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Question from Griffiths book?

  1. Sep 22, 2013 #1
    In a normalization chapter theres an equation(1.21) which says: d/dt ∫|ψ(x,t)|[itex]^{2}[/itex]dx=∫∂/∂t |ψ(x,t)|[itex]^{2}[/itex]dx
    there was a description:(Note that integral is a function only of t,so I use a total derivative (d/dt) in the first expression,but the integrand is a function of x as well as t , so it's a partial derivative in the second one (∂/∂t) )
    so this text book started very simple and intuitive, but now I'm really confused.First of all why did d/dt appear and why did it "transform" to a partial derivative as it "entered" the integral?
     
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  3. Sep 22, 2013 #2

    vanhees71

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    The integral
    [tex]N(t)=\int_{\mathbb{R}} \mathrm{d} x \; |\psi(x,t)|^2[/tex]
    is a function of [itex]t[/itex] only, because you integrate over [itex]x[/itex]. Thus you can take only the derivative wrt. [itex]t[/itex].

    Now, because the integration range [itex]\mathbb{R}[/itex] doesn't change with time, you can as well take first the time derivative of the integrand and then integrate wrt. [itex]x[/itex]. Now, since you have a function of [itex]x[/itex] and [itex]t[/itex] you must indicate that you take the time derivative at fixed [itex]x[/itex]. That's why you have to use a partial derivative:
    [tex]\frac{\mathrm{d}N(t)}{\mathrm{d} t} = \int_{\mathbb{R}} \mathrm{d} x \frac{\partial}{\partial t} |\psi(x,t)|^2.[/tex]
     
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