Help on the expectation value of two added operators

[grzegorz19]

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.

 

[Simon Bridge]

Definition of a linear operator?

 

[Ravi Mohan]

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.

 

[grzegorz19]

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!