$ \Psi^* \frac{\partial^2 \Psi}{\partial t \partial x} = -\frac{\partial \Psi^*}{\partial x} \frac{\partial \Psi}{\partial t} $

Question

I'm currently through a derivation in my introductory Quantum Mechanics course. I'm having trouble understanding the steps behind the following partial derivative.

For a function $\Psi (x,t)$ and its complex conjugate $\Psi^*(x,t)$,why is the following true? $$ \Psi^* \frac{\partial^2 \Psi}{\partial t \partial x} = -\frac{\partial \Psi^*}{\partial x} \frac{\partial \Psi}{\partial t} $$

In case I made some error, the full problem is:

$$ \frac{d\left < p \right >}{dt} = -i \hbar \int_{-\infty}^\infty \left ( \frac{\partial \Psi^*}{\partial t} \frac{\partial \Psi}{\partial x} + \Psi^* \frac{\partial^2 \Psi}{\partial t \partial x} \right ) dx \\ = \int_{-\infty}^\infty \left ( i\hbar \frac{\partial \Psi}{\partial t} \right )^* \frac{\partial \Psi}{\partial x} + \frac{\partial \Psi^*}{\partial x}\left ( i\hbar \frac{\partial \Psi}{\partial t} \right )dx$$

Thank you

Answer 1

The first equation you wrote is not correct, but the integral expression is, using integration by parts:

$$ \int_{-\infty}^{\infty} \Psi^* \frac{\partial^2\Psi}{\partial t\partial x} dx = -\int_{-\infty}^\infty \frac{\partial \Psi^*}{\partial x}\frac{\partial \Psi}{\partial t}dx $$ (Assuming the boundary terms vanish). The rest is algebra.

Answer 2

The first equality isn't necessarily true. You should integrate by parts the second summand of your first integral: $$ \frac{\partial}{\partial x}\left(\Psi^*\frac{\partial\Psi}{\partial t}\right)=\frac{\partial\Psi^*}{\partial x}\frac{\partial\Psi}{\partial t}+\Psi^*\frac{\partial^2\Psi}{\partial x\partial t}. $$ Assuming the first summand in this expression vanishes at $\pm\infty$ you get the result.