Integration of the gradient of a vector

[dyn]

Hi.
Is it true to say that the integral over all volume of ∇ψ where ψ is a scalar function of position and time is just ψ ?
Thanks

 

[BvU]

No

 

[dyn]

Thanks. I'm trying to integrate over all volume ψ^*∇ψ
When I integrate by parts I need to know the integral of ∇ψ. I assumed it was ψ . Can you tell me what it is ?
Thanks

 

[BvU]

Not so well-versed in telepathy, I'm afraid. So a bit of context might be helpful. My initial two-letter answer was more a feeble attempt at irony. But if I have to guess: you are in introductory quantum mechanics and looking at probability density and probability current density ?

Do you realize that, if $\ \psi(\vec r, t)\$ is a scalar, then $\ \nabla \psi(\vec r, t)\$ is a vector ? Meaning that $\ \displaystyle \int \psi^* \,\nabla \psi(\vec r, t)\ dV$ is also a vector and can therefore never be equal to the scalar $\psi$ ?

 

[dyn]

Yes your guess is correct. It is QM and showing that the expectation value of momentum is real.
[QUOTE="
Do you realize that, if $\ \psi(\vec r, t)\$ is a scalar, then $\ \nabla \psi(\vec r, t)\$ is a vector ? Meaning that $\ \displaystyle \int \psi^* \,\nabla \psi(\vec r, t)\ dV$ is also a vector and can therefore never be equal to the scalar $\psi$ ?[/QUOTE]
Yes I understand that ψ is a scalar and ∇ψ is a vector but to integrate by parts I need to know the integral of ∇ψ and assuming it is just ψ does end up giving me the correct answer in the end. How do I integrate ∇ψ over all volume ?
Thanks

 

[Chestermiller]

What is $\psi^*$ supposed to be? The complex conjugate of $\psi$?

 

[BvU]

Yes

 

[BvU]

showing that the expectation value of momentum is real

Well, if it has three components, maybe you might want to do it one component at the time ... ?


Another thing that comes to mind is Gauss' theorem -- probably a much better path to explore in the case you have already seen things like probability density ( $\rho = \psi^* \psi$ ) -- which is clearly real, and its conservation requirement $\left ( \ \displaystyle {\partial \rho \over \partial t } + \vec \nabla\cdot\vec j = 0 \right )$ .

Is this homework ? What does your textbook say ?

to integrate by parts I need to know the integral of ∇ψ and assuming it is just ψ does end up giving me the correct answer

Doesn't sound good at all to me, but I can be mistaken. Could you elaborate ? (Note that you don't integrate $\nabla\psi$ but $\psi^*\nabla\psi$...)

Normally integration by parts just lets you write (very useful) things like $\displaystyle {\int \bigl [ \psi^* \nabla \psi - (\nabla \psi^*)\psi\bigr ] \ dV = 2 \int \psi^* \nabla \psi \; dV}\ \$ -- where you use that $\psi^*\psi=0$ at infinity.

 

[dyn]

I am just trying to follow some lecture notes I found online and to be honest , I don't like them. As part of the integration by parts I need to know what the integral of ∇ψ is but that doesn't seem to be very straightforward

 

[eys_physics]

Presumably, it would help to post the link to the lecture notes. If you really need to integrate $\nabla\Psi$, my suggestion would be to do it component-wise.

 

[dyn]

To integrate ψ^*(∇ψ)over all volume by parts I need to know what the integral of ∇ψ is. Does the integral of ∇ψ not have a simple form ?
Obviously in 1-d its the integral of dψ/dx which is just ψ. Does the 3-d version not have a similar form ?

 

[eys_physics]

Yes, in cartesian coordinates you have
$$\nabla \psi = (\partial\psi/\partial x,\partial\psi/\partial y, \partial\psi/\partial z)=\frac{\partial\psi}{\partial x} e_x +\frac{\partial\psi}{\partial y} e_y+\frac{\partial\psi}{\partial z} e_z$$, where $e_x$, $e_y$ and $e_z$ are the basis vectors. So, you can do the integration term by term.

 

[dyn]

Yes, in cartesian coordinates you have
$$\nabla \psi = (\partial\psi/\partial x,\partial\psi/\partial y, \partial\psi/\partial z)=\frac{\partial\psi}{\partial x} e_x +\frac{\partial\psi}{\partial y} e_y+\frac{\partial\psi}{\partial z} e_z$$, where $e_x$, $e_y$ and $e_z$ are the basis vectors. So, you can do the integration term by term.

Thanks. It just all looks so messy. I was hoping that a simple elegant way existed to perform the integration by parts as it would in 1-d

 

[BvU]

some lecture notes I found online

Have a link for us ?