Understanding operators for Green's function derivation

[askhetan]

Dear All,

I am trying to understand what operators actually mean when deriving the definition of green's function. Is this integral representation of an operator in the $x-basis$ correct ?

$D = <x|\int dx|D|x>$

I am asking this because the identity operator for non-denumerable or infinite dimensional basis sets is given as

$I = \int |x><x| dx$

But i cannot seem to use this knowledge directly. For eg. if i start by saying $D.f(x)=g(x)$, I can first write this in the bra-ket notation as:

$D |f> = |g>$

where $D$ is a differential operator. I can express the same, (given $D$ is invertible), as:

$|f> = D^{-1}|g>$

Now if i project all of them on the $x-basis$, I can simply write the same equation as:

$<x|f> = <x|D^{-1}|g>$

Now, I am allowed to introduce an Identity operator in the middle as:

$<x|f> = <x|D^{-1}|I|g> = <x|D^{-1}| \int dx'|x'><x'|g> = \int dx'<x|D^{-1}|x'><x'|g>$

Clearly $<x|f> = f(x)$, $<x'|g> = g(x')$ and I introduce $<x|D^{-1}|x'> = G(x,x')$. then I get:

$f(x) = \int dx' G(x,x') g(x')$

Now, the books say that to solve for $G(x,x')$, one can actually note that:

$D|G> = D<x|D^{-1}|x'> = \delta (x-x')$

... which is the step i couldn't understand! I know that $<x'|x> = <x|x'>= \delta (x-x')$, but how to reproduce these from the expression of the operator ? When does an integral kick in and when not ?

When i use my definition of the operator in terms of an integral, which i defined right at the top then I can get:

$D|G> = <x| \int D|x>dx <x|D^{-1}|x'> = <x|D \int dx |x><x| D^{-1}|x'> = <x|x'>$

Which will lead me to the delta function. But this was just cooked up by looking back. Whats the definition of an operator when expressed in terms on an integral? if this definition is incorrect (which it 100% is), how do I get to the last step that $D|G>= \delta (x-x')$

Please help me. I love operator mechanics until the matrix representation, but these integrals are doing my head in.

 

[jvicens]

Thank you for your question. Operators in quantum mechanics can be quite tricky to understand, especially when they are expressed in terms of integrals. Let me try to explain the steps you mentioned in your post.

First of all, the integral representation of an operator in the x-basis is correct. This is because in quantum mechanics, we often work with wavefunctions which are functions of position (x). So, an operator acts on the wavefunction to give us another wavefunction, which can be represented as an integral over the position basis.

Now, let's look at the identity operator for non-denumerable or infinite dimensional basis sets. The expression you have mentioned is correct, but it is not directly applicable in all cases. In your example, you have a differential operator D acting on a function f(x) to give another function g(x). In this case, you can use the identity operator as you have shown, but this may not always be the case. The identity operator is a useful tool, but it should be used with caution and only when it is applicable.

Next, let's look at the step where you introduce the identity operator in the middle. This step is valid because the identity operator is defined as the sum of all possible projections onto the basis states. So, when you write <x|I|g>, you are essentially projecting the state |g> onto the position basis, which gives you back the function g(x). Similarly, when you write <x|D^{-1}|I|g>, you are projecting the function g(x) onto the position basis and then applying the inverse of the operator D, which gives you back the function f(x).

Now, let's look at the step where you introduce the Green's function G(x,x'). The Green's function is defined as the inverse of the operator D. So, when you write <x|x'>, you are essentially projecting the state |x> onto the position basis and then applying the inverse of the operator D, which gives you back the delta function. This is how you get the expression D|G> = δ(x-x').

To summarize, the definition of an operator in terms of an integral is valid, but it should be used with caution and only when it is applicable. The identity operator is a useful tool, but it should be used only when it is applicable. And finally, the Green's function is defined as the inverse of the operator and it