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Orbital and Spin Angular momentum of light derivation

  1. Sep 5, 2015 #1
    1. The problem statement, all variables and given/known data
    The momentum carried by an electromagnetic field is [;\vec{P}(\vec{x}, t) = \frac{1}{4\pi c} \int d\vec{x}\vec{E}(\vec{x},t) \times \vec{B}(\vec{x},t);]

    show that for a finite field extension
    [;\vec{J}(\vec{x}, t) = \frac{1}{4\pi c}\int -i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t) + \vec{E}(\vec{x},t) \times \vec{A}(\vec{x},t) d\vec{x};] where [;\vec{J};] is the angular momentum, [;L = -ir \times \nabla;], and [;\vec{A};] is the vector potential in radiation gauge, i.e. [;A^0 = 0;] and [;\nabla \cdot \vec{A} = 0;] (hint: using partial integration)
    2. Relevant equations
    coulomb gauge [;\vec{B} = \nabla \times \vec{A};]

    3. The attempt at a solution
    There are some things in the wording in the problem that I do not understand so I am making some assumptions that may not even be true.

    1) Finite field extension means that the field does NOT go to infinite so [;\nabla \cdot \vec{E} = 0;]
    2) The subscripts in the [;-i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t);] represents a sum.
    3) I have no clue what partial integration is.
    [;\int \vec{x} \times (\vec{E} \times \vec{B}) d\vec{x};]
    [;\int \vec{x} \times (\vec{E} \times (\nabla \times \vec{A})) d\vec{x};]
    using triple product expansion
    [;\int \vec{x} \times (\nabla(\vec{E} \cdot \vec{A}) - \vec{A}(\vec{E}\cdot \nabla)) d\vec{x};]
    distributing the [;\vec{x} \times;]
    [;\int \vec{x} \times \nabla(\vec{E} \cdot \vec{A}) - \vec{x} \times \vec{A}(\vec{E}\cdot \nabla) d\vec{x};]

    This is where I get stuck. If I assume that [;\nabla \cdot \vec{E} = 0;], I have no idea where to get the [;\vec{E} \times \vec{A};] term in the answer I am supposed to get. If I don't assume that, I try to expand my [;\nabla(\vec{E} \cdot \vec{A});] term using some vector identity the [;\nabla \cdot \vec{E};] cancels anyway and I simply get the equation in my second line of work.

    My gut tells me that I missing something having to do with this "partial integration" business. I have googled this and I get it as an alternative term to integration by parts or as integrating a partial derivative. If I am to integrate by parts, what am I integrating by parts? I don't have a product of two functions I have different vector products of different vector fields. If I am supposed to integrate some partial derivative, which one? I have so many!

    please help, thanks!
     
  2. jcsd
  3. Sep 5, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Your TeX formulas get displayed properly if you enclose them in $$ or ## (inline mode). Currently they are hard to read.
     
  4. Sep 5, 2015 #3
    I can't seem to edit the post so I am re-posting here.

    1. The problem statement, all variables and given/known data
    The momentum carried by an electromagnetic field is $$\vec{P}(\vec{x}, t) = \frac{1}{4\pi c} \int d\vec{x}\vec{E}(\vec{x},t) \times \vec{B}(\vec{x},t)$$

    show that for a finite field extension
    $$\vec{J}(\vec{x}, t) = \frac{1}{4\pi c}\int -i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t) + \vec{E}(\vec{x},t) \times \vec{A}(\vec{x},t) d\vec{x}] where $\vec{J}$$ is the angular momentum, $$L = -ir \times \nabla$, and [\vec{A}$$ is the vector potential in radiation gauge, i.e. $$A^0 = 0$$ and $$\nabla \cdot \vec{A} = 0$$ (hint: using partial integration)
    2. Relevant equations
    coulomb gauge $$\vec{B} = \nabla \times \vec{A}$$

    3. The attempt at a solution
    There are some things in the wording in the problem that I do not understand so I am making some assumptions that may not even be true.

    1) Finite field extension means that the field does NOT go to infinite so $$\nabla \cdot \vec{E} = 0$$
    2) The subscripts in the $$-i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t)$$ represents a sum.
    3) I have no clue what partial integration is.
    $$\int \vec{x} \times (\vec{E} \times \vec{B}) d\vec{x}$$
    $$\int \vec{x} \times (\vec{E} \times (\nabla \times \vec{A})) d\vec{x}$$
    using triple product expansion
    $$\int \vec{x} \times (\nabla(\vec{E} \cdot \vec{A}) - \vec{A}(\vec{E}\cdot \nabla)) d\vec{x}$$
    distributing the $$\hat{x}\times$$
    $$\int \vec{x} \times \nabla(\vec{E} \cdot \vec{A}) - \vec{x} \times \vec{A}(\vec{E}\cdot \nabla) d\vec{x}$$

    This is where I get stuck. If I assume that $$\nabla \cdot \vec{E} = 0$$, I have no idea where to get the $$\vec{E} \times \vec{A}$$ term in the answer I am supposed to get. If I don't assume that, I try to expand my $$\nabla(\vec{E} \cdot \vec{A})$$ term using some vector identity the $$\nabla \cdot \vec{E}$$ cancels anyway and I simply get the equation in my second line of work.

    My gut tells me that I missing something having to do with this "partial integration" business. I have googled this and I get it as an alternative term to integration by parts or as integrating a partial derivative. If I am to integrate by parts, what am I integrating by parts? I don't have a product of two functions I have different vector products of different vector fields. If I am supposed to integrate some partial derivative, which one? I have so many!

    please help, thanks!
     
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