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QuantumBunnii

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## Homework Statement

This is a much more general question regarding differential equations; however, since it was presented in a quantum mechanics text (and physicists often make appeals to empirical considerations in their mathematics), I thought it might be appropriate to post here.

The following is the first step in Griffith's endeavor to prove the constant normalization of an evolving wavefunction:

[itex] \frac{d}{dt} \int |ψ(x, t)|^2|\,dx [/itex] = [itex] \int \frac{\partial }{\partial t}|ψ(x, t)|^2|\,dx [/itex]

He notes "that the 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.

Firstly, I don't understand why he says that the intregral is a function only of t. To me, it seems that nothing in this equation is a function *only* of t, as the wavefunction is a function of x and t evaluated with respect to x.

Because of this, I also don't see what justifies going from a total derivative to a partial one. Consider the simple case of integrating the function [itex] ψ= xt[/itex]:

[itex] \frac{d}{dt} \int xt\,dx = \frac{d}{dt} \frac{1}{2} x^2 t + C = \frac{1}{2} (x^2 + \frac{dx^2}{dt}t) [/itex]

[itex] \int \frac{\partial }{\partial t}xt\,dx = \int x\,dx = \frac{1}{2} x^2 + C [/itex]

The two cases clearly aren't the same.

## Homework Equations

N/A

## The Attempt at a Solution

My thoughts are:

(1) Perhaps [itex] \frac{dx^2}{dt} [/itex] is simply constant with respect to time, which would in fact make the two cases equal. However, this doesn't seem very reasonable, as is it a common feat of quantum mechanics to calculate the average velocity of the particle.

(2) Maybe my attempt at pursuing the integral in the first case is flawed, as t is itself a variable which should not be ignored in the integral. This mistake may be what deprives the equality (of course, I'm not sure).

This may seem a very elementary or otherwise intuitive concern, but it's been bothering me for quite some time. Any help is appreciated. :)