Linear Operators And Observables Flashcards
A linear operator Ο^^ on a vector space V is
A linear map
Express linearity of operator on vector space V
Express identity operator
Show that operators form a vector space
Product of linear operators?
Densely defined linear operator
Expectation value of linear operator on Hilbert space in a state |Ψ> € L^2(R)
Expectation value of Ο^^ in state |Ψ> € L^2(R)
For Ο^^ € L^2(R)
Where expectation is normalised
Think of (below) as
Coordinates of |Ψ> in the basis {|e_1, …, }
Given an orthonormal basis in H, every (below) can be written as (sum)
In basis
Where |e_k> are orthonormal basis for H and <e_k|Ο^^|Ψ> are coordinates of Ο^^|Ψ> in that basis
In matrix form?
Matrix elements of Ο^^
Matrix elements of Ο^^
Properties of Hermitian adjoint operator
Defining property of Hermitian adjoint operator
For every operator, there exists Hermitian adjoint operator such that
To check basis is orthonormal
Inner product of each base vector with itself = 1
Inner product of each base vector with another = 0
Cronecker Delta function
A domain X is dense if
It’s closure (the domain itself together with its limit points) is the entire space X
Bra vectors
<a| , elements of dual space H^*
Given a linear operator and a bra vector, define new bra vector
Hermitian matrix
It is equal to its conjugate transpose
An observable on a Hilbert space is
A self adjoint linear operator on Hilbert space
Show that the Hamiltonian operator is Hermitian for a free particle
Show that a Hamiltonian (with a nonzero potential) is Hermitian
Where the last identity follows from the fact that U(x^^> is Hermitian
