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Math Help - Cross product - non commutative algebra

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    Cross product - non commutative algebra

    In quantum mechanics I am told that the cross product of angular momentum with itself is not zero and this is because r and momentum (p) do not commute.

    I have \underline l = \underline r \times \underline p and \underline p = (h/i) \nabla and I want to calculate l cross l.

    I cannot write \underline l \times \underline l = \hat x_i \epsilon_{ijk}r_jp_k because this definition of the cross product requires the algebra to be commutative

    Similarly I cannot use the determinant mnemonic for the cross product because it makes the same assumption?

    So how do I show that:

    \underline l \times \underline l = ih \underline l ?
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    Re: Cross product - non commutative algebra

    Hey Kiwi_Dave.

    Can you expand the definition by substituting in p = (h/i)del so you get l x l = (r x (h/i)del) x (r x (h/i)del) = r x -l x (h/i)del = -p x (h/i)del. -(h/i)*i/i = ih so you combine that to get ih(p x del). But what is the definition of angular momentum?

    (My physics is knowledge is very limited, but hopefully this can help you solve the problem).
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    Re: Cross product - non commutative algebra

    is this from Griffiths?
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    Solved: Cross product - non commutative algebra

    \underline l \times \underline l = (\underline r \times \underline p) \times (\underline r \times \underline p) = \underline a \times \underline b where:

    \underline b = (yP_z-zP_y)\hat x + (zP_x-xP_z)\hat y +(xP_y-yP_x)\hat z and

    \underline a = \frac hi \left[(y\frac \partial {\partial z}-z\frac \partial {\partial y})\hat x + (z\frac \partial {\partial x}-x\frac \partial {\partial z})\hat y +(x\frac \partial {\partial y}-y\frac \partial {\partial x})\hat z \right]

    So

    \underline l \times \underline l = \frac hi  \left| \begin{array}{ccc} \hat x & \hat y & \hat z \\ (y\frac \partial {\partial z}-z\frac \partial {\partial y}) & (z\frac \partial {\partial x}-x\frac \partial {\partial z}) & (x\frac \partial {\partial y}-y\frac \partial {\partial x})  \\ (yP_z-zP_y) & (zP_x-xP_z) & (xP_y-yP_x)\hat z \end{array} \right| So:

    \underline l \times \underline l = \frac hi [(zP_y-yP_z) \hat x - (zP_x-xP_z) \hat y + (yP_x-xP_y) \hat z]

    \underline l \times \underline l = \frac hi  \left| \begin{array}{ccc} \hat x & \hat y & \hat z \\ P_x & P_y & P_z \\ x & y & z \end{array} \right| =ih \underline l
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    Re: Cross product - non commutative algebra

    Quote Originally Posted by mathlover10 View Post
    is this from Griffiths?
    No, I am studying Noether's theorem (self study) and it delves into many areas of Physics.
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    Re: Cross product - non commutative algebra

    After a large amount of tedious work I have to say that I disagree with your final solution that \overline{L} \times \overline{L} = ih \overline{L}.

    I'll give you the relevant steps and leave it to you to work through the details to see if I have made a mistake.

    First:
    \overline{L} \times \overline{L} \propto \left ( \overline{r} \times \overline{ \nabla } \right ) \times \left ( \overline{r} \times \overline{ \nabla } \right )

    Since the cross product is associative we may regroup this as:
     = \left \[ \overline{r} \times \left ( \overline{ \nabla } \times \overline{r} \right )  \right \] \times \overline{ \nabla }

    The term in the brackets is known so I'll just quote it:
     = \left \[ \frac{1}{2} \overline{ \nabla } \left ( \overline{r} \cdot \overline{r} \right ) - \left ( \overline{r} \cdot \overline{ \nabla } \right ) \overline{r} \right \] \times \overline{ \nabla }

    Let's look at each term separately:
    \frac{1}{2} \overline{ \nabla } \left ( \overline{r} \cdot \overline{r} \right ) = \left ( x \hat{i} + y \hat{j} + z \hat{k} \right )

    So
    \left \[ \frac{1}{2} \overline{ \nabla } \left ( \overline{r} \cdot \overline{r} \right ) \right \] \times \overline{ \nabla } = \left ( x \hat{i} + y \hat{j} + z \hat{k} \right ) \times \overline{ \nabla }

     = L_x \hat{i} - L_y \hat{j} + L_z \hat{k}

    Now for the second term:
    \left \[ \left ( \overline{r} \cdot \overline{ \nabla } \right ) \overline{r} \right \] \times \overline{ \nabla }

     = \left \[  \left ( x \frac{ \partial }{ \partial x } + y \frac{ \partial }{ \partial y } + z \frac{ \partial }{ \partial z} \right ) \overline{r} \right \] \times \overline{ \nabla }

     = L_x \hat{i} - L_y \hat{j} + L_z \hat{k}

    Finishing up we get:
     = \left \[ \overline{r} \times \left ( \overline{ \nabla } \times \overline{r} \right )  \right \] \times \overline{ \nabla }  = \left ( L_x \hat{i} - L_y \hat{j} + L_z \hat{k} \right ) - \left ( L_x \hat{i} - L_y \hat{j} + L_z \hat{k} \right )

    Thus
    \overline{L} \times \overline{L} = 0

    There is a more physical basis of this result. \overline{L} = m \overline{r} \times \overline{p} is a pseudovector. That means that \overline{L} \times \overline{L} is a vector. But if we have \overline{L} \times \overline{L} = ih \overline{L} then we are equating a vector with a pseudovector, a contradiction. Thus \overline{L} \times \overline{L} = 0.

    -Dan
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    Re: Cross product - non commutative algebra

    Quote Originally Posted by topsquark View Post
    There is a more physical basis of this result. \overline{L} = m \overline{r} \times \overline{p} is a pseudovector. That means that \overline{L} \times \overline{L} is a vector. But if we have \overline{L} \times \overline{L} = ih \overline{L} then we are equating a vector with a pseudovector, a contradiction. Thus \overline{L} \times \overline{L} = 0.
    Please disregard this statement. It is incorrect. That's what I get for staying up and doing Physics at 3 in the morning!

    -Dan
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    Re: Cross product - non commutative algebra

    Thanks Dan

    First up I am getting malware warnings when visiting MHF. Google Crome suddenly refuses to come here.

    I have two "concerns" with your working. Probably just my understanding:

    1. You say that the cross product is associative. I can't find that written anywhere but find sites (like Cross Product Not Associative - ProofWiki) that say it is not associative.
    2. You have used an identity that I worry may only be valid when the multiplication commutes?



    The text I am working from says l cross l = ihl without any working, he just stated it as fact. That said I have found so many errors in this particular text that I don't trust anything that I cannot prove for myself (otherwise it is a great text).
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    Re: Cross product - non commutative algebra

    Quote Originally Posted by Kiwi_Dave View Post
    I have two "concerns" with your working. Probably just my understanding:

    1. You say that the cross product is associative. I can't find that written anywhere but find sites (like Cross Product Not Associative - ProofWiki) that say it is not associative.
    2. You have used an identity that I worry may only be valid when the multiplication commutes?



    The text I am working from says l cross l = ihl without any working, he just stated it as fact. That said I have found so many errors in this particular text that I don't trust anything that I cannot prove for myself (otherwise it is a great text).
    Oh dear. I never knew cross products weren't associative. You learn something new every day, I guess.

    Well, I guess that takes care of my derivation! Hmmm...I wonder if the assumption of associativity would imply commutivity? I'll have to look that one up. Thanks for the catch!

    -Dan
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    Re: Cross product - non commutative algebra

    Quote Originally Posted by Kiwi_Dave View Post
    First up I am getting malware warnings when visiting MHF. Google Crome suddenly refuses to come here.
    I know, I've had to turn off the malware warning in Chrome and trust Norton 360 to safeguard my computer. I don't know what's going on yet, nor how to fix it. Thanks for letting me know.

    -Dan
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