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Chapter 42
Atomic Physics (cont.)
Dr. Jie Zou
PHY 1371
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Outline
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The quantum model of the hydrogen
atom
The wave functions for hydrogen
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PHY 1371
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The quantum model of the
hydrogen atom
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Difficulties with Bohr theory:
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Many of the lines in the Balmer and other
series of hydrogen are not single lines at
all.
Splitting of spectral lines when atoms are
placed in a strong magnetic filed.
Need a full quantum model involving
the Schrödinger equation to describe
the hydrogen atom.
Dr. Jie Zou
PHY 1371
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The quantum model of the
hydrogen atom
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General strategy: Solving the Schrödinger
equation for the hydrogen atom
The three-dimensional time-independent
Schrödinger equation:
 2   2  2  2
 2  2  2

2m  x
y
z
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Formal procedure for solving the problem:
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Dr. Jie Zou

  U  E

The potential energy function for the hydrogen
atom:
e2
U r   ke
r
Substitute U(r) into the Schrödinger equation
and find the appropriate solutions to the
equation satisfied by appropriate boundary
conditions. PHY 1371
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Results for a Hydrogen atom
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Three different quantum numbers for each
allowed state of the hydrogen atom ( n,l ,m ):
l
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Principal quantum number, n: The energies
of the allowed states for the hydrogen atom
depend only on n,
2
ke e
En  
2a0
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13.606
 1
eV
 2
2
n
n 
n  1,2,3,...
Orbital quantum number, l
Orbital magnetic quantum number, ml
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PHY 1371
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Relationship among the three
quantum numbers
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PHY 1371
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Atomic shell and subshell
notations
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Example: The n=2 level of
Hydrogen
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For a hydrogen atom, determine the
number of allowed states corresponding
to the principal quantum number n = 2,
and calculate the energies of these
states.
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PHY 1371
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The wave functions for
Hydrogen
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Wave function in the ground state 1s:
 1s r  
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1
a03
e  r / a0
Radial probability density function
P(r) = 4r2||2.
For the hydrogen atom in its ground state:
 4r 2  2 r / a0
P1s r    3 e
 a0 
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PHY 1371
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Hydrogen atom in its ground
state
The charge of the electron is extended throughout a
diffuse region of space - Electron cloud
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Example: Probabilities for the
electron in hydrogen
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Calculate the probability that the
electron in the ground state of
hydrogen will be found outside the first
Bohr radius, a0.
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PHY 1371
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Example: The ground state of
hydrogen
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Calculate the most probable value of r
for an electron in the ground state of
the hydrogen atom.
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PHY 1371
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Homework

Chapter 42, P. 1393, Problems: #19,
20.
Dr. Jie Zou
PHY 1371
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