Help on the expectation value of two added operators

  • Thread starter grzegorz19
  • Start date
  • #1
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 any one 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.
 

Answers and Replies

  • #2
Simon Bridge
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Definition of a linear operator?
 
  • #3
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17
I feel like I am completely overlooking something. Does any one 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!
 

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