- #1

- 532

- 2

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

This expression can be simplified using integration by-parts:

[tex]\frac{d\left\langle x \right\rangle}{dt}= - \frac{i\hbar}{2m}\int \left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial x}\Psi \right) dx[/tex]

(I used the fact that [tex]\partial x / \partial x = 1[/tex], and threw away the boundary term, on the ground that [tex]\Psi[/tex] goes to zero at (+/-) infinity.)

**My two questions**

1. I obtained the following intermediate form between these two equations:

[tex]\frac{d\left\langle x \right\rangle}{dt}= \frac{i\hbar}{2m} \left[ -\int \left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial x}\Psi \right) dx + x\left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial x}\Psi \right) \right|^{+\infty}_{-\infty} \right][/tex]

Is this correct?

EDIT: The second part doesn't quite make sense according to my current arguments. I will have to get back to you all. It was clear before I left my house but apparently not when I got home. Problems with Latex stole my focus. =)

2. Assuming the response is correct, how can the author make his claim that the second term equals 0?

[tex]x\left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial x}\Psi \right) = 0[/tex]

I ask this because of the following supposition:

If [tex] \Psi [/tex] and [tex] \Psi^{*} [/tex] are even functions, then [tex]\partial \Psi / \partial x[/tex] and [tex] \partial \Psi^{*} / \partial x [/tex] must be odd functions (unless there is some function that defies this rule(?)). Therefore, the products [tex] x\Psi \partial \Psi / \partial x [/tex] and [tex] x\Psi^{*} \partial \Psi / \partial x [/tex] are even. But if [tex] f \left( x \right) [/tex] if an even function, then

[tex]\int^{+a}_{-a} f \left( x \right) \neq 0[/tex]

always.

Mathematically, his conclusion only makes sense to me if [tex] \Psi [/tex] is of the general form

[tex] \Psi = A \psi \left( x \right) \psi \left( t \right) [/tex]

where

[tex] \psi \left( x \right) = e^{-ax^{n}} [/tex]

where

[tex] A [/tex], [tex] a [/tex], and [tex] n [/tex]

are constants.

From my experience with QM, this general form is a common description for particle wave-functions. Is his claim based on physical grounds, which is analogous to how the potential energy of a gravitational or electromagnetic field equals 0 at infinity and allows him to neglect the other mathematical functions that are in contradiction, such as [tex] \Psi = Ax [/tex] ?

Thanks in advance.