PHYS 3313 – Section 001
Lecture #13
Monday, March 23, 2015
Dr. Jaehoon Yu
•
•
•
Bohr Radius
Bohr’s Hydrogen Model and Its Limitations
Characteristic X-ray Spectra
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1
Announcements
• Homework #3
– End of chapter problems on CH4: 5, 14, 17, 21, 23 and 45
– Due: Monday, March 30
• Quiz Wednesday, April 1
– At the beginning of the class
– Covers CH4.1 through what we finish Monday, March 30
– BYOF with the same rule as before
• Colloquium at 4pm this Wednesday, in SH101
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
Uncertainties
• Statistical Uncertainty: A naturally occurring
uncertainty due to the number of measurements
– Usually estimated by taking the square root of the
number of measurements or samples, √N
• Systematic Uncertainty: Uncertainty occurring
due to unintended biases or unknown sources
– Biases made by personal measurement habits
– Some sources that could impact the measurements
• In any measurement, the uncertainties provide
the significance to the measurement
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
3
Bohr’s Quantized Radius of Hydrogen
• The angular momentum is
L = r ´ p = mvr = n
n
• So the speed of an orbiting e can be written ve =
me r
• From the Newton’s law for a circular motion
2
e
1 e2 meve
Þ ve =
Fe =
=
2
r
4pe 0 r
4pe 0 mer
• So from above two equations, we can get
n
e
4pe 0 n 2
ve =
=
Þ r=
2
mer
4pe 0 mer
mee
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
4
Bohr Radius
• The radius of the hydrogen atom for the nth stationary state is
4pe 0 2 n 2
2
rn =
=
a0 n
2
mee
Where the Bohr radius for a given stationary state is:
1.055 ´10 J × s )
4pe 0
(
-10
=
0.53
´10
m
=
a0 =
2
2
9
2
2
-31
-19
me e
( 8.99 ´10 N × m C ) × ( 9.11 ´10 kg ) × (1.6 ´10 C )
2
-34
2
• The smallest diameter of the hydrogen atom is
d = 2r1 = 2a0 »10-10 m » 1A
– OMG!! The fundamental length!!
• n = 1 gives its lowest energy state (called the “ground” state)
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
5
Ex. 4.6 Justification for nonrelativistic treatment of orbital e
• Are we justified for the non-relativistic treatment
of the orbital electrons?
– When do we apply relativistic treatment?
• When v/c>0.1
• Orbital speed:
• Thus
4pe 0 mer
(1.6 ´ 10 ) × ( 9 ´ 10 )
» 2.2 ´10 ( m s )< 0.01c
( 9.1 ´ 10 ) × ( 0.5 ´ 10 )
-16
ve =
ve =
e
Monday, March 23, 2015
9
6
-31
-10
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
6
The Hydrogen Atom
• Recalling the total E of an e in an atom, the nth stationary states En
e2
e2
E1
En = = =
- 2
8pe 0 rn
8pe 0 a0 n 2
n
E0 = -
e2
(
)(
9
2
2
-19
= - 8.99 ´10 N × m C × 1.6 ´10 C
8pe 0 a0
2 0.53 ´10 -10 m
(
)
)
2
= -13.6eV
where E0 is the ground state energy
 Emission of light occurs when the atom is
in an excited state and decays to a lower
energy state (nu → nℓ).
hf = Eu - El
where f is the frequency of a photon.
æ 1 1ö
f Eu - El E0 æ 1 1 ö
= =
=
- 2 ÷ = R¥ ç 2 - 2 ÷
2
ç
l c
hc
hc è nl nu ø
è nl nu ø
1
R∞ is the Rydberg constant. R¥ = E0 hc
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
7
Transitions in the Hydrogen Atom
• Lyman series: The atom will
remain in the excited state for a
short time before emitting a
photon and returning to a lower
stationary state. All hydrogen
atoms exist in n = 1 (invisible).
• Balmer series: When sunlight
passes through the atmosphere,
hydrogen atoms in water vapor
absorb the wavelengths (visible).
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
8
Fine Structure Constant
• The electron’s speed on an orbit in the Bohr model:
n
ve =
=
mern
n
4pe 0 n 2
me
2
me e
=
2
1 e
n 4pe 0
2
• On the ground state, v1 = 2.2 × 106 m/s ~ less than
1% of the speed of light
• The ratio of v1 to c is the fine structure constant,  .
2
e
v1
=
=
=
aº
ma0 c 4pe 0 c
c
( 8.99 ´ 10 N × m C ) × (1.6 ´ 10 C ) » 1
137
(1.055 ´ 10 J × s ) × ( 3 ´ 10 m s )
9
2
-34
Monday, March 23, 2015
-19
2
2
8
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
9
The Correspondence Principle
Classical electrodynamics
+
Bohr’s atomic model
Determine the properties
of radiation
Need a principle to relate the new modern results with classical ones.
Bohr’s correspondence
principle
Monday, March 23, 2015
In the limits where classical and quantum
theories should agree, the quantum theory
must produce the classical results.
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
10
The Correspondence Principle
• The frequency of the radiation emitted fclassical is equal to the orbital
frequency forb of the electron around the nucleus.
fclassical = fobs
e
æ
ö
e
w
1 v
= 1
=
=
=
3
2p 2p r 2p r 4pe 0 mer 2p çè 4pe 0 me r ÷ø
1
2
12
me e 4 1
=
4 e 02 h 3 n 3
• The frequency of the photon in the transition from n + 1 to n is
fBohr
E0 æ 1
1 ö E0 n 2 + 2n +1- n 2
E0 é 2n + 1 ù
= ç 2=
ê 2
2÷ =
2 ú
2
2
h è ( n ) ( n +1) ø h
h
n
n
+
1
) úû
êë (
n ( n +1)
• For large n the classical limit,
Substitute E0:
fBohr
2nE0 2E0
fBohr »
= 3
4
hn
hn
2 æ e 2 ö me e 4 1
2E0
= 2 3 3 = fClassical
= 3 =
3ç
÷
hn è 8pe 0 a0 ø 4 e 0 h n
hn
So the frequency of the radiated E between classical theory and Bohr
model agrees in large n case!!
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
11
Importance of Bohr’s Model
• Demonstrated the need for Plank’s constant in
understanding the atomic structure
• Assumption of quantized angular momentum which
led to quantization of other quantities, r, v and E as
follows
4pe 0 2 2
2
• Orbital Radius: rn =
a
n
n = 0
2
mee
• Orbital Speed:
• Energy levels:
Monday, March 23, 2015
n
1
v=
=
mrn ma0 n
e2
E0
En =
= 2
2
8pe 0 a0 n
n
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
12
Successes and Failures of the Bohr Model
• The electron and hydrogen nucleus actually revolve about their
mutual center of mass  reduced mass correction!!
• All we need is to replace me with atom’s reduced mass.
me M
me
me =
=
me + M 1+ me M
• The Rydberg constant for infinite nuclear mass, R∞ is replaced
by R.
me
me e 4
1
R=
R¥ =
R¥ =
2
3
me
1+ me M
4p c ( 4pe 0 )
For H: RH = 1.096776 ´ 10 7 m-1
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
13
Limitations of the Bohr Model
The Bohr model was a great step of the new quantum
theory, but it had its limitations.
1) Works only to single-electron atoms
–
–
Works even for ions  What would change?
1
1ö
2 æ 1
= Z Rç 2 - 2 ÷
The charge of the nucleus
l
èn n ø
l
u
2) Could not account for the intensities or the fine
structure of the spectral lines
–
–
Fine structure is caused by the electron spin
Under a magnetic field, the spectrum splits by the spin
3) Could not explain the binding of atoms into molecules
Monday, March 23, 2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
14