Integral Notations in Quantum Mechanics












1












$begingroup$


I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the right side):
$$int f(x)dx$$

In this manner of notation, I can easily see the integrand as it is sandwiched by the integral sign and the $dx$.
However, I often see integrals written in this way (with $dx$ beside the integral sign):
$$int dx f(x)$$
Is this notation not ambiguous? This is especially confusing for me when used in products, as I cannot identify what is the integrand sometimes. For example, I don't understand which is true in the following (when evaluating time evolution operator): $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$



The last line is especially confusing for me as I'm not sure if the integrand changes. Could I please get clarification for these different notations? Is there a reason for such notation? (If I'm not wrong, it is to group the integrals and the integrands in separate places for convenience? I'm not sure if it sacrifices clarity for this though.)



EDIT: There is also an issue of when operators are involved:
$$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$
How do you know which operator is in the integrand? And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
    $endgroup$
    – Qmechanic
    1 hour ago


















1












$begingroup$


I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the right side):
$$int f(x)dx$$

In this manner of notation, I can easily see the integrand as it is sandwiched by the integral sign and the $dx$.
However, I often see integrals written in this way (with $dx$ beside the integral sign):
$$int dx f(x)$$
Is this notation not ambiguous? This is especially confusing for me when used in products, as I cannot identify what is the integrand sometimes. For example, I don't understand which is true in the following (when evaluating time evolution operator): $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$



The last line is especially confusing for me as I'm not sure if the integrand changes. Could I please get clarification for these different notations? Is there a reason for such notation? (If I'm not wrong, it is to group the integrals and the integrands in separate places for convenience? I'm not sure if it sacrifices clarity for this though.)



EDIT: There is also an issue of when operators are involved:
$$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$
How do you know which operator is in the integrand? And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
    $endgroup$
    – Qmechanic
    1 hour ago
















1












1








1





$begingroup$


I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the right side):
$$int f(x)dx$$

In this manner of notation, I can easily see the integrand as it is sandwiched by the integral sign and the $dx$.
However, I often see integrals written in this way (with $dx$ beside the integral sign):
$$int dx f(x)$$
Is this notation not ambiguous? This is especially confusing for me when used in products, as I cannot identify what is the integrand sometimes. For example, I don't understand which is true in the following (when evaluating time evolution operator): $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$



The last line is especially confusing for me as I'm not sure if the integrand changes. Could I please get clarification for these different notations? Is there a reason for such notation? (If I'm not wrong, it is to group the integrals and the integrands in separate places for convenience? I'm not sure if it sacrifices clarity for this though.)



EDIT: There is also an issue of when operators are involved:
$$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$
How do you know which operator is in the integrand? And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?










share|cite|improve this question











$endgroup$




I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the right side):
$$int f(x)dx$$

In this manner of notation, I can easily see the integrand as it is sandwiched by the integral sign and the $dx$.
However, I often see integrals written in this way (with $dx$ beside the integral sign):
$$int dx f(x)$$
Is this notation not ambiguous? This is especially confusing for me when used in products, as I cannot identify what is the integrand sometimes. For example, I don't understand which is true in the following (when evaluating time evolution operator): $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$



The last line is especially confusing for me as I'm not sure if the integrand changes. Could I please get clarification for these different notations? Is there a reason for such notation? (If I'm not wrong, it is to group the integrals and the integrands in separate places for convenience? I'm not sure if it sacrifices clarity for this though.)



EDIT: There is also an issue of when operators are involved:
$$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$
How do you know which operator is in the integrand? And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?







quantum-mechanics notation integration






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago







Hexiang Chang

















asked 1 hour ago









Hexiang ChangHexiang Chang

100211




100211












  • $begingroup$
    Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
    $endgroup$
    – Qmechanic
    1 hour ago




















  • $begingroup$
    Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
    $endgroup$
    – Qmechanic
    1 hour ago


















$begingroup$
Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
$endgroup$
– Qmechanic
1 hour ago






$begingroup$
Related: physics.stackexchange.com/q/200378/2451, math.stackexchange.com/q/387572/11127
$endgroup$
– Qmechanic
1 hour ago












2 Answers
2






active

oldest

votes


















3












$begingroup$

I started seeing $$int dx f(x)$$ in my freshman year of undergraduate.
It's pretty common and the more you learn about integration the more it makes sense.



Now, regarding this part:




$$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$




All of the equals signs there are correct.
Integrals factor like this:
$$
int dx int dy , f(x) , g(y) =
left( int dx , f(x) right) left( int dy ,g(y) right) , ,
$$

which is all you did there.
In fact, these are all the same:
begin{align}
int int dx , dy , f(x) g(y)
&= int dx int dy , f(x) g(y) \
&= int dx , dy , f(x) g(y) \
&= left( int dx , f(x) right) left( int dy , g(y) right) \
&= left( int dx , f(x) right) left( int dx , g(x) right) \
end{align}

Note, however, that you cannot factor something like this:
$$
int_0^t f(t') left( int_0^{t'} dt'' f(t'') right) dt'
$$

because the limit of the second integral depends on the first integral's integration variable.
You can, however, write it as
$$
int_0^t dt' f(t') int_0^{t'} dt'' f(t'') , .
$$



Operators




There is also an issue of when operators are involved:
$$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$




There's really no difference.
The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
By convention we tend to write the $dx$ either at the front or at the end.
I've never seen it written in the middle like that.
I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.




How do you know which operator is in the integrand?




Ok that's a good question!
It really comes down to the fact that notation has to be clear.
If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
For example, this makes no sense:
$$ g(x) = int_0^1 sin(x) dx$$
because there's no "free" $x$ on the right hand side.




And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?




Well, you certainly would not write
$$ int dx hat F(x) hat G(x) neq left( int dx hat F(x) right) hat G(x) , .$$
That just makes no sense.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you for a great answer! Could you please address my edit as well?
    $endgroup$
    – Hexiang Chang
    1 hour ago










  • $begingroup$
    I'm so sorry, I forgot that I needed to ask that as well
    $endgroup$
    – Hexiang Chang
    1 hour ago



















1












$begingroup$

The notation is not ambiguous; it's purely convention. The correspondence is



$$
left(
int_{t_1}^{t_2} H(t) dt right) left( int_{t'_1}^{t'_2}H(t',t)dt' right)
iff int_{t_1}^{t_2}int_{t'_1}^{t'_2}H(t)H(t',t)dt' dt.
$$



That is instead of evaluating "inside out" we evaluate the integrals from right to left.





If you square an integral as



$$ left(int ^t_{t_0} dt' H(t')right)^2 $$



you should know that in general these two integrals don't talk to one another, except in very special cases. That is, they are completely separate entities.






share|cite|improve this answer











$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "151"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: false,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: null,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f467758%2fintegral-notations-in-quantum-mechanics%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    I started seeing $$int dx f(x)$$ in my freshman year of undergraduate.
    It's pretty common and the more you learn about integration the more it makes sense.



    Now, regarding this part:




    $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$




    All of the equals signs there are correct.
    Integrals factor like this:
    $$
    int dx int dy , f(x) , g(y) =
    left( int dx , f(x) right) left( int dy ,g(y) right) , ,
    $$

    which is all you did there.
    In fact, these are all the same:
    begin{align}
    int int dx , dy , f(x) g(y)
    &= int dx int dy , f(x) g(y) \
    &= int dx , dy , f(x) g(y) \
    &= left( int dx , f(x) right) left( int dy , g(y) right) \
    &= left( int dx , f(x) right) left( int dx , g(x) right) \
    end{align}

    Note, however, that you cannot factor something like this:
    $$
    int_0^t f(t') left( int_0^{t'} dt'' f(t'') right) dt'
    $$

    because the limit of the second integral depends on the first integral's integration variable.
    You can, however, write it as
    $$
    int_0^t dt' f(t') int_0^{t'} dt'' f(t'') , .
    $$



    Operators




    There is also an issue of when operators are involved:
    $$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$




    There's really no difference.
    The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
    By convention we tend to write the $dx$ either at the front or at the end.
    I've never seen it written in the middle like that.
    I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.




    How do you know which operator is in the integrand?




    Ok that's a good question!
    It really comes down to the fact that notation has to be clear.
    If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
    It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
    For example, this makes no sense:
    $$ g(x) = int_0^1 sin(x) dx$$
    because there's no "free" $x$ on the right hand side.




    And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?




    Well, you certainly would not write
    $$ int dx hat F(x) hat G(x) neq left( int dx hat F(x) right) hat G(x) , .$$
    That just makes no sense.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Thank you for a great answer! Could you please address my edit as well?
      $endgroup$
      – Hexiang Chang
      1 hour ago










    • $begingroup$
      I'm so sorry, I forgot that I needed to ask that as well
      $endgroup$
      – Hexiang Chang
      1 hour ago
















    3












    $begingroup$

    I started seeing $$int dx f(x)$$ in my freshman year of undergraduate.
    It's pretty common and the more you learn about integration the more it makes sense.



    Now, regarding this part:




    $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$




    All of the equals signs there are correct.
    Integrals factor like this:
    $$
    int dx int dy , f(x) , g(y) =
    left( int dx , f(x) right) left( int dy ,g(y) right) , ,
    $$

    which is all you did there.
    In fact, these are all the same:
    begin{align}
    int int dx , dy , f(x) g(y)
    &= int dx int dy , f(x) g(y) \
    &= int dx , dy , f(x) g(y) \
    &= left( int dx , f(x) right) left( int dy , g(y) right) \
    &= left( int dx , f(x) right) left( int dx , g(x) right) \
    end{align}

    Note, however, that you cannot factor something like this:
    $$
    int_0^t f(t') left( int_0^{t'} dt'' f(t'') right) dt'
    $$

    because the limit of the second integral depends on the first integral's integration variable.
    You can, however, write it as
    $$
    int_0^t dt' f(t') int_0^{t'} dt'' f(t'') , .
    $$



    Operators




    There is also an issue of when operators are involved:
    $$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$




    There's really no difference.
    The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
    By convention we tend to write the $dx$ either at the front or at the end.
    I've never seen it written in the middle like that.
    I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.




    How do you know which operator is in the integrand?




    Ok that's a good question!
    It really comes down to the fact that notation has to be clear.
    If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
    It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
    For example, this makes no sense:
    $$ g(x) = int_0^1 sin(x) dx$$
    because there's no "free" $x$ on the right hand side.




    And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?




    Well, you certainly would not write
    $$ int dx hat F(x) hat G(x) neq left( int dx hat F(x) right) hat G(x) , .$$
    That just makes no sense.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Thank you for a great answer! Could you please address my edit as well?
      $endgroup$
      – Hexiang Chang
      1 hour ago










    • $begingroup$
      I'm so sorry, I forgot that I needed to ask that as well
      $endgroup$
      – Hexiang Chang
      1 hour ago














    3












    3








    3





    $begingroup$

    I started seeing $$int dx f(x)$$ in my freshman year of undergraduate.
    It's pretty common and the more you learn about integration the more it makes sense.



    Now, regarding this part:




    $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$




    All of the equals signs there are correct.
    Integrals factor like this:
    $$
    int dx int dy , f(x) , g(y) =
    left( int dx , f(x) right) left( int dy ,g(y) right) , ,
    $$

    which is all you did there.
    In fact, these are all the same:
    begin{align}
    int int dx , dy , f(x) g(y)
    &= int dx int dy , f(x) g(y) \
    &= int dx , dy , f(x) g(y) \
    &= left( int dx , f(x) right) left( int dy , g(y) right) \
    &= left( int dx , f(x) right) left( int dx , g(x) right) \
    end{align}

    Note, however, that you cannot factor something like this:
    $$
    int_0^t f(t') left( int_0^{t'} dt'' f(t'') right) dt'
    $$

    because the limit of the second integral depends on the first integral's integration variable.
    You can, however, write it as
    $$
    int_0^t dt' f(t') int_0^{t'} dt'' f(t'') , .
    $$



    Operators




    There is also an issue of when operators are involved:
    $$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$




    There's really no difference.
    The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
    By convention we tend to write the $dx$ either at the front or at the end.
    I've never seen it written in the middle like that.
    I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.




    How do you know which operator is in the integrand?




    Ok that's a good question!
    It really comes down to the fact that notation has to be clear.
    If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
    It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
    For example, this makes no sense:
    $$ g(x) = int_0^1 sin(x) dx$$
    because there's no "free" $x$ on the right hand side.




    And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?




    Well, you certainly would not write
    $$ int dx hat F(x) hat G(x) neq left( int dx hat F(x) right) hat G(x) , .$$
    That just makes no sense.






    share|cite|improve this answer











    $endgroup$



    I started seeing $$int dx f(x)$$ in my freshman year of undergraduate.
    It's pretty common and the more you learn about integration the more it makes sense.



    Now, regarding this part:




    $$begin{align}left(int ^t_{t_0} dt' H(t')right)^2stackrel{?}{=}int ^t_{t_0} H(t') dt'int ^t_{t_0} H(t'')dt''\stackrel{?}{=}int ^t_{t_0} dt' H(t')int ^t_{t_0} dt'' H(t'')\stackrel{?}{=}int ^t_{t_0} dt'int ^t_{t_0} dt'' H(t') H(t'') end{align}$$




    All of the equals signs there are correct.
    Integrals factor like this:
    $$
    int dx int dy , f(x) , g(y) =
    left( int dx , f(x) right) left( int dy ,g(y) right) , ,
    $$

    which is all you did there.
    In fact, these are all the same:
    begin{align}
    int int dx , dy , f(x) g(y)
    &= int dx int dy , f(x) g(y) \
    &= int dx , dy , f(x) g(y) \
    &= left( int dx , f(x) right) left( int dy , g(y) right) \
    &= left( int dx , f(x) right) left( int dx , g(x) right) \
    end{align}

    Note, however, that you cannot factor something like this:
    $$
    int_0^t f(t') left( int_0^{t'} dt'' f(t'') right) dt'
    $$

    because the limit of the second integral depends on the first integral's integration variable.
    You can, however, write it as
    $$
    int_0^t dt' f(t') int_0^{t'} dt'' f(t'') , .
    $$



    Operators




    There is also an issue of when operators are involved:
    $$begin{align}int dx hat{F}(x) hat{G}(x)stackrel{?}{=}int hat{F}(x)dxhat{G}(x)\stackrel{?}{=}int hat{F}(x)hat{G}(x)dxend{align}$$




    There's really no difference.
    The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
    By convention we tend to write the $dx$ either at the front or at the end.
    I've never seen it written in the middle like that.
    I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.




    How do you know which operator is in the integrand?




    Ok that's a good question!
    It really comes down to the fact that notation has to be clear.
    If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
    It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
    For example, this makes no sense:
    $$ g(x) = int_0^1 sin(x) dx$$
    because there's no "free" $x$ on the right hand side.




    And assuming the general case where $hat{F}$ and $hat{G}$ do not commute, you cannot write the integral with $hat{G}(x)$ on the left of the integral. How is this not ambiguous?




    Well, you certainly would not write
    $$ int dx hat F(x) hat G(x) neq left( int dx hat F(x) right) hat G(x) , .$$
    That just makes no sense.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 1 hour ago

























    answered 1 hour ago









    DanielSankDanielSank

    17.6k45178




    17.6k45178












    • $begingroup$
      Thank you for a great answer! Could you please address my edit as well?
      $endgroup$
      – Hexiang Chang
      1 hour ago










    • $begingroup$
      I'm so sorry, I forgot that I needed to ask that as well
      $endgroup$
      – Hexiang Chang
      1 hour ago


















    • $begingroup$
      Thank you for a great answer! Could you please address my edit as well?
      $endgroup$
      – Hexiang Chang
      1 hour ago










    • $begingroup$
      I'm so sorry, I forgot that I needed to ask that as well
      $endgroup$
      – Hexiang Chang
      1 hour ago
















    $begingroup$
    Thank you for a great answer! Could you please address my edit as well?
    $endgroup$
    – Hexiang Chang
    1 hour ago




    $begingroup$
    Thank you for a great answer! Could you please address my edit as well?
    $endgroup$
    – Hexiang Chang
    1 hour ago












    $begingroup$
    I'm so sorry, I forgot that I needed to ask that as well
    $endgroup$
    – Hexiang Chang
    1 hour ago




    $begingroup$
    I'm so sorry, I forgot that I needed to ask that as well
    $endgroup$
    – Hexiang Chang
    1 hour ago











    1












    $begingroup$

    The notation is not ambiguous; it's purely convention. The correspondence is



    $$
    left(
    int_{t_1}^{t_2} H(t) dt right) left( int_{t'_1}^{t'_2}H(t',t)dt' right)
    iff int_{t_1}^{t_2}int_{t'_1}^{t'_2}H(t)H(t',t)dt' dt.
    $$



    That is instead of evaluating "inside out" we evaluate the integrals from right to left.





    If you square an integral as



    $$ left(int ^t_{t_0} dt' H(t')right)^2 $$



    you should know that in general these two integrals don't talk to one another, except in very special cases. That is, they are completely separate entities.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      The notation is not ambiguous; it's purely convention. The correspondence is



      $$
      left(
      int_{t_1}^{t_2} H(t) dt right) left( int_{t'_1}^{t'_2}H(t',t)dt' right)
      iff int_{t_1}^{t_2}int_{t'_1}^{t'_2}H(t)H(t',t)dt' dt.
      $$



      That is instead of evaluating "inside out" we evaluate the integrals from right to left.





      If you square an integral as



      $$ left(int ^t_{t_0} dt' H(t')right)^2 $$



      you should know that in general these two integrals don't talk to one another, except in very special cases. That is, they are completely separate entities.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        The notation is not ambiguous; it's purely convention. The correspondence is



        $$
        left(
        int_{t_1}^{t_2} H(t) dt right) left( int_{t'_1}^{t'_2}H(t',t)dt' right)
        iff int_{t_1}^{t_2}int_{t'_1}^{t'_2}H(t)H(t',t)dt' dt.
        $$



        That is instead of evaluating "inside out" we evaluate the integrals from right to left.





        If you square an integral as



        $$ left(int ^t_{t_0} dt' H(t')right)^2 $$



        you should know that in general these two integrals don't talk to one another, except in very special cases. That is, they are completely separate entities.






        share|cite|improve this answer











        $endgroup$



        The notation is not ambiguous; it's purely convention. The correspondence is



        $$
        left(
        int_{t_1}^{t_2} H(t) dt right) left( int_{t'_1}^{t'_2}H(t',t)dt' right)
        iff int_{t_1}^{t_2}int_{t'_1}^{t'_2}H(t)H(t',t)dt' dt.
        $$



        That is instead of evaluating "inside out" we evaluate the integrals from right to left.





        If you square an integral as



        $$ left(int ^t_{t_0} dt' H(t')right)^2 $$



        you should know that in general these two integrals don't talk to one another, except in very special cases. That is, they are completely separate entities.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 1 hour ago

























        answered 1 hour ago









        InertialObserverInertialObserver

        3,2181027




        3,2181027






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Physics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f467758%2fintegral-notations-in-quantum-mechanics%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Knooppunt Holsloot

            Altaar (religie)

            Gregoriusmis