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I'm trying to learn quantum physics (chemistry) on my own so that my work with Gaussian and Q-Chem for electronic structural modeling is less of a black box for me. I've reached the harmonic oscillator point in McQuarrie's Quantum Chemistry book and I'm having trouble justifying a step in his math. It's the integral of force with respect to x.

Integrate[m*(d2x/dt2), dx]

This says integrate the second derivative of time with respect to t for the integration variable x.

Changing the variable of integration to time gives:

Integrate[m*(d2x/dt2)*(dt/dt), dx]

or

Integrate[m*(d2x/dt2)*(dx/dt), dt]

Then something occurs here:

Integrate[m*(d(dx/dt)/dt)*(dx/dt), dt]

Integrate[(m/2)*d((dx/dt)*(dx/dt))/dt, dt]

Integrate[(m/2)*d((dx/dt)^2)/dt, dt]

What calculus rule have I forgotten that says that

(d2x/dt2)*(dx/dt)=(1/2)*d((dx/dt)^2)/dt

I can't seem to find it in any of my old text books or online. I hope that my notation is clear. I appreciate your help. If you can direct me to a web site that explains this rule, I'd appreciate it.

Thanks,

Heath

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# Question about a harmonic oscillator integral

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