5.61 Fall 2007 Lecture #4 page 1
The ATOM of NIELS BOHR
Niels Bohr, a Danish physicist who established the Copenhagen school.
(a) Assumptions underlying the Bohr atom
(1) Atoms can exist in stable “states” without radiating. The states have
discrete energies En, n = 1, 2, 3,..., where n = 1 is the lowest energy
state (the most negative, relative to the dissociated atom at zero
energy), n = 2 is the next lowest energy state, etc. The number “n” is
an integer, a quantum number, that labels the state.
(2) Transitions between states can be made with the absorption or
ΔE
emission of a photon of frequency ν where ν = .
h
En1
hν hν
or
Absorption Emission
En2
These two assumptions “explain” the discrete spectrum of atomic vapor
emission. Each line in the spectrum corresponds to a transition between two
particular levels. This is the birth of modern spectroscopy.
h
(3) Angular momentum is quantized: ! = n" where " =
2π
Angular momentum
! ! ! !
L=r×p "= L
L! !
For circular motion:
L
! ! !
L is constant if r and p are constant !
r
l = mrv is a constant of the motion ! !
p = mv
Other useful properties
1
, 5.61 Fall 2007 Lecture #4 page 2
v!
!
( )
= 2π r ⋅ ν rot = r ω rot
! !
velocity
(m/s) circumference frequency angular
(m/cycle) (cycles/s) frequency
(rad/s)
⇒ " = mvr = mr 2ω rot
Recall the moment of inertia I = ∑ mi ri2
i
2
∴ For our system I = mr
⇒ ! = I ω rot
Note: Linear motion vs. Circular motion
mass m ↔ I moment of inertia
velocity v ↔ ω rot angular velocity
momentum p = mv ↔ ! = Iω angular momentum
Kinetic energy is often written in terms of momentum:
1 2 p2 1 m2 r 2 v 2 ! 2
K.E. = mv = K.E. = =
2 2m 2 mr 2 2I
Introduce Bohr’s quantization into the Rutherford’s planetary model.
For a 1-electron atom with
r e-
a nucleus of charge +Ze
+Ze
Ze2 n2 !2
r=
4πε 0 mv 2
⇒ r=
Z
(
4πε 0
me2
) The radius is quantized!!
!2
( 4πε ) 0
me2
≡ a0 the Bohr radius
2
The ATOM of NIELS BOHR
Niels Bohr, a Danish physicist who established the Copenhagen school.
(a) Assumptions underlying the Bohr atom
(1) Atoms can exist in stable “states” without radiating. The states have
discrete energies En, n = 1, 2, 3,..., where n = 1 is the lowest energy
state (the most negative, relative to the dissociated atom at zero
energy), n = 2 is the next lowest energy state, etc. The number “n” is
an integer, a quantum number, that labels the state.
(2) Transitions between states can be made with the absorption or
ΔE
emission of a photon of frequency ν where ν = .
h
En1
hν hν
or
Absorption Emission
En2
These two assumptions “explain” the discrete spectrum of atomic vapor
emission. Each line in the spectrum corresponds to a transition between two
particular levels. This is the birth of modern spectroscopy.
h
(3) Angular momentum is quantized: ! = n" where " =
2π
Angular momentum
! ! ! !
L=r×p "= L
L! !
For circular motion:
L
! ! !
L is constant if r and p are constant !
r
l = mrv is a constant of the motion ! !
p = mv
Other useful properties
1
, 5.61 Fall 2007 Lecture #4 page 2
v!
!
( )
= 2π r ⋅ ν rot = r ω rot
! !
velocity
(m/s) circumference frequency angular
(m/cycle) (cycles/s) frequency
(rad/s)
⇒ " = mvr = mr 2ω rot
Recall the moment of inertia I = ∑ mi ri2
i
2
∴ For our system I = mr
⇒ ! = I ω rot
Note: Linear motion vs. Circular motion
mass m ↔ I moment of inertia
velocity v ↔ ω rot angular velocity
momentum p = mv ↔ ! = Iω angular momentum
Kinetic energy is often written in terms of momentum:
1 2 p2 1 m2 r 2 v 2 ! 2
K.E. = mv = K.E. = =
2 2m 2 mr 2 2I
Introduce Bohr’s quantization into the Rutherford’s planetary model.
For a 1-electron atom with
r e-
a nucleus of charge +Ze
+Ze
Ze2 n2 !2
r=
4πε 0 mv 2
⇒ r=
Z
(
4πε 0
me2
) The radius is quantized!!
!2
( 4πε ) 0
me2
≡ a0 the Bohr radius
2
