Is A a Linear Operator and Hermitian in Quantum Mechanics Postulate 2?

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SUMMARY

The discussion centers on the nature of the operator A in quantum mechanics, specifically regarding its linearity and Hermitian properties. The participant clarifies that the operator A acts on the wave function ψ(x) and emphasizes the importance of using the notation (Aψ)(x) for clarity in mathematical operations. The conclusion drawn is that understanding the operator's action on functions rather than numerical values is crucial for grasping the concepts of linearity and Hermitian characteristics in quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics postulates, particularly Postulate 2 (superposition principle).
  • Familiarity with linear operators in the context of functional analysis.
  • Knowledge of Hermitian operators and their significance in quantum mechanics.
  • Proficiency in mathematical notation used in quantum mechanics, specifically operator notation.
NEXT STEPS
  • Research the properties of linear operators in quantum mechanics.
  • Study the definition and implications of Hermitian operators in quantum systems.
  • Learn about the mathematical framework of functional analysis as it applies to quantum mechanics.
  • Explore examples of operators acting on wave functions in quantum mechanics.
USEFUL FOR

Students of quantum mechanics, physicists exploring operator theory, and anyone seeking to deepen their understanding of linear and Hermitian operators in quantum systems.

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Hi I am new here and also new to QM.

I have encountered a problem about postulate 2 which is the superposition principle. Basically I understand the concept. However, when it comes to real question, I get stuck and can't proceed. Could anyone enlightens me how to get started with the problem below?

Question:
A psi(x) = [psi(x)]^2

Is A a linear operator? Is A a Hermitian?
 
Last edited:
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Heh, I found the way to solve it already! :smile:
 
BTW: You should start using the notation [tex](A\psi )(x)[/tex] instead of [tex]A\,\psi(x)[/tex]. It will help you to understand better the mathematical operations involved.

[tex](A\psi )(x)[/tex] reads: the value of the function [tex]A\psi[/tex] at the point x.

Operators act on functions (as objects) and not on values of these functions (numbers).
 

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