Question

Consider the following wave function:

Psi(x,t) = Asin(2piBx)e^(-iCt) for 0<x<1/2B

Psi(x,t) = 0 for all other x

where A,B and C are some real, positive constants.

a) Normalize Psi(x,t)

b) Calculate the expectation values of the position operator and its square. Calculate the standard deviation of x.

c) Calculate the expectation value of the momentum operator and its square. Calculate the standard deviation of p.

d) Is what you found in b) and c) consistent with the uncertainty principle? Explain.

2. Show the time derivative of the expectation value of momentum is equal to (up to a minus sign) the expectation value of the x-derivative of the potential energy.

Thank you!

Answer #1

Normalize the following wave function (3)
Ψ(x, t) = (
Ce−γx+iδt, x ≥ 0
0, x < 0
where γ and δ are some real constants and γ > 0.

A free particle has the initial wave function Ψ(x, 0) = Ae−ax2
where A and a are real and positive constants. (a) Normalize it.
(b) Find Ψ(x, t). (c) Find |Ψ(x, t)| 2 . Express your result in
terms of the quantity w ≡ p a/ [1 + (2~at/m) 2 ]. At t = 0 plot |Ψ|
2 . Now plot |Ψ| 2 for some very large t. Qualitatively, what
happens to |Ψ| 2 , as time goes on? (d)...

The ground state of a particle is given by the time‐dependent
wave function
Ψ0(x, t) =
Aeαx^2+iβt
with an energy eigenvalue of E0 =
ħ2α/m
a. Determine the potential in which this particle exists. Does this
potential resemble any that you have seen before?
b. Determine the normalization constant A for this wave
function.
c. Determine the expectation values of x,
x2, p, and
p2.
d. Check the uncertainty principle Δx and Δp. Is
their product consistent with the uncertainty...

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2)
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(1) What is the normalization constant, C?
(2) Express ψ(x,0) as a linear combination of the eigenstates of
the infinite square well on the interval, 0 < x < 1. (You
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(3) The energies of the eigenstates are En =
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(4) Compute the expectation...

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a) Draw this wavefunction, labeling the axes in terms of A and
L.
b) Find the relationship between A and L that satisfies the
normalization condition.
c) Calculate the approximate probability of finding the particle
between positions x = −L and x = L.
d) What are 〈x〉, 〈x^2〉, and σ_x ? (Hint: use shortcuts where
possible).
e) Find the minimum uncertainty...

For the wave function ψ(x) defined by
ψ(x)= A(a-x),x<a
ψ(x)=0, x>=a
(a) Sketch ψ(x) and calculate the normalization constant A.
(
b) Using Zettili’s conventions, find its Fourier transform φ(k)
and sketch it.
(c) Using the ψ(x) and φ(k), make reasonable estimates for ∆x
and ∆k. Using p = ħk, check whether your results are compatible
with the uncertainty principle.

question #1: Consider the following function.
f(x) =
16 − x2,
x ≤ 0
−7x,
x > 0
(a) Find the critical numbers of f. (Enter your answers
as a comma-separated list.)
x =
(b) Find the open intervals on which the function is increasing or
decreasing. (Enter your answers using interval notation. If an
answer does not exist, enter DNE.)
increasing
decreasing
question#2:
Consider the following function.
f(x) =
2x + 1,
x ≤ −1
x2 − 2,
x...

Consider the vector space P2 := P2(F) and
its standard basis α = {1,x,x^2}.
1Prove that β = {x−1,x^2 −x,x^2 + x} is also a basis of
P2
2Given the map T : P2 → P2 deﬁned by T(a + bx + cx2) = (a + b +
c) + (a + 2b + c)x + (b + c)x2
compute [T]βα.
3 Is T invertible? Why
4 Suppose the linear map U : P2 → P2 has the matrix
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Let a, c be positive constants and assume that a/ 2πc is a
positive integer. Consider the equation Utt +
aut = c^2Uxx , which represents a damped
version of the wave equation (telegrapher’s equation). Assuming
Dirichlet boundary conditions u(0, t) = u(1, t) = 0, on the
infinite strip 0 ≤ x ≤ 1, t ≥ 0, with initial conditions u(x, 0) =
f(x), ut(x, 0) = 0, complete the following:
(a) Find all separable solutions (of the form...

Given a nonlinear system x(t)" + cx(t)' + sin(x(t))
=
0
(3-1)
a) Consider its phase plane by assuming proper parameters of
c.
Do you have singular points with the cases of CENTER FOCUS, NODE
and SADDLE ? Are those stable? asymptotic stable or unstable?
b) When c = 0, plot the phase plane and find these singular
points

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