8.50 Write down the radial density P(r) for the 2s and 2p states of hydrogen. [See (8.90) and (8.97)| Find the most probable radius for each of these states r) of the for the 1s plain the probable P(r) [Hint: If P(r) is maximum, so is in the bability nd out- SECTION 8.10 (Hydrogen-Like lons) What is the most probable radius for a 1s electron in the hydrogen-like ion Ni? What is its binding energy 8.51 for the 8.52 An inner electron ina heavy atom is affected rela- tively little by the other electrons and hence has a wave function very like that for a single electron in orbit around the same nucleus. Approximately. what is the most probable radius for a 1s electron lead? What is this electron's ann 9.1 9.2 9.3 dition imate binding ene allowed val- potential energy U at all. There- re exactly the same for any hydrogen-like ion as for hydrogen itself. Second, the Schrödinger equation for hydrogen has acceptable solutions ways he nth shell, re clustered given shell word "shell' shell) or to for the allowed energies, only m.(key 1 ER E = - 2h2 Replacing ke by Zke, we find for the allowed energies of a hydrogen-like ion: mc(Zke) 1 n one elec- ates of any hree ER -Z2 quan- ore, just as at peak at ated from of spatial iven n, in E = (8.99) 212 n2 n2 Third, the spatial extent of the hydrogen wave functions is determined by the Bohr radius meke n a multi- icular, we thus the corresponding parameter for a hydrogen-like ion is ot neces- nto ener- ав (8.100) 1 r than to Z meZke2 t corre- different xample, aite well his case For example, the exp(-r/ag); therefore, that for a hydrogen-like ion, with ag replaced by aB/Z, is ground-state wave function of hydrogen is is= A = Ae Zr/ap 1s her. alifica- e have 10 We Since all wave functions are modified in the same way, each state of a factor 1/7. compared to the corre-
8.50 Write down the radial density P(r) for the 2s and 2p states of hydrogen. [See (8.90) and (8.97)| Find the most probable radius for each of these states r) of the for the 1s plain the probable P(r) [Hint: If P(r) is maximum, so is in the bability nd out- SECTION 8.10 (Hydrogen-Like lons) What is the most probable radius for a 1s electron in the hydrogen-like ion Ni? What is its binding energy 8.51 for the 8.52 An inner electron ina heavy atom is affected rela- tively little by the other electrons and hence has a wave function very like that for a single electron in orbit around the same nucleus. Approximately. what is the most probable radius for a 1s electron lead? What is this electron's ann 9.1 9.2 9.3 dition imate binding ene allowed val- potential energy U at all. There- re exactly the same for any hydrogen-like ion as for hydrogen itself. Second, the Schrödinger equation for hydrogen has acceptable solutions ways he nth shell, re clustered given shell word "shell' shell) or to for the allowed energies, only m.(key 1 ER E = - 2h2 Replacing ke by Zke, we find for the allowed energies of a hydrogen-like ion: mc(Zke) 1 n one elec- ates of any hree ER -Z2 quan- ore, just as at peak at ated from of spatial iven n, in E = (8.99) 212 n2 n2 Third, the spatial extent of the hydrogen wave functions is determined by the Bohr radius meke n a multi- icular, we thus the corresponding parameter for a hydrogen-like ion is ot neces- nto ener- ав (8.100) 1 r than to Z meZke2 t corre- different xample, aite well his case For example, the exp(-r/ag); therefore, that for a hydrogen-like ion, with ag replaced by aB/Z, is ground-state wave function of hydrogen is is= A = Ae Zr/ap 1s her. alifica- e have 10 We Since all wave functions are modified in the same way, each state of a factor 1/7. compared to the corre-
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For problem 8.51, how do I find the most probable radius and the binding energy?
Transcribed Image Text:8.50 Write down the radial density P(r) for the 2s and
2p states of hydrogen. [See (8.90) and (8.97)| Find
the most probable radius for each of these states
r) of the
for the 1s
plain the
probable
P(r)
[Hint: If P(r) is maximum, so is
in the
bability
nd out-
SECTION 8.10 (Hydrogen-Like lons)
What is the most probable radius for a 1s electron in
the hydrogen-like ion Ni? What is its binding energy
8.51
for the
8.52 An inner electron ina heavy atom is affected rela-
tively little by the other electrons and hence has a
wave function very like that for a single electron in
orbit around the same nucleus. Approximately.
what is the most probable radius for a 1s electron
lead? What is this electron's ann
9.1
9.2
9.3
dition
imate binding
ene
Transcribed Image Text:allowed val-
potential energy U at all. There-
re exactly the same for any hydrogen-like ion
as for hydrogen itself.
Second, the Schrödinger equation for hydrogen has acceptable solutions
ways
he nth shell,
re clustered
given shell
word "shell'
shell) or to
for the allowed energies,
only
m.(key
1
ER
E = -
2h2
Replacing ke by Zke, we find for the allowed energies of a hydrogen-like ion:
mc(Zke) 1
n one elec-
ates of
any
hree
ER
-Z2
quan-
ore, just as
at peak at
ated from
of spatial
iven n, in
E =
(8.99)
212
n2
n2
Third, the spatial extent of the hydrogen wave functions is determined
by the Bohr radius
meke
n a multi-
icular, we
thus the corresponding parameter for a hydrogen-like ion is
ot neces-
nto ener-
ав
(8.100)
1
r than to
Z
meZke2
t corre-
different
xample,
aite well
his case
For example, the
exp(-r/ag); therefore, that for a hydrogen-like ion, with ag replaced by
aB/Z, is
ground-state wave function of hydrogen is is= A
= Ae Zr/ap
1s
her.
alifica-
e have
10 We
Since all wave functions are modified in the same way, each state of a
factor 1/7. compared to the corre-
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