Help on the expectation value of two added operators

In summary, the conversation is about the mathematical formalism of quantum mechanics and the question of whether the expression < \psi | (A+B) | \psi > = < \psi | A | \psi > + < \psi | B | \psi > is always true. The speaker is attempting to prove this and is seeking guidance and literature on the topic. They find the book "Principles of Quantum Mechanics" by P.A.M. Dirac to be helpful in understanding linear operators.
  • #1
grzegorz19
4
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 | [itex]\psi[/itex] >

I am wondering if
< [itex]\psi[/itex] | (A+B) | [itex]\psi[/itex] > = < [itex]\psi[/itex] | A | [itex]\psi[/itex] > + < [itex]\psi[/itex] | B | [itex]\psi[/itex] >
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) | [itex]\psi[/itex] > = A | [itex]\psi[/itex] > + B | [itex]\psi[/itex] >
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
  • #2
Definition of a linear operator?
 
  • #3
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.
 
  • #4
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!
 

Related to Help on the expectation value of two added operators

1. What is an expectation value in quantum mechanics?

In quantum mechanics, an expectation value refers to the average value of a physical quantity that is measured in a quantum system. It is calculated by taking the sum of all possible outcomes multiplied by their respective probabilities.

2. How is the expectation value of two added operators calculated?

The expectation value of two added operators is calculated by taking the sum of the expectation values of each individual operator, multiplied by their respective coefficients. This can be written as E(A+B) = c1E(A) + c2E(B), where c1 and c2 are the coefficients of the operators A and B respectively.

3. What is the significance of the expectation value of two added operators?

The expectation value of two added operators is significant because it allows us to predict the average value of a physical quantity when both operators are applied to a quantum system. This can provide valuable insights into the behavior and properties of the system.

4. Can the expectation value of two added operators be negative?

Yes, the expectation value of two added operators can be negative. This occurs when the individual expectation values of the operators have opposite signs, resulting in a negative overall value. This can happen in quantum systems where the operators do not commute.

5. How does the expectation value of two added operators relate to the uncertainty principle?

The expectation value of two added operators is related to the uncertainty principle in that it provides a way to calculate the standard deviation of a physical quantity. The uncertainty principle states that the product of the standard deviations of two non-commuting operators must be greater than or equal to the absolute value of their expectation value. This can be represented mathematically as ΔAΔB ≥ |E(A+B)|.

Similar threads

I What does ## \psi^{*}\hat{H}\psi ## mean?
    • Quantum Physics
Replies
9
Views
984
I How to apply potential operator ##V(\hat{x})##
    • Quantum Physics
Replies
2
Views
895
I Very basic questions about operators in QM
    • Quantum Physics
Replies
31
Views
2K
I Vector operator acting on a ket gives a ket out of the state space
    • Quantum Physics
Replies
11
Views
1K
I Invariance of a system under symmetry operations
    • Quantum Physics
Replies
14
Views
947
I Dirac Notation for Operators: Ambiguity in Expectation Values?
    • Quantum Physics
Replies
5
Views
1K
I Constant phase factors in wavefunctions
    • Quantum Physics
Replies
5
Views
541
Relation between Mutual information and Expectation Values
    • Introductory Physics Homework Help
Replies
2
Views
378
I How to interpret integration by parts
    • Quantum Physics
Replies
11
Views
1K
I How to "derive" momentum operator in position basis using STE?
    • Quantum Physics
Replies
3
Views
288

Hot Threads

Recent Insights

Back
Top