**Quantum Mechanics Exercise: Operators on Functions**Let's explore an interesting exercise in quantum mechanics using operators. You have two functions defined as follows: - \( g(x) = e^{-ikx} \)- \( f(x) = e^{-ax^2} \)There's a function called \( Q(\hat{x}, \hat{p}) \) with some specific rules:- \( \hat{x} = x \)- \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \)Your task is to apply the function \( Q \) by following the standard rules of calculus. Compute the following operations on the specified functions:a) \( \hat{x}^2 g(x) \)b) \( \hat{p}^2 f(x) \)c) \( \hat{p} g(x) \)d) \( \hat{p} f(x) \)e) \( \hat{x} \hat{p} g(x) \)f) \( \hat{p} \hat{x} g(x) \)h) \( \hat{x} \hat{p} f(x) \)i) \( \hat{p} \hat{x} f(x) \)k) Analyze whether the operation of \( \hat{x} \hat{p} \) on a function is always equal to \( \hat{p} \hat{x} \). Provide a justification for your answer.**Note:** When performing these operations, remember to apply the rules of differentiation and multiplication as dictated by quantum mechanics operators typically pertain to position and momentum in physical systems.

Answered: Image /qna-images/answer/99ecf0a7-b62c-474d-91f4-3ff97dbabe79.jpg

e-ikx and f (x) I give you a function called Q(£,p) and tell you the rules are: î = x, and p = -iħ-. Simply plug in the function Q and Let us play a game. I give you two functions. Let g(x) = e-ax? follow the standard rules of calculus! Work out the following operations on the specified functions. a) £?g(x) b) p²f(x) c) pg(x) d) pf (x) e) &ôg(x) f) pâg(x) h) &pƒ(x) i) p&ƒ (x) k) Is it true that the operation of âp on a function is always equal to pâ? Justify your answer.

e-ikx and f (x) I give you a function called Q(£,p) and tell you the rules are: î = x, and p = -iħ-. Simply plug in the function Q and Let us play a game. I give you two functions. Let g(x) = e-ax? follow the standard rules of calculus! Work out the following operations on the specified functions. a) £?g(x) b) p²f(x) c) pg(x) d) pf (x) e) &ôg(x) f) pâg(x) h) &pƒ(x) i) p&ƒ (x) k) Is it true that the operation of âp on a function is always equal to pâ? Justify your answer.