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Vector Integral Notation/ Substition question Watch

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    (Apologies vector intergral may not be the terminology here)

    So I am looking at a subsituton of the form:  \int d^3 \vec{k} \to (\frac{L}{2\pi})^3 \int d^3 \vec{n}

    where  \vec{k}=\frac{2\pi}{L} \vec{n}

    I just want to check I have some definitions correct and why we need the factor cubed correct, since I've always took this integral notation for granted.

    So just to confirm that   d^3 \vec{k} = dk_x \hat{i} + dk_y \hat{j} + dk_z \hat{k} ?

    So therefore I have   d^3 \vec{k} = \frac{L}{2\pi}^dn_x \hat{i} + \frac{L}{2\pi}^dn_y \hat{j} + \frac{L}{2\pi}^dn_z \hat{k}

    Now explicitly, most likely this is a question/answer on notation, if I was to write it out in (unecessary ) detail, what would the next line(or two) be before getting:

     (\frac{L}{2\pi})^3  \int d^3 \vec{n}

    Many thanks in advance
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    (Original post by nologiceretoday)
    (Apologies vector intergral may not be the terminology here)

    So I am looking at a subsituton of the form:  \int d^3 \vec{k} \to (\frac{L}{2\pi})^3 \int d^3 \vec{n}

    where  \vec{k}=\frac{2\pi}{L} \vec{n}

    I just want to check I have some definitions correct and why we need the factor cubed correct, since I've always took this integral notation for granted.

    So just to confirm that   d^3 \vec{k} = dk_x \hat{i} + dk_y \hat{j} + dk_z \hat{k} ?
    This doesn't look at all right. You seem to be mixing up a volume element with the differential of a vector. It's like confusing dV = dx\ dy\ dz with d\vec{r}=dx \vec{i} + dy \vec{j} + dz \vec{k}

    I guess that you are trying to represent a volume element in k-space for a quantum particle in a 3D box of size L so you have:

    k_x=\dfrac{\pi n_x}{L}, k_y=\dfrac{\pi n_y}{L}, k_z=\dfrac{\pi n_z}{L}

    where the n_x come from the requirement that the wave function goes to 0 at the sides of the box, and then we would have:

    d^3 \vec{k} = dk_x\ dk_y\ dk_z = (\frac{\pi}{L})^3 dn_x\ dn_y\ dn_z

    So unless I'm confused, you seem to have the reciprocal of the correct constant, and an extra factor of 2 - that may come from your box, I guess, if you have -L <x,y,z < L
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    (Original post by atsruser)
    This doesn't look at all right. You seem to be mixing up a volume element with the differential of a vector. It's like confusing dV = dx\ dy\ dz with d\vec{r}=dx \vec{i} + dy \vec{j} + dz \vec{k}

    I guess that you are trying to represent a volume element in k-space for a quantum particle in a 3D box of size L so you have:

    k_x=\dfrac{\pi n_x}{L}, k_y=\dfrac{\pi n_y}{L}, k_z=\dfrac{\pi n_z}{L}

    where the n_x come from the requirement that the wave function goes to 0 at the sides of the box, and then we would have:

    d^3 \vec{k} = dk_x\ dk_y\ dk_z = (\frac{\pi}{L})^3 dn_x\ dn_y\ dn_z

    So unless I'm confused, you seem to have the reciprocal of the correct constant, and an extra factor of 2 - that may come from your box, I guess, if you have -L <x,y,z < L
    ok thank you I guess that would make more sense.

    But the notation is inconsistent then? As in we don't write [tex] d \vec{V} [/tex[ when we are concerned with this volume element.

    I wouldnt have thought you'd use the vector notation for any sort of volume element?
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    (Original post by nologiceretoday)
    But the notation is inconsistent then? As in we don't write [tex] d \vec{V} [/tex[ when we are concerned with this volume element.

    I wouldnt have thought you'd use the vector notation for any sort of volume element?
    I'm not exactly sure to what you're referring but here are two short threads on volume notation that may be helpful:

    https://math.stackexchange.com/quest...ntial-notation

    https://math.stackexchange.com/quest...ments-notation

    As far as I can tell, the notation for volume elements is pretty variable and non-standard, and differs between maths and physics, and in physics, between different branches.
 
 
 
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