Why can’t you do integration-by-parts directly on the middle expression in Equation 1.29—pull the time derivative over onto x, note that ∂ x /∂ t = 0, and conclude that d(x) /dt = 0?

Question

Why can’t you do integration-by-parts directly on the middle expression in Equation 1.29—pull the time derivative over onto x, note that ∂ x /∂ t = 0, and conclude that [latex] d\left\langle x\right\rangle /dt = 0 [/latex]?.

[latex] \frac{d\left\langle x\right\rangle }{dt} = \int{x \frac{\partial}{\partial t} \left|\Psi \right|^2 } dx = \frac{i \hbar }{2m} \int{x \frac{\partial}{\partial x}\biggl(\Psi ^*\frac{\partial \Psi }{\partial x} - \frac{\partial \Psi ^*}{\partial x} \Psi \biggr) } dx [/latex] (1.29).

Accepted Answer

For integration by parts, the differentiation has to be with respect to the integration variable - in this case the differentiation is with respect to t, but the integration variable is x. It's true that

[latex] \frac{∂}{∂t} (x \left|\Psi \right|^2 ) = \frac{∂ x}{∂t} \left|\Psi \right|^2 + x \frac{∂}{∂t} \left|\Psi \right|^2 = x \frac{∂}{∂t} \left|\Psi \right|^2 [/latex].

but this does not allow us to perform the integration:

[latex] \int_{a}^{b}{x \frac{∂}{∂t}\left|\Psi \right|^2}dx = \int_{a}^{b} {\frac{∂}{∂t}(x \left|\Psi \right|^2 )} dx ≠ (x \left|\Psi \right|^2 )\mid ^b_a [/latex].

Question 1.6: Why can’t you do integration-by-parts directly on the middle...

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