# Integration of the gradient of a vector

• I

## Main Question or Discussion Point

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
Homework Helper
2019 Award
No

• Chestermiller
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
Homework Helper
2019 Award
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 $\psi$ ?

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

BvU
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2019 Award
Yes

BvU
Homework Helper
2019 Award
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.

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

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.

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 ?

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.

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 Have a link for us ?