]

is Hermitian.

[Graphics:../Images/index_gr_17.gif]
[Graphics:../Images/index_gr_18.gif]

We conclude by clearing the definitions associated with SuperDagger, SuperStar, [Graphics:../Images/index_gr_19.gif], [Graphics:../Images/index_gr_20.gif], and Subscript.

[Graphics:../Images/index_gr_21.gif]


Converted by Mathematica      September 30, 1999

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