6.86. Making use of the solution of the foregoing problem, deter-
mine the probability of the particle with energy E = U 012 to be
h 2
located in the region x > 1, if 12 U 0 T
)
.6.87. Find the possible values of energy of a particle of mass m
located in a spherically symmetrical potential well U (r) = 0 for
r < 7. 0 and U (r) = oo for r = r 0 , in the case when the motion of
the particle is described by a wave function* (r) depending only on r.
Instruction. When solving the Schrodinger equation, make the
substitution 11) (r) = x (r)Ir.
6.88. From the conditions of the foregoing problem find:
(a) normalized eigenfunctions of the particle in the states for
which (r) depends only on r;
(b) the most probable value r
pr
for the ground state of the particle
and the probability of the particle to be in the region r < rpr.
6.89. A particle of mass m is located in a spherically symmetrical
potential well U (r) = 0 for r < r 0 and U (r) = U 0 for r >
(a) By means of the substitution (r) = x (r)Ir find the equation
defining the proper values of energy E of the particle for E < U 0 ,
when its motion is described by a wave function (r) depending
only on r. Reduce that equation to the form
sin kro =+kro Vh 2 12mrsUo , where k.li2mElh.(b) Calculate the value of the quantity r°U 0 at which the first
level appears.
6.90. The wave function of a particle of mass in in a unidimension-
al potential field U (x) = kx 212 has in the ground state the form
(x) = A e -coc 2 , where A is a normalization factor and a is a positive
constant. Making use of the Schrodinger equation, find the constant a
and the energy E of the particle in this state.
6.91. Find the energy of an electron of a hydrogen atom in a sta-
tionary state for which the wave function takes the form p (r)
= A (1 + ar) e -ar, where A, a, and a are constants.
6.92. The wave function of an electron of a hydrogen atom in the
ground state takes the form (r) = A e-r/ri, where A is a certain
constant, r 1 is the first Bohr radius. Find:
(a) the most probable distance between the electron and the
nucleus;
(b) the mean value of modulus of the Coulomb force acting on the
electron;
(c) the mean value of the potential energy of the electron in the
field of the nucleus.
6.93. Find the mean electrostatic potential produced by an
electron in the centre of a hydrogen atom if the electron is in the
ground state for which the wave function is (r) = A e -r/ri, where A
is a certain constant, r^1 is the first Bohr radius.
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