Help on the expectation value of two added operators

Click For Summary

Discussion Overview

The discussion revolves around the mathematical formalism of quantum mechanics, specifically the properties of linear operators and their expectation values. Participants explore whether the equation < \psi | (A+B) | \psi > = < \psi | A | \psi > + < \psi | B | \psi > holds true for any linear operators A and B acting on a state ket | \psi >.

Discussion Character

  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant questions whether the equation for the expectation value of the sum of two operators is always valid.
  • Another participant asks for clarification on the definition of a linear operator.
  • A suggestion is made to refer to Dirac's "Principles of Quantum Mechanics" for a deeper understanding of linear operators and their properties.
  • A later reply expresses gratitude for the reference provided, indicating it was helpful in addressing their confusion.

Areas of Agreement / Disagreement

The discussion does not reach a consensus on the validity of the expectation value equation, as the initial question remains open and participants have not confirmed or denied its universality.

Contextual Notes

The discussion highlights a potential gap in the participant's understanding of linear operators and their properties, which may depend on specific definitions and contexts within quantum mechanics.

grzegorz19
Messages
4
Reaction score
0
Hi everyone,

I was just working on some problems regarding the mathematical formalism of QM, and while trying to finish a proof, I realized that I am not sure if the following fact is always true:

Suppose that we have two linear operators A and B acting over some vector space. Consider a state ket | \psi >

I am wondering if
< \psi | (A+B) | \psi > = < \psi | A | \psi > + < \psi | B | \psi >
is always true?

I am thinking that it IS true.

My attempt at the problem, is of course to try and show that
(A+B) | \psi > = A | \psi > + B | \psi >
But I am having trouble finding a definition which will confirm this to always be true.

I feel like I am completely overlooking something. Does anyone have a helpful hint for me? ANy literature to point me to? My linear algebra books are failing me on this one, at first glance.
 
Physics news on Phys.org
Definition of a linear operator?
 
grzegorz19 said:
I feel like I am completely overlooking something. Does anyone have a helpful hint for me? ANy literature to point me to?
Checkout Principles of Quantum Mechanics (P.A.M. Dirac) chapter II, Dynamical Variables and Observables, section 7, Linear Operators.
 
THANK YOU! I don't know why this was so hard to find, but this is exactly the sort of thing I was looking for!
 

Similar threads

Graduate What happens when an operator maps a vector out of the Hilbert space?
  • · Replies 59 ·
2
Replies
59
Views
6K
Undergrad Motivation behind the Operator-Formalism in QM
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad QED replacing photon field with current in 3-point function
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Different Approaches to the Time Dependent Variational Principle
  • · Replies 0 ·
Replies
0
Views
1K
Undergrad How to apply potential operator ##V(\hat{x})##
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Vector operator acting on a ket gives a ket out of the state space
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Invariance of a system under symmetry operations
  • · Replies 14 ·
Replies
14
Views
2K
High School A stupidly basic question about expectation value
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Angular momentum uncertainty principle and the particle on a ring
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Dirac Notation for Operators: Ambiguity in Expectation Values?
  • · Replies 5 ·
Replies
5
Views
2K