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March 31

Dx - xD = 1, why does this not commute?

This equation: Dx - xD = 1 from the differential operator article says its basic to quantum physics. Obviously there is no commutativity otherwise Dx=xD, but I don't understand why. Any explanation? I've taken calculus and I would like to know what specific things I need to learn in order to understand this equation. --Idontnodi (talk) 02:03, 31 March 2013 (UTC)

To help others who wish to respond, here's a link to the relevant section of that article: Differential_operator#Properties_of_differential_operators. StuRat (talk) 02:13, 31 March 2013 (UTC)

• This is really nothing more than the product rule in action. Apply both sides to a function f(x) and then use the product rule; hey presto. Looie496 (talk) 03:19, 31 March 2013 (UTC)

How is that done? I do understand the product rule d(uv)=udv+vdu which is the notation I am accustom to. But I'm not familiar with the operator notation being used. Can someone rewrite this equation using Leibniz notation perhaps? Doesn't (d(y)/d(x)) x = x (d(y)/d(x))? I'm confused. --Idontnodi (talk) 03:42, 31 March 2013 (UTC)

Using the notation in the article and apply to f(x). D(x f(x)) - x D(f(x))= (Dx) * f(x) + x D(f(x)) - x D(f(x)) = (Dx) * f(x) = 1 * f(x). Or in you notation d(x * f) - x*d(f) = d(x)*f + x*d(f) - x*d(f)=d(x)*f=1*f.--Salix (talk): 04:33, 31 March 2013 (UTC)

According to these equations: Dx = 1 and xD = 0. Correct? --Idontnodi (talk) 05:27, 31 March 2013 (UTC)

They are both operators and need to be applied to a function. $Dx=\frac{d}{dx}(x*?)$, $xD=x*\frac{d}{dx}(?)$ where ? is replaced by some function. They don't have any particular value until the function is specified. If you apply them to the constant function 1 $Dx(1)=\frac{d}{dx}(x*1)=1$, $xD(1)=x*\frac{d}{dx}(1)=0$. --Salix (talk): 05:46, 31 March 2013 (UTC)

Thanks. --Idontnodi (talk) 08:03, 31 March 2013 (UTC)

Here's amother couple of operators to show how something like that can work. With $\mathrm{T}=(\times 2),\mathrm{S}=(+1)$, you can see that $(\times 2)(+1)f-(+1)(\times 2)f=1$. We could write that as $\mathrm{T}\mathrm{S}-\mathrm{S}\mathrm{T}=\mathrm{Constant}1$. In the original the 1 was the identity operator rather than producing the constant 1 when applied to any function. Dmcq (talk) 11:19, 31 March 2013 (UTC)

Not having ever used operators such as these, I almost wasted my time with that one. Thanks again!! --Idontnodi (talk) 15:04, 31 March 2013 (UTC)

This is perhaps what Dmcq is trying to say.

With $\mathrm{T}(x)=2x,\mathrm{S}(x)=1+x$, you can see that $(\mathrm{T}\mathrm{S}-\mathrm{S}\mathrm{T})x=2(1+x)-(1+2x)=1$.

Bo Jacoby (talk) 15:21, 31 March 2013 (UTC).