Question

Consider a finite potential barrier of height U_0 = 3eV between 0<x<L where L=2nm. The energy of an electron incident on this barrier is E=2eV.

A) What is the general form of the wavefunction Ψ (x) in the barrier region? Compute numerical values for constants when possible.

B) Sketch the wavefunction of the electron before, inside, and after the barrier.

Answer #1

Particles of mass m are incident from the positive x axis
(moving to the left) onto a potential energy step at x=0. At the
step the potential energy drops from the positive value U_0 for all
x>0 to the value 0 for all x<0. The energy of the particles
is greater than U_0.
A) Sketch the potential energy U(x) for this system.
B) How would the wavelength of a particle change in the x<0
region compared to the x>0 region?...

The infinite potential well has zero potential energy between 0
and a, and is infinite elsewhere.
a) What are the energy eigenstates of this quantum system, and
what are their energies? In the case of a discrete spectrum,
explain where the quantization comes from.
b) Suppose we take the wavefunction at a given time to be an
arbitrary function of x that is symmetric around the center of the
well (at x = a/2). Is this a stationary state in...

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...

A square barrier of height Vo = 5.0 eV and width of a = 1.0nm
has a beam of electrons incident on it having kinetic energy E =
4.0 eV. The wave function for x less than or equal to 0 can be
written, Psi1 = Aeik1x +
Be-ik1x and for the region inside the
barrier, the wave function can be written Psi2 =
Cek2x + Dek2x.
Determine the equations describing the continuity conditions at
x=0.

An electron is confined between x = 0 and x =
L. The wave function of the electron is
ψ(x) = A sin(2πx/L).
The wave function is zero for the regions x < 0 and
x > L. (a) Determine the normalization
constant A. (b) What is the probability of finding the
electron in the region 0 ≤ x ≤ L/8? {
(2/L)1/2, 4.54%}

A beam of electrons with kinetic energy 25 eV encounter a
potential barrier of height 20 eV. Some electrons reflect from the
barrier, and some are transmitted. Find the wave number k of the
transmitted electrons.
You can take U = 0 for x < 0, and U = 15 eV for x > 0

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...

A beam of particles is incident from the negative x
direction on a potential energy step at x = 0. When
x < 0, the potential energy of the particles is zero,
and for x > 0 the potential energy has the constant
positive value U0. In the region x
< 0, the particles have a kinetic energy K that is
smaller than U0. What should the form of the
wave function be in the region x > 0?

Particles with energy E, are incident from the left, on the
step-potential of height V0 = 2E as shown: a. What are the wave
numbers in the two regions, 1 k and 2 k , in terms of E? b. Write
down the most general solutions for the Schrodinger Equation in
both regions? Identify, with justification, if any of the
coefficients are zero. c. Write down the equations that result for
applying the boundary conditions for the wave functions at...

A particle in a strange potential well has the following two
lowest-energy stationary states:
ψ1(x) = C1 sin^2 (πx) for − 1 ≤ x ≤ 1
ψ2(x) = C2 sin^2 (πx) for − 1 ≤ x ≤ 0
ψ2(x) = −C2 sin^2 (πx) for 0 ≤ x ≤ 1
The energy of ψ1 state is ¯hω. The energy of ψ2 state is 2¯hω.
The particle at t = 0 is in a superposition state Ψ(x, t = 0) = 1/√...

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