Commutators

What's up I am working on some quantum chemistry problems and I am somewhat confused about commutators.

The question asks to calculate the commutators, the questions are:

[x,d/dx]
[sqrt,()^2]

There is also one concerning nuclear spins that you guys may not be able to help me with, but if I can see how to do these two I think I can figure that one out on my own.

Thanks for the help

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Quote:

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Originally Posted by csuiter

The question asks to calculate the commutators, the questions are:

[x,d/dx]
[sqrt,()^2]

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First, you have to ask what sort of objects these commutators act on.

The commutator [x,d/dx] has to act on a function f(x). Then $\displaystyle [x,\tfrac d{dx}]f(x) = x\tfrac d{dx}f(x) - \tfrac d{dx}\bigl(xf(x)\bigr) = xf'(x) - \bigl(f(x) + xf'(x)\bigr) = {\color{red}-}f(x)$. So [x,d/dx] is actually the negative of the identity operator: it takes the function f(x) to –f(x).

The commutator [√,^2] has to act on a nonnegative real number. You should be able to work out for yourself that this is the zero operator: it takes each number to 0.

thank you very much for your help!