Hi, I'm doing a Quantum mechanics and one of my question is to determine if [tex]\frac{d^2}{dx^2}[/tex] (a second derivative wrt to x) is a Hermitian Operator or not.(adsbygoogle = window.adsbygoogle || []).push({});

An operator is Hermitian if it satisfies the following:

[tex]\int_{-\infty}^{\infty}\Psi^{*}A\Psi = \int_{-\infty}^{\infty}\left(A\Psi\right)^{*}\Psi[/tex]

where [tex] \Psi^{*} [/tex] is a complex conjugate of the wavefunction psi and A is the operator.

I do the LHS of the equation with A= [tex]\frac{d^2}{dx^2}[/tex]. I get:

[tex]\left[\Psi^{*} \frac{d}{dx}\Psi - \frac{d}{dx}\Psi^{*}\Psi\right]_{-\infty}^{\infty} + \int_{-\infty}^{\infty}\left(A\Psi\right)^{*}\Psi [/tex]

using integration by parts twice.

The second term obviously is the RHS, and in order for [tex]\frac{d^2}{dx^2}[/tex] to be Hermitian (and I'm pretty sure it is), the first term has to be equal to zero. Can anyone justify?

Thanks a bunch!

-Rick

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# Homework Help: Is (d^2/dx^2) a Hermitian Operator?

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