Question

An electron in a carbon nanotube of length "L" is described by the wavefunction psi(x)= sqrt(2/L)*sin(pix/L). Compute the expectation value of the kinetic energy of the electron.

Answer #1

Consider a wave packet of a particle described by the
wavefunction ψ(x,0) = Axe^−(x^2/L^2), -∞ ≤ x ≤ ∞.
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...

Assume the wavefunction Ψ(x)=Axe^(-bx^2) is a solution to
Schrodinger’s equation for an electron in some potential U(x) over
the range -∞<x< ∞.
A) Write an expression which would enable you to find the value
of the constant A in terms of the constant b.
B) What is (x)_avg, the average value of x?
C) Write an expression which would enable you to find (x^2)_avg,
the average value of x^2 in terms of the constant b.
D) Write an expression which...

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2)
cos(πx/2) on the interval 0 ≤ x ≤ 1.
(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
will only need two terms.)
(3) The energies of the eigenstates are En =
h̄2π2n2/(2m) for a = 1. What is
ψ(x, t)?
(4) Compute the expectation...

7. A particle of mass m is described by the wave function ψ ( x)
= 2a^(3/2)*xe^(−ax) when x ≥ 0
0 when x < 0
(a) (2 pts) Verify that the normalization constant is correct.
(b) (3 pts) Sketch the wavefunction. Is it smooth at x = 0? (c) (2
pts) Find the particle’s most probable position. (d) (3 pts) What
is the probability that the particle would be found in the region
(0, 1/a)? 8. Refer to the...

II(20pts). Short Problems
a) The lowest energy of a particle in an infinite one-dimensional
potential well is 4.0 eV. If the width of the well is doubled, what
is its lowest energy?
b) Find the distance of closest approach of a 16.0-Mev alpha
particle incident on a gold foil.
c) The transition from the first excited state to the ground
state in potassium results in the emission of a photon with = 310
nm. If the potassium vapor is...

A standing wave on a string fixed at both ends is described by
y(x,t)=2 sin((π/3)x)cos((π/3)t), where x and y are given in cm and
time t is given in s. Answer the following questions
a) Find the two simplest travelling waves which form the above
standing wave
b) Find the amplitude, wave number, frequency, period and speed of
each wave(Include unit in the answer)
c) When the length of the string is 12 cm, calculate the distance
between the nodes...

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