Solving a Hermitian Problem: Showing that C is Hermitian

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In summary, the conversation discusses proving that C, a noncommuting quantum mechanical operator, is Hermitian. The approach involves isolating C and showing that it satisfies the equation for a Hermitian operator. The appropriate boundary conditions are assumed to be satisfied.
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Homework Statement



A and B are noncommuting quantum mechanical operators:
AB - BA = iC

Show that C is Hermitian. Assume all the appropriate boundary conditions are satisfied.

I do not understand how to show this. I isolated C as:

C = i(BA-AB)

..and I want to show that C is Hermitian showing:
[tex]\int\stackrel{a}{b}\psi^{*}C\psi d\tau[/tex]
is equal to:
[tex]\int\stackrel{a}{b}\psi(C\psi)^{*} d\tau[/tex]

Can someone help? Thanks!

 
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Homework Equations Hermitian:\int\stackrel{a}{b}\psi^{*}C\psi d\tau = \int\stackrel{a}{b}\psi(C\psi)^{*} d\tau The Attempt at a Solution I know that the equation above is true for Hermitian operators, but I do not know how to show this from the statement given.
 

FAQ: Solving a Hermitian Problem: Showing that C is Hermitian

What is a Hermitian problem?

A Hermitian problem is a mathematical problem that involves showing that a matrix or operator is Hermitian. A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the matrix is equal to its transpose with the complex conjugate of each element. A Hermitian operator is an operator that follows the same principle, where the operator is equal to its adjoint operator.

Why is it important to show that a matrix or operator is Hermitian?

Showing that a matrix or operator is Hermitian is important because it is a necessary condition for the matrix or operator to have real eigenvalues. Real eigenvalues are important in many applications, such as in quantum mechanics and signal processing, and can provide valuable information about the system being studied.

How do you determine if a matrix is Hermitian?

To show that a matrix is Hermitian, you must compare it to its conjugate transpose. This can be done by taking the complex conjugate of each element in the matrix and then transposing the resulting matrix. If the original matrix is equal to its conjugate transpose, then it is Hermitian.

What are the steps involved in solving a Hermitian problem?

The first step in solving a Hermitian problem is to determine if the matrix or operator is Hermitian. This can be done by comparing it to its conjugate transpose. If it is not Hermitian, the problem cannot be solved. If it is Hermitian, the next step is to find the eigenvalues and eigenvectors using standard methods. Finally, the eigenvalues and eigenvectors can be used to solve the problem and show that the matrix or operator is Hermitian.

Can a non-square matrix be Hermitian?

No, a non-square matrix cannot be Hermitian. A matrix must be square in order to have a conjugate transpose, which is necessary for it to be Hermitian. Non-square matrices may have similar properties, such as being skew-Hermitian, but they cannot be considered Hermitian.

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