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Question about a harmonic oscillator integral

  1. Jun 19, 2009 #1
    Hi,
    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
     
  2. jcsd
  3. Jun 19, 2009 #2
    If you work backwards it's easy to see: it's just the chain rule!

    [tex]\frac{d}{dt} \left(\frac{1}{2} \left( \frac{dx}{dt}\right)^2 \right) = 2 \times \left( \frac{1}{2} \frac{dx}{dt} \right) \times \frac{d}{dt} \frac{dx}{dt} = \frac{dx}{dt} \frac{d^2x}{dt^2}[/tex]

    Substituting [itex]\frac{dx}{dt} = f(t)[/itex] maybe helps:
    [tex]\frac{d}{dt} \left( \frac{1}{2} \left( f(t) \right)^2 \right) = 2 \times \left(\frac{1}{2} f(t) \right) \times \frac{df}{dt} = f(t) \frac{df}{dt}[/tex]

    Or even easier in 'words':
    The derivative of [itex]f^2[/itex] is [itex]2f \, f'[/itex]. Of course, we don't want the two, so we use 1/2 in front of the [itex]f^2[/itex] term (which doesn't change the differentiation process since it's a constant).


    It's a pretty common differentiation 'trick' .
     
    Last edited: Jun 19, 2009
  4. Jun 19, 2009 #3
    Thanks very much Nick. The chain rule! How embarrassing. :blushing:
     
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