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I Green's function and the evolution operator

  1. Jun 17, 2017 #1
    The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator:

    \begin{equation}

    \begin{split}

    \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0)

    \end{split}

    \end{equation}


    However, I have seen the following description of the Green's function (which contradicts the above definition):

    \begin{equation}

    \begin{split}

    \delta(\textbf{r}-\textbf{r}_0) &= \langle \textbf{r}| \textbf{r}_0 \rangle

    \end{split}

    \end{equation}


    Where ##\textbf{r}## is a 4-vector with components of 3-space ##\vec{x}## and 1-time ##t##:

    \begin{equation}

    \begin{split}

    \textbf{r}&= [\vec{x},t]

    \end{split}

    \end{equation}


    Such that, where ##\hat{U}(t,t_0)## denotes the time evolution operator, ##\hat{U}(t,t_0)\doteq e^{-2 \pi i \omega(t-t_0)}##:

    \begin{equation}

    \begin{split}

    \langle \textbf{r}| \textbf{r}_0 \rangle&=\langle \vec{x},t| \vec{x}_0,t_0 \rangle

    \\

    &=\langle \vec{x}|\hat{U}(t,t_0)| \vec{x}_0 \rangle

    \\

    &=\langle \vec{x}|e^{-2 \pi i \omega(t-t_0)}| \vec{x}_0 \rangle

    \end{split}

    \end{equation}


    Where the Green's function ##\hat{G}(\vec{x},t|\vec{x}_0,t_0)## is defined such that:

    \begin{equation}

    \begin{split}

    \hat{G}(\vec{x},t|\vec{x}_0,t_0) &\doteq \langle \vec{x}|e^{-2 \pi i \omega(t-t_0)}| \vec{x}_0 \rangle

    \end{split}

    \end{equation}


    What am I missing?
     
  2. jcsd
  3. Jun 18, 2017 #2

    hilbert2

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    There are Green's functions for many kinds of differential operators, all of them are based on the idea of solving a differential equation with an integral transform where the Green's function is the kernel. In quantum mechanics the Green function is called "propagator" and its physical interpretation is that it gives the probability of a particle moving from point ##\mathbf{x}_0## to point ##\mathbf{x}## during a time interval ##t - t_0##. It's also used when forming the path integral representation of quantum dynamics.

    So, the definition in (1) is that of a general Green's function while the latter equations describe the Green function in the special case of quantum time evolution.
     
  4. Jun 18, 2017 #3
    Yes; that is why I assign such importance to understanding Green's functions properly. Whatever the specific application of a Green's function, it should be consistent with the general definition. In this case, the latter definition is not consistent with the first. In definition (1): ##\hat{G}(\textbf{r},\textbf{r}_0) = \hat{L}_{\textbf{r}}^{-1} \delta(\textbf{r}-\textbf{r}_0)## while in the latter definition, ##\hat{G}(\textbf{r},\textbf{r}_0) = \delta(\textbf{r}-\textbf{r}_0) = \langle \vec{x} | e^{-2 \pi i \omega(t-t_0)}| \vec{x}_0 \rangle##.
     
  5. Jun 18, 2017 #4

    Orodruin

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    Because your differential operator is not an operator in space only. You also have the time variable that needs to be included.
     
  6. Jun 18, 2017 #5
    Yes; that is true, but I don't see how that solves the problem. Could you please explain?
     
  7. Jun 18, 2017 #6

    Orodruin

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    Have you thought anything about what operator your function is the Green's function for and what differential equation it should satisfy?
     
  8. Jun 18, 2017 #7
    My motive is to understand the mathematical foundations of the propagator.

    In my opinion, solutions of various differential equations via a Green's function approach are really just variations on a theme. For any application, the underlying mathematics should be rigorous and consistent, which brings me back to my original question....
     
  9. Jun 19, 2017 #8

    Orodruin

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    This really does not answer my question in #6. If you have not thought about it, I suggest that you do. If you have, what are your conclusions?
     
  10. Jun 20, 2017 #9
    Is this what you mean?

    Given a linear differential operator on ##\textbf{r}##, i.e., ##\hat{L}_{\textbf{r}}##, the Green's function ##\hat{G}(\textbf{r},\textbf{r}_0)## is defined such that:

    \begin{equation}

    \hat{L}_{\textbf{r}}\hat{G}(\textbf{r},\textbf{r}_0)=\delta(\textbf{r}-\textbf{r}_0)

    \end{equation}


    Thus:

    \begin{equation}

    \hat{L}_{\textbf{r}} u(\textbf{r})=f(\textbf{r})

    \end{equation}


    Where:

    \begin{equation}

    u(\textbf{r})= \int G(\textbf{r},\textbf{r}_0) f(\textbf{r}_0) d\textbf{r}_0

    \end{equation}


    Since:

    \begin{equation}

    \begin{split}

    \hat{L}_{\textbf{r}} u(\textbf{r})&=\hat{L}_{\textbf{r}}\int G(\textbf{r},\textbf{r}_0) f(\textbf{r}_0) d\textbf{r}_0

    \\

    &=\int \hat{L}_{\textbf{r}}G(\textbf{r},\textbf{r}_0) f(\textbf{r}_0) d\textbf{r}_0

    \\

    &=\int \delta(\textbf{r}-\textbf{r}_0) f(\textbf{r}_0) d\textbf{r}_0

    \\

    &=f(\textbf{r})

    \end{split}

    \end{equation}

    The Green's function can further considered in the context of the Fourier transform. I begin by noting the following:

    \begin{equation}

    \begin{split}

    \delta(\textbf{r}-\textbf{r}_0)&=\int_{-\infty}^{\infty} e^{2 \pi i \textbf{k}(\textbf{r}-\textbf{r}_0)} d\textbf{k}

    \end{split}

    \end{equation}



    And:

    \begin{equation}

    \hat{G}(\textbf{r},\textbf{r}_0)=\int_{-\infty}^{\infty} \hat{G}(\textbf{k},\textbf{r}_0) e^{2 \pi i \textbf{k}\textbf{r}} d\textbf{k}

    \end{equation}


    Such that:

    \begin{equation}

    \begin{split}

    \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0)

    \\

    \hat{L}_{\textbf{r}}\int_{-\infty}^{\infty} \hat{G}(\textbf{k},\textbf{r}_0) e^{2 \pi i \textbf{k} \textbf{r}} d\textbf{k}&= \int_{-\infty}^{\infty} e^{2 \pi i \textbf{k} (\textbf{r}-\textbf{r}_0)} d\textbf{k}

    \end{split}

    \end{equation}


    Given ##\hat{L}_{\textbf{r}}=\frac{\partial}{\partial \textbf{r}}##

    \begin{equation}

    \begin{split}

    \hat{L}_{\textbf{r}}\int_{-\infty}^{\infty} \hat{G}(\textbf{k},\textbf{r}_0) e^{2 \pi i \textbf{k} \textbf{r}} d\textbf{k}&=\frac{\partial}{\partial \textbf{r}} \left[\int_{-\infty}^{\infty} \hat{G}(\textbf{k},\textbf{r}_0) e^{2 \pi i \textbf{k} \textbf{r}} d\textbf{k}\right ]

    \\

    &=\int_{-\infty}^{\infty} \hat{G}(\textbf{k},\textbf{r}_0)\frac{\partial}{\partial \textbf{r}}\left [e^{2 \pi i \textbf{k} \textbf{r}}\right ] d\textbf{k}

    \\

    &=\int_{-\infty}^{\infty} (2 \pi i \textbf{k}) \hat{G}(\textbf{k},\textbf{r}_0)e^{2 \pi i \textbf{k} \textbf{r}} d\textbf{k}

    \end{split}

    \end{equation}


    Thus:

    \begin{equation}

    \hat{L}_{\textbf{r}} \rightarrow \hat{L}_{\textbf{k}}

    \end{equation}


    Where:

    \begin{equation}

    \hat{L}_{\textbf{k}} = 2 \pi i \textbf{k}

    \end{equation}


    Thus:

    \begin{equation}

    \int_{-\infty}^{\infty} (2 \pi i \textbf{k}) \hat{G}(\textbf{k},\textbf{r}_0)e^{2 \pi i \textbf{k} \textbf{r}} d\textbf{k}=\int_{-\infty}^{\infty} e^{2 \pi i \textbf{k}(\textbf{r}-\textbf{r}_0)} d\textbf{k}

    \end{equation}


    Such that:

    \begin{equation}

    \begin{split}

    (2 \pi i \textbf{k}) \hat{G}(\textbf{k},\textbf{r}_0)&= e^{-2 \pi i \textbf{k}\textbf{r}_0}

    \end{split}

    \end{equation}


    Where:

    \begin{equation}

    \begin{split}

    \hat{G}(\textbf{k},\textbf{r}_0)&= \frac{e^{-2 \pi i \textbf{k}\textbf{r}_0}}{(2 \pi i \textbf{k}) }

    \end{split}

    \end{equation}


    This can be generalized as follows:

    \begin{equation}

    \hat{L}_{\textbf{r}^n}=\frac{\partial^n}{\partial \textbf{r}^n}

    \end{equation}


    Such that:

    \begin{equation}

    \hat{L}_{\textbf{k}^n}=(2 \pi i \textbf{k})^n

    \end{equation}


    And:

    \begin{equation}

    \begin{split}

    \hat{G}(\textbf{k}^n,\textbf{r}_0) &= \frac{e^{-2 \pi i \textbf{k}\textbf{r}_0}}{(2 \pi i \textbf{k})^n}

    \end{split}

    \end{equation}


    Thus, the ##n##th derivative in position space transforms to multiplication by ##\textbf{k}^n## in wavenumber space, and the ##n##th integration in position space transforms to division by ##\textbf{k}^n## in wavenumber space.

    To emphasize, all of this is based on the understanding of the Green's function in definition 1, i.e., ##\hat{G}(\textbf{r},\textbf{r}_0) = \hat{L}_{\textbf{r}}^{-1} \delta(\textbf{r}-\textbf{r}_0) ##. Unfortunately, I still don't see how it answers my question. Can you be clearer in your guidance?
     
  11. Jun 21, 2017 #10

    Orodruin

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    I am talking about the other Green's function you described. What sort of problem do you use it to solve?

    Hint: Not all differential equations are differential equations in space only.
     
  12. Jun 21, 2017 #11
    The equations I presented are sufficiently general to work in Minkowski spacetime where ##\textbf{r} = [\vec{x},i t]##, such that in flat space, ##\textbf{r}^2 = \vec{x}^2-t^2##.



    Nevertheless, I think I understand your question, which interestingly, leads to the second part of question. I was going to save it for another thread, but no need now.


    I consider the case of a 3-space vector ##\vec{x}## that is a function of time, such that:

    \begin{equation}

    \begin{split}

    \vec{x}&\doteq \vec{x}(t)

    \\

    \vec{x}_0&\doteq \vec{x}(t_0)

    \end{split}

    \end{equation}


    I note the following:

    \begin{equation}

    \begin{split}

    \delta(\vec{x} - \vec{x}_0)&=\langle \vec{x} | \vec{x}_0 \rangle

    \end{split}

    \end{equation}


    Where:

    \begin{equation}

    \begin{split}

    \langle\vec{x}|\psi\rangle &=|\psi(\vec{x})\rangle

    \end{split}

    \end{equation}


    Such that:

    \begin{equation}

    \begin{split}

    \delta(\vec{x} - \vec{x}_0)&=\langle \vec{x} | \vec{x}_0 \rangle

    \\

    &=\sum_n \langle \vec{x} |\psi_n \rangle \langle \psi_n | \vec{x}_0 \rangle

    \\

    &=\sum_n \langle \psi_n(\vec{x}_0) | \psi_n(\vec{x}) \rangle

    \end{split}

    \end{equation}


    Furthermore, given ##\delta(\vec{x} - \vec{x}_0)= \delta(\vec{x}_0 - \vec{x})##:

    \begin{equation}

    \begin{split}

    \langle \vec{x}_0 | \vec{x} \rangle&=\langle \vec{x} | \vec{x}_0 \rangle

    \\

    \sum_n \langle \vec{x}_0 |\psi_n \rangle \langle \psi_n | \vec{x} \rangle&=\sum_n \langle \vec{x} |\psi_n \rangle \langle \psi_n | \vec{x}_0 \rangle

    \\

    \sum_n \langle \psi_n(\vec{x}) | \psi_n(\vec{x}_0) \rangle&=\sum_n \langle \psi_n(\vec{x}_0) | \psi_n(\vec{x}) \rangle

    \end{split}

    \end{equation}



    Assuming ##\hat{G}(\vec{x},t|\vec{x}_0,t_0)\doteq\langle \vec{x}| \hat{U}(t,t_0)\vec{x}_0\rangle##:

    \begin{equation}

    \begin{split}

    \hat{G}(\vec{x},t|\vec{x}_0,t_0)&\doteq\langle \vec{x}| \hat{U}(t,t_0) \vec{x}_0\rangle

    \\

    &=\sum_n \langle \vec{x}|\psi_n \rangle \langle \psi_n | \hat{U}(t,t_0)\vec{x}_0\rangle

    \\

    &=\sum_n \langle \vec{x}|\psi_n \rangle \hat{U}^{\dagger}(t,t_0)\langle \psi_n |\vec{x}_0\rangle

    \\

    &=\sum_n \langle \psi_n(\vec{x}_0) | \hat{U}^{\dagger}(t,t_0)| \psi_n (\vec{x})\rangle

    \\

    &=\sum_n \langle \psi_n(\vec{x}) | \hat{U}(t,t_0)| \psi_n (\vec{x}_0)\rangle

    \end{split}

    \end{equation}


    Given ##\psi(\vec{x})=\hat{U}(t,t_0)\psi(\vec{x}_0)##:

    \begin{equation}

    \begin{split}

    \sum_n \langle \psi_n(\vec{x}) | \hat{U}(t,t_0)| \psi_n (\vec{x}_0)\rangle&=\sum_n \langle \psi_n(\vec{x}) | \psi_n (\vec{x})\rangle

    \\

    &=\langle \vec{x}|\vec{x}\rangle

    \\

    &=\delta(\vec{x}-\vec{x})

    \\

    &=\delta(0)

    \\

    &=1

    \\

    &=\hat{G}(\vec{x},t|\vec{x}_0,t_0)

    \end{split}

    \end{equation}



    What am I missing?
     
  13. Jun 21, 2017 #12

    Orodruin

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    Your states ##|\vec x\rangle## are time independent (they are just a time independent basis of the Hilbert space) and build the state space (let us start working with regular QM), they do not depend on time. You can apply the time evolution operator ##U(t,t_0)## to the state ##|\vec x_0\rangle## to find out what the state is at a later time. This is what the Green's function tells you, the amplitude of evolving ##|\vec x_0\rangle## at time ##t_0## into ##|\vec x\rangle## at time ##t##. Note that ##\vec x## and ##\vec x_0## are not connected such that ##U(t,t_0) |\vec x_0\rangle = |\vec x\rangle## (furthermore ##\delta(0) \neq 1##). The Green's function in this case is the Green's function of the Schrödinger equation and satisfies
    $$
    (-i\partial_t + \hat H) G(\vec x,\vec x_0,t,t_0) = -i \delta(\vec x - \vec x_0) \delta(t-t_0),
    $$
    where ##\hat H## is the position space representation of the Hamiltonian (a bit depending on how you define the RHS of the equation). You can quite easily check that it satisfies this relation if you let ##G(\vec x,\vec x_0, t, t_0) = \langle \vec x|U(t-t_0)|\vec x_0\rangle \theta(t-t_0)## (this is the retarded Green's function and so it is zero if ##t_0 > t##).

    Note that
    $$
    \partial_t G(\vec x,\vec x_0, t,t_0) = \theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle + \delta(t-t_0) \langle \vec x|\vec x_0\rangle
    = \theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle + \delta(t-t_0) \delta(\vec x - \vec x_0).
    $$
    The second term here takes care of the inhomogeneity for the Green's function and the first enters the differential equation satisfied by the operator ##U(t-t_0)##.
     
  14. Jun 22, 2017 #13

    Sorry, dumb mistake stating ##\delta(0)=1##.


    A derivation of the connection between ##\vec{x}## and ##\vec{x}_0##:

    \begin{equation}

    \begin{split}

    \psi_{\vec{x}}(t)

    &=\hat{U}(t,t_0)\psi_{\vec{x}}(t_0)

    \\

    \langle \vec{x} | \psi \rangle &=\langle \vec{x}_{0} |\hat{U}(t,t_0) | \psi \rangle

    \end{split}

    \end{equation}


    Such that:

    \begin{equation}

    \begin{split}

    \langle \psi |\vec{x}\rangle &=\langle \psi|\hat{U}^{\dagger}(t,t_0) | \vec{x}_0 \rangle

    \end{split}

    \end{equation}


    Thus:

    \begin{equation}

    \begin{split}

    |\vec{x}\rangle &=|\hat{U}^{\dagger}(t,t_0) | \vec{x}_0 \rangle

    \end{split}

    \end{equation}




    Where is the mistake in this derivation?


    Additionally, I'm not clear on how you derived the following:

    \begin{equation}

    \begin{split}

    \partial_t G(\vec x,\vec x_0, t,t_0) &= \theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle + \delta(t-t_0) \langle \vec x|\vec x_0\rangle

    \end{split}

    \end{equation}


    Could you please be more explicit? Thanks.
     
  15. Jun 23, 2017 #14

    Orodruin

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    Unfortunately, there is not much that is right about it. You are being sloppy with the use of ##|\psi\rangle##. The ##|\psi\rangle## on the LHS represents the state at time ##t## and the one on the RHS the state at time ##t_0##. Furthermore, it is unclear how you start. Where does the ##x_0## come from in the first place?

    I suggest you consider the connection between the position states and a basis of a finite dimensional Hilbert space (ie, a system with a finite number of eigenstates).

    Which part is not clear? It is just the product rule for derivatives applied to the previous expression.
     
  16. Jun 27, 2017 #15
    I see your point on ##|\psi \rangle##.


    The derivative is still not clear to me. See the following:

    \begin{equation}

    \begin{split}

    \frac{\partial}{\partial t} \hat{G}(\vec{x},\vec{x}_0,t,t_0)&=\frac{\partial}{\partial t} \left[\theta(t-t_0) \langle \vec x|U(t-t_0)|\vec x_0\rangle\right]

    \\

    &=\frac{\partial}{\partial t} \left[\theta(t-t_0) \right ]\langle \vec x|U(t-t_0)|\vec x_0\rangle +\theta(t-t_0) \frac{\partial}{\partial t}\left[ \langle \vec x|U(t-t_0)|\vec x_0\rangle\right]

    \\

    &=\delta(t-t_0) \langle \vec x|U(t-t_0)|\vec x_0\rangle +\theta(t-t_0) \langle \vec x|\acute{U}(t-t_0)|\vec x_0\rangle

    \end{split}

    \end{equation}


    How does ##\delta(t-t_0) \langle \vec x|U(t-t_0)|\vec x_0\rangle = \delta(t-t_0) \delta(\vec{x}-\vec{x}_0)##?
     
  17. Jun 27, 2017 #16

    Orodruin

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    The delta function is non-zero only when ##t = t_0## and ##U(0) = 1##.
     
  18. Jun 28, 2017 #17
    Nice. Got it.


    A related question: Given the (non-retarded) Green's function ##\hat{G}(\vec{x},\vec{x}_0,t,t_0)=\langle \vec x|U(t-t_0)|\vec x_0\rangle##, is the following true?:

    \begin{equation}

    \begin{split}

    \langle \vec x|U(t-t_0)|\vec x_0\rangle&= U(t-t_0)\delta(\vec x-\vec x_0)

    \end{split}

    \end{equation}


    Or is it the following?:

    \begin{equation}

    \begin{split}

    \langle \vec x|U(t-t_0)|\vec x_0\rangle&= \delta(\vec x-U(t-t_0)\vec x_0)

    \\

    &=\delta(U^{\dagger}(t-t_0)\vec x-\vec x_0)

    \end{split}

    \end{equation}
     
  19. Jun 28, 2017 #18

    Orodruin

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    First, that is the retarded GF. Just without the heaviside function, which is fine as long as you assume ##t \geq t_0##.

    No, the relations you wrote down make no sense. ##U(t-t_0)## is an operator that acts on states in the Hilbert space. It seems you have trouble distinguishing between the states themselves, their representations, and the bases of those representations. I would suggest reading up on this because as it appears in this thread, you are only guessing.

    (Please take this the right way, it is not intended as any sort of personal insult, just an observation of something you seem to have problems with and would benefit from thinking more about instead of using guesswork.)
     
  20. Jul 2, 2017 #19
    I'm not insulted; I am admittedly much more comfortable in coordinate representation as opposed to Dirac notation; In any case, I appreciate your responses, and if you have any suggested references, I would gladly check them out.


    A quick question regarding ##\hat{H}## before I get to my larger question. I recall the following:

    \begin{equation}

    \begin{split}

    (-i \partial_t + \hat{H})G(\vec x,\vec x_0, t,t_0)&=-i \partial_t G(\vec x,\vec x_0, t,t_0)+ \hat{H} G(\vec x,\vec x_0, t,t_0)

    \\

    &=-i \delta(\vec x- \vec x_0)\delta(t-t_0)

    \end{split}

    \end{equation}


    Such that:

    \begin{equation}

    \begin{split}

    \hat{H} G(\vec x,\vec x_0, t,t_0)&=-i \delta(\vec x- \vec x_0)\delta(t-t_0)+i \partial_t G(\vec x,\vec x_0, t,t_0)

    \\

    i\hat{H} G(\vec x,\vec x_0, t,t_0)&= \delta(\vec x- \vec x_0)\delta(t-t_0)- \partial_t G(\vec x,\vec x_0, t,t_0)

    \end{split}

    \end{equation}


    Given ##\partial_t G(\vec x,\vec x_0, t,t_0) = \theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle + \delta(t-t_0) \delta(\vec x - \vec x_0)##

    \begin{equation}

    \begin{split}

    i\hat{H} G(\vec x,\vec x_0, t,t_0)&= -\theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle

    \\

    \hat{H} G(\vec x,\vec x_0, t,t_0)&= i\theta(t-t_0) \langle \vec x|U'(t-t_0)|\vec x_0\rangle

    \end{split}

    \end{equation}


    Is that correct regarding ##\hat{H}##?
     
  21. Jul 3, 2017 #20

    Orodruin

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    Yes, but it tells you nothing new. It is just a rewriting of the original equation and tells you that ##\langle\vec x | U(t-t_0)|\vec x_0\rangle## satisfies the Schrödinger equation for ##t > t_0##, which you already know as that was our starting assumption that gave us the solution ##G = \theta(t-t_0)\langle\vec x | U(t-t_0)|\vec x_0\rangle##.
     
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