1. Determine the variance in position and momentum,
Δx2=<x2>−<x>2 and
Δp2=<p2>−<p>2 for the
ground-state SHO and...
1. Determine the variance in position and momentum,
Δx2=<x2>−<x>2 and
Δp2=<p2>−<p>2 for the
ground-state SHO and show that they satisfy the Heisenberg
uncertainty relation, (Δx)(Δp)≥ℏ2
Compute the average of x and x^2 and determine ∆x for the ground
state of the harmonic oscillator. b) Determine the average
potential energy and describe how it is related to the total
energy?
particle in a well
1. Using the uncertainty principle for position and momentum,
estimate the ground state energy
for an infinite well of width a, compare the obtained result with
the one found in the lecture.
2. Compute <p> and <x> for the particle in a box.
3. Prove that the eigenfunctions ψn(x) are orthogonal .
4. What are the energies of the ground state, first excited and
second excited states for an
electron trapped in an infinite well of...
Consider the linear transformation T : P2 ? P2 given by T(p(x))
= p(0) + p(1) + p 0 (x) + 3x 2p 00(x). Let B be the basis {1, x,
x2} for P2.
(a) Find the matrix A for T with respect to the basis B.
(b) Find the eigenvalues of A, and a basis for R 3 consisting of
eigenvectors of A.
(c) Find a basis for P2 consisting of eigenvectors for T.
2a. Confirm that the position and
momentum, p=-ℏ ∂/∂x, operators
do not commute.
2b. Show how the statement above
limits the ability for a state that to be both an eigenstate of
position and momentum concurrently.
2c. For an eigenstate where the
momentum is zero, show that x than p leads to a result, xp, that is
different than that of p then x.
2. For the function : f(x) = x2 − 30x − 2
a) State where f is increasing and where f is
decreasing
b) Identify any local maximum or local minimum values.
c) Describe where f is concave up or concave down d) Identify any
points of inflection (in coordinate form)
3. For the function f (x)= x4 − 50 2
a) Find the intervals where f is increasing and where f
is decreasing.
b) Find any local extrema and...
Let
X1,X2,...,Xn
be i.i.d. (independent and identically distributed) from the
Bernoulli distribution
f(x)=p^x(1-p)^1-x,
x=0,1,p∈(0,1) where p is unknown parameter. Find the UMVUE of
p parameter and calculate MSE (Mean Square Error) of this UMVUE
estimator.