# [x,p] = xp - px = ih

## Main Question or Discussion Point

xp - px = x * (-i*hbar*d/dx) - (-i * hbar * d/dx (x)) = i * hbar

I can see that the px = i * hbar
But, why does xp = 0 ... ? I just get - i * hbar everytime I try, Wolfram Alpha agrees...

Can anyone show me why this isn't the case please?

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Doc Al
Mentor
I can see that the px = i * hbar
Really? Show how you got that.

The annoying thing about commutators is that, because they are operators, when you do it out explicitly you have to apply them to a function to make them work out, so apply [x,p] to f and see what happens.

Ok,

What I've been doing is just d/dx ( const * x ) = x
That works when we apply p to x. But not x to p I suppose.

Doc Al
Mentor
Ok,

What I've been doing is just d/dx ( const * x ) = x
That works when we apply p to x. But not x to p I suppose.
These are operator relationships. To calculate what you call "px", you need to evaluate:
$$\left( -i \hbar \frac{\partial}{\partial x} \right) x \psi$$
Note that you apply x first, then the derivative. So you'll need to use the product rule.

Bit aside, how arises I in p and x?

Fredrik
Staff Emeritus
Gold Member
Bit aside, how arises I in p and x?
It shows up when you apply the product rule for derivatives:
$$(\hat p\hat x f)(x)=(\hat p(\hat x f))(x) = -i(\hat x f)'(x) =-i\frac{d}{dx}(\hat x f(x)) =-i\frac{d}{dx}(xf(x))=\cdots.$$

Last edited:
dextercioby
Homework Helper
Bit aside, how arises I in p and x?
Have you heard about a Poisson bracket?

Hi,

Roughtly speaking, the derivative operator "p" acts over everything to the right and the wave function it's always implicitly the last term, so you really have:

pxΨ= i * hbar ∂x(xΨ) = i * hbar * [x * ∂x(Ψ) + Ψ * ∂x(x) ]

Where Ψ is the wave function in the evaluated point, implicitly, Ψ(x).

x(x) = 1 so the conmutation is [x,p]Ψ = (xp - px)Ψ = i*h_bar Ψ.

Best regards,
Sergio

bhobba
Mentor
Derivative is Ok, but what for I ? Moreover, how imaginary valued are connected with physical?

Fredrik
Staff Emeritus
There are many such connections. For example, $|\psi(x)|^2\Delta x$ is approximately equal to the probability that a particle detector near $x$ that's covering a region of size $\Delta x$ will detect a particle.
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