Verify the commutation relations for x and p by definition.

[pdxautodidact]

Homework Statement

Verify $\left[ x^{i} , p_{k}\right] = i \hbar \delta^{i}_{k}$

Homework Equations

$p_{j} = -i \hbar \partial_{j}$

The Attempt at a Solution

Writing it out i get
$$ i \hbar \left( \partial_{k} x^{j} - x^{j} \partial_{k} \right)$$
The Kronecker makes perfect sense, it's identically zero unless k=j. Assuming it does, I arrive at:
$$ i \hbar \delta^{j}_{k} \left( \partial_{k} x^{j} - x^{j} \partial_{k} \right)$$

I assume I am missing something obvious, because most of the problem in this book are pretty straight forward, but this one's been a pain. I'm not doing any coursework, I already did my undergrad and am in limbo.

*Note Einstein convention is in use*
cheers.

 

[CompuChip]

You should apply the commutator to a test function f(x^k) (which is shorthand for f(x¹, x², x³) because k is a free index) and work out what the partials do with it.

 

[pdxautodidact]

You should apply the commutator to a test function f(x^k) (which is shorthand for f(x¹, x², x³) because k is a free index) and work out what the partials do with it.

I don't follow, do you mean what the term in parenthesis becomes when i=J? ($1- x \bullet \nabla$). The book's question specifically says verify the commutation relation using the definition of momentum given. Sorry, I should have been more precise maybe?

 

[CompuChip]

I mean that "$(\partial_k x^j - x^j \partial_k)$" by itself does not make sense. You should consider a test function f and work out what
$$(\partial_k x^j - x^j \partial_k)f$$
is, taking into account things like the product (Leibniz) rule.

I assume that by "use the definition of momentum" they just mean you should use $p_{j} = -i \hbar \partial_{j}$ as you have already done.

 

[pdxautodidact]

I mean that "$(\partial_k x^j - x^j \partial_k)$" by itself does not make sense. You should consider a test function f and work out what
$$(\partial_k x^j - x^j \partial_k)f$$
is, taking into account things like the product (Leibniz) rule.

I assume that by "use the definition of momentum" they just mean you should use $p_{j} = -i \hbar \partial_{j}$ as you have already done.

I understand that it works, the problem comes from a chapter titled "Lie Groups and Lie Algebras", so I assumed it was something more fundamental. Thank you.