Question

Please first prove A to be a Hermitian operator, and after please prove <A^2> to be positive.

Answer #1

prove positive operators are hermitian

Show, without using stationary states, that the Hamiltonian is a
Hermitian operator.

Prove that an orthogonal projection is a positive operator.

Are the operators L+ = Lx + iLy and L− = Lx − iLy Hermitian?
Prove your answer.

An operator Qˆ is given by Qˆ = xpy − ypx where px = −ih∂/∂x ¯
and py = −ih∂/∂y ¯ . Determine whether Qˆ is Hermitian. What does
Qˆ represent?
Please expand

2. Please justify and prove each statement
a) Prove that a finite positive linear combination of metrics is
a metric. If it is infinite, will it be metric?
b) Is the difference of two metrics a metric?

2. Please justify and prove each statement
a) Prove that a finite positive linear combination of metrics is
a metric. If it is infinite, will it be metric?
b) Is the difference between two metrics a metric?

Suppose V is a vector space and T is a linear operator. Prove by
induction that for all natural numbers n, if c is an eigenvalue of
T then c^n is an eigenvalue of T^n.

Consider a one-dimensional real-space wave-function ψ(x) and let
Pˆ denote the parity operator such that P ψˆ (x) = ψ(−x).
a)Starting from the Rodrigues formula for Hermitian polynomials,
Hn(y) = (−1)^n*e^y^2*(d^n/dy^n)e^-y^2 with n ∈ N, show that the
eigenfunctions ψn(x) of the one-dimensional harmonic oscillator,
with mass m and frequency ω, are also eigenfunctions of the parity
operator. What are the eigenvalues?
b)Define the operator Π = exp [ iπ (( 1 /2α) *pˆ 2 +
α xˆ 2/ (h/2π)^2-1/2)] ,...

Suppose V is a ﬁnite dimensional inner product space. Prove that
every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V
, is an isomorphism.

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