6.70. Employing the uncertainty principle, estimate the mini-
mum kinetic energy of an electron confined within a region whose
size is
1 =
0.20 nm.
6.71. An electron with kinetic energy
T
4 eV is confined
within a region whose linear dimension is
1 = 1
µm.
Using the
uncertainty principle, evaluate the relative uncertainty of its velo-
city.
6.72. An electron is located in a unidimensional square potential
well with infinitely high walls. The width of the well is
1.
From
the uncertainty principle estimate the force with which the electron
possessing the minimum permitted energy acts on the walls of the well.
6.73. A particle of mass m moves in a unidimensional potential
field
U = kx
2
I2
(harmonic oscillator). Using the uncertainty prin-
ciple, evaluate the minimum permitted energy of the particle in
that field.
6.74. Making use of the uncertainty principle, evaluate the mini-
mum permitted energy of an electron in a hydrogen atom and its
corresponding apparent distance from the nucleus.
6.75. A parallel stream of hydrogen atoms with velocity v
600 m/s falls normally on a diaphragm with a narrow slit behind
which a screen is placed at a distance
1 =
1.0 m. Using the uncer-
tainty principle, evaluate the width of the slit S at which the width
of its image on the screen is minimum.
6.76. Find a particular solution of the time-dependent Schrodinger
equation for a freely moving particle of mass m.
6.77. A particle in the ground state is located in a unidimensional
square potential well of length
1
with absolutely impenetrable walls
(0 <
x < 1).
Find the probability of the particle staying within
1
2
a region -
3
-
1 < x < -
3
-1.
6.78. A particle is located in a unidimensional square potential
well with infinitely high walls. The width of the well is
1.
Find the
normalized wave functions of the stationary states of the particle,
taking the midpoint of the well for the origin of the
x
coordinate.
6.79. Demonstrate that the wave functions of the stationary states
of a particle confined in a unidimensional potential well with infi-
nitely high walls are orthogonal, i.e. they satisfy the condition
1)
7
,11)„,•
dx =
0 if n' n. Here
1
is the width of the well, n are
integers.
6.80. An electron is located in a unidimensional square potential
well with infinitely high walls. The width of the well equal to
1
is
such that the energy levels are very dense. Find the density of energy
levels
dN/dE,
i.e. their number per unit energy interval, as a func-
tion of
E.
Calculate
dNIdE
for
E =
1.0 eV if
1 =
1.0 cm.
6.81. A particle of mass m is located in a two-dimensional square
potential well with absolutely impenetrable walls. Find:
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