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
(1)Eigenstates equation for a self adjoint linear operator (Dirac notation) ?
(2)Take Hermitian conjugate?
(1) above
(2) below
Prove that all eigenvalues of a Hermitian operator are real
What does this mean
Eigenstates of a Hermitian operator with distinct eigenvalues are
Orthogonal to eachother
Prove orthogonality of distinct eigenstates of a Hermitian Operator with distinct eigenstates
Eigen value equations are
Ο^^|Ψ> = λ|Ψ>
<Ψ|Ο^^ = λ*<Ψ|
The spectrum of Ο^^ ?
All eigenvalues of Ο^^
Which are all € R
Eigenstates of Ο^^ form
A complete basis of H
Spectral theorem
DISCRETE, If |λ_k> form an orthonormal basis , then any state |Ψ> € H can be written as
DISCRETE, Denote linearly independent eigenstates all with same eigenvalue λ_k
DISCRETE, Where eigenvevtors λ_i and λ_k are orthonormal and these are representations of eigenstates a and b
DISCRETE, Write state of an eigenfunction in terms of its orthonormal eigenvectors and associated (linearly independent) eigenstates
Continuous spectrum for a linear operator
Continuous, write any state of an eigenfunction as integral function of orthonormal eigenvectors
3 properties/equations for Dirac delta function
When using eigen vector equation always check
That |Ψ> is non zero
Eigenvalue are analogous to
Expectations
Distinct eigenstates are
Orthogonal
If you normalise continuous eigenvectors
Write any |Ψ> in H using continuous normalised eigen vectors
Fourier transform of momentum
If geometric multiplicity of eigenvectors is greater than 1
Look at entire eigenstate to determine which vectors to choose for orthonormal basis
