What is the Hermitian Conjugate of 5+6i?

In summary, the Hermitian conjugate of a complex number is its complex conjugate. However, when it comes to matrices, the Hermitian conjugate is defined as the adjoint and is not the same as the complex conjugate. This is especially important in quantum mechanics for infinite-dimensional operators.
  • #1
yzou_ua
11
0
What is the Hermitian conjugate of a complex #, say, 5+6i??
 
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  • #2
As far as I know the name Hermitian conjugate is alternate name for the conjugate transpose of a matrix with complex entries. I think the only type of conjugate for a complex number is the regular one:

[tex]
\overline{5-6 i} = 5 + 6i
[/tex]
 
  • #3
"Hermitian conjugate" is usually used for matrices, not numbers. However, if you think of a+ bi as a "1 by 1 matrix" then its Hermitian conjugate is just its complex conjugate, a- bi.
 
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  • #4
statdad said:
As far as I know the name Hermitian conjugate is alternate name for the conjugate transpose of a matrix with complex entries.

Actually the Hermitian conjugate A* of an operator A is defined by

<x,Ay> = <A*x,y>

in other words the Hermitian conjugate is what mathematicians call an adjoint. It turns out that for finite dimensional operators (matrices) the Hermitian conjugate is simply equal to the transpose conjugate, as you state, but the more general definition is highly important in quantum mechanics, where the momentum operator is infinite-dimensional.
 

FAQ: What is the Hermitian Conjugate of 5+6i?

What is the Hermitian conjugate of 5-6i?

The Hermitian conjugate of 5-6i is 5+6i. It is the complex conjugate of a given complex number, where the imaginary part is multiplied by -1.

How do you find the Hermitian conjugate of a complex number?

To find the Hermitian conjugate of a complex number, simply change the sign of the imaginary part. If the complex number is in the form a+bi, the Hermitian conjugate is a-bi. If it is in the form a-bi, the Hermitian conjugate is a+bi.

What is the significance of Hermitian conjugates in quantum mechanics?

In quantum mechanics, Hermitian conjugates are used to calculate the expectation values of operators. They are also used to determine the probability amplitudes for physical observables.

Can the Hermitian conjugate of a real number be a complex number?

No, the Hermitian conjugate of a real number is always itself. Since real numbers do not have an imaginary part, there is no need for a complex conjugate.

What is the relationship between Hermitian conjugates and adjoint operators?

In mathematics, Hermitian conjugates can be thought of as the adjoint of a linear operator. The adjoint operator is the transpose of the operator's matrix representation, with the complex conjugates of the matrix elements. In quantum mechanics, the adjoint operator is used to find the Hermitian conjugate of a given operator.

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