I Relation between operators and their variables' signs

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TL;DR Summary
Do differential operators change corresponding to their variables' signs?
Consider the operator ##\hat{A} (x,t) =\frac{d}{dx} +f(x,t)## where ##\hat{A}## is an operator and ##f(x,t)## a function. I don't know which of the following expression is correct for ##\hat{A} (-x,-t)##?
first one: $$\hat{A} (-x,-t) =\frac{d}{dx} +f(-x,-t)$$ and the second: $$\hat{A} (-x,-t) =-\frac{d}{dx} +f(-x,-t)$$
 
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My take:
$$\hat{A}(-x,-t) = \left.\frac{d}{dx}\right|_{-x} + f(-x,-t)$$

Unless the function acted upon by ##\hat{A}(x,t)## possesses any symmetries, I'd say neither of your suggestions are correct.
 
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hokhani said:
TL;DR Summary: Do differential operators change corresponding to their variables' signs?

Consider the operator ##\hat{A} (x,t) =\frac{d}{dx} +f(x,t)## where ##\hat{A}## is an operator and ##f(x,t)## a function. I don't know which of the following expression is correct for ##\hat{A} (-x,-t)##?
first one: $$\hat{A} (-x,-t) =\frac{d}{dx} +f(-x,-t)$$ and the second: $$\hat{A} (-x,-t) =-\frac{d}{dx} +f(-x,-t)$$
An operator is a mapping from functions to functions. It's not itself a function of ##x## and ##t##.
 
PeroK said:
An operator is a mapping from functions to functions. It's not itself a function of ##x## and ##t##.
Of course in general an operator can be a function of the coordinates; e.g., the 3D Laplacian in spherical coordinates:$$\nabla^{2}=\dfrac{1}{r^{2}}\dfrac{\partial}{\partial r}\left(r^{2}\dfrac{\partial}{\partial r}\right)+\dfrac{1}{r^{2}\sin\theta}\dfrac{\partial}{\partial\theta}\left(\sin\theta\dfrac{\partial}{\partial\theta}\right)+\dfrac{1}{r^{2}\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}}$$
 
renormalize said:
Of course in general an operator can be a function of the coordinates; e.g., the 3D Laplacian in spherical coordinates:$$\nabla^{2}=\dfrac{1}{r^{2}}\dfrac{\partial}{\partial r}\left(r^{2}\dfrac{\partial}{\partial r}\right)+\dfrac{1}{r^{2}\sin\theta}\dfrac{\partial}{\partial\theta}\left(\sin\theta\dfrac{\partial}{\partial\theta}\right)+\dfrac{1}{r^{2}\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}}$$
That's not a function of coordinates. It uses a coordinate function such as ##r^2## as part of its action on a function. The input to the Laplacian is a function and the output is another function.
 
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PeroK said:
That's not a function of coordinates. It uses a coordinate function such as ##r^2## as part of its action on a function. The input to the Laplacian is a function and the output is another function.
OK, mea culpa! Yes, I was using "function" in the colloquial sense of "having a dependence on" the coordinates. Since a function by definition takes numbers into numbers, something taking functions into functions is indeed a different beast. Labeling it a "mapping", I modify my previous statement to simply note that this function-to-function "map" may well depend upon the coordinates.
 
If ##g(x,t) = f(-x,-t)##, then we could relate ##\hat Ag## to ##\hat Af##.
 
PeroK said:
That's not a function of coordinates. It uses a coordinate function such as ##r^2## as part of its action on a function. The input to the Laplacian is a function and the output is another function.
In mathematical language, as you said, an operator is not a function of ##(x,t)##. However, the operator depends on x and t. If we cannot write this dependence as ##H(x,t)##, how to show this dependence in an operator equation?
 
hokhani said:
In mathematical language, as you said, an operator is not a function of ##(x,t)##. However, the operator depends on x and t. If we cannot write this dependence as ##H(x,t)##, how to show this dependence in an operator equation?
Can you be more specific about the context of the question?
 
  • #10
PeroK said:
Can you be more specific about the context of the question?
Of course; in the Schrodinger equation we have something like: ##H(x,t)\psi(x,t)=i\hbar \frac{\partial }{\partial t} \psi(x,t)## which at each space-time point, ##(x,t)##, only the Hamiltonian ##H(x,t)## acts on ##\psi(x,t)##. I try to understand why ##\psi(x,-t)## is not the answer of ##H(x,-t)##?
 
  • #11
hokhani said:
Of course; in the Schrodinger equation we have something like: ##H(x,t)\psi(x,t)=i\hbar \frac{\partial }{\partial t} \psi(x,t)## which at each space-time point, ##(x,t)##, only the Hamiltonian ##H(x,t)## acts on ##\psi(x,t)##. I try to understand why ##\psi(x,-t)## is not the answer of ##H(x,-t)##?
The Hamiltonian may depend on time, which means you do not have a simple operator equation. The Hamiltonian itself does not depend on position. The Hamiltonian is an operator. It's not a function.

Let's take the example of a time-independent Hamiltonian. This can be written in the form of an operator:
$$\hat H = -\frac{\hbar^2}{2m}\hat D^2 + \hat V$$We have used ##\hat D## to denote the differential operator and ##\hat V## to represent the operator generated by multiplication by some (potential) function ##V##.

The action of this operator on a function, ##\psi## is:
$$\hat H\psi = -\frac{\hbar^2}{2m}\hat D^2\psi + \hat V\psi$$Finally, we can use ##x## and ##t## as the position and time variables for our functions and express this equation for every point ##x## and time ##t##:
$$(\hat H\psi)(x, t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}(x, t) + V(x)\psi(x, t)$$If the potential is also a function of ##t##, then we have a time-varying equation:
$$(\hat H\psi)(x, t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}(x, t) + V(x,t)\psi(x, t)$$In this case, you could write the Hamiltonian itself as a function of time:
$$\hat H(t) = -\frac{\hbar^2}{2m}\hat D^2 + \hat V(t)$$
 
  • #12
PeroK said:
The Hamiltonian may depend on time, which means you do not have a simple operator equation. The Hamiltonian itself does not depend on position. The Hamiltonian is an operator. It's not a function.

Let's take the example of a time-independent Hamiltonian. This can be written in the form of an operator:
$$\hat H = -\frac{\hbar^2}{2m}\hat D^2 + \hat V$$We have used ##\hat D## to denote the differential operator and ##\hat V## to represent the operator generated by multiplication by some (potential) function ##V##.

The action of this operator on a function, ##\psi## is:
$$\hat H\psi = -\frac{\hbar^2}{2m}\hat D^2\psi + \hat V\psi$$Finally, we can use ##x## and ##t## as the position and time variables for our functions and express this equation for every point ##x## and time ##t##:
$$(\hat H\psi)(x, t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}(x, t) + V(x)\psi(x, t)$$If the potential is also a function of ##t##, then we have a time-varying equation:
$$(\hat H\psi)(x, t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}(x, t) + V(x,t)\psi(x, t)$$In this case, you could write the Hamiltonian itself as a function of time:
$$\hat H(t) = -\frac{\hbar^2}{2m}\hat D^2 + \hat V(t)$$
Great answer, many thanks.
 
  • #13
Sorry for returning again to this question, I would like to be certain. So, I raise another question:
For the general form of the Hamiltonian in one dimension, ##H=P^2/2m +V(x)##, which of the following is correct for ##H\psi(-x)##?
$$ (1): \\
H\psi(-x)=(P^2/2m +V(x)) \psi(-x)$$ or $$ (2): \\
H\psi(-x)=(P^2/2m +V(-x)) \psi(-x)?$$
 
  • #14
The way to resolve such problems is to let:
$$\phi(x) \equiv \psi(-x)$$That's technically what ##\psi(-x)## means. That the wavefunction is a composition of the function ##\psi## with the function ##x\to -x##. That is case (1).

However, you might also have the case of a general spatial inversion ##x \ to -x##. That is case (2) where you have a general change of coordinates.
 
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