- 125

- 1

**1. Homework Statement**

A is a non-Hermitian operator. Show that

[tex]i(A-A^t)[/tex]

is a Hermitian operator.

**2. Homework Equations**

[tex]\int \psi_1^*\L\psi_2 d\tau=\int (\L\psi_1)^*\psi_2 d\tau[/tex]

[tex]\int \psi_1^*A^t\psi_2 d\tau=\int (A\psi_1)^*\psi_2 d\tau[/tex]

**3. The Attempt at a Solution**

[tex]\int \psi_1^*i(A-A^t)\psi_2 d\tau[/tex]

[tex]=\int \psi_1^*iA\psi_2 d\tau + \int \psi_1^*(-iA^t)\psi_2 d\tau[/tex]

[tex]=\int (iA^t\psi_1)^*\psi_2 d\tau + \int ((-iA)\psi_1)^*\psi_2 d\tau[/tex]

[tex]=\int i((A-A^t)\psi_1)^*\psi_2 d\tau[/tex]

Is this right? The signs are wrong in the third line, but taking the i out of the complex conjugate brackets fix them. Can I do that?

Last edited: