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 and and I want to calculate l cross l.

I cannot write 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:

?

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).

Re: Cross product - non commutative algebra

Solved: Cross product - non commutative algebra

Re: Cross product - non commutative algebra

Quote:

Originally Posted by

**mathlover10** is this from Griffiths?

No, I am studying Noether's theorem (self study) and it delves into many areas of Physics.

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 .

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:

Since the cross product is associative we may regroup this as:

The term in the brackets is known so I'll just quote it:

Let's look at each term separately:

So

Now for the second term:

Finishing up we get:

Thus

There is a more physical basis of this result. is a pseudovector. That means that is a vector. But if we have then we are equating a vector with a pseudovector, a contradiction. Thus .

-Dan

Re: Cross product - non commutative algebra

Quote:

Originally Posted by

**topsquark** There is a more physical basis of this result.

is a pseudovector. That means that

is a vector. But if we have

then we are equating a vector with a pseudovector, a contradiction. Thus

.

Please disregard this statement. It is incorrect. (Doh) That's what I get for staying up and doing Physics at 3 in the morning!

-Dan

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).

Re: Cross product - non commutative algebra

Quote:

Originally Posted by

**Kiwi_Dave** 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. (Crying) 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

Re: Cross product - non commutative algebra

Quote:

Originally Posted by

**Kiwi_Dave** 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