Phys 172 Modern Mechanics
Summer 2010
r
r
Δp sys = Fnet Δt
ΔE sys = Wsurr + Q
r
r
ΔLsys = τ net Δt
Lecture 14 – Energy Quantization
Read:Ch 8
Reading Quiz 1
An electron volt (eV) is a measure of:
A)
B)
C)
D)
E)
Electricity
F
Force
Energy
Momentum
None of the above
1 eV = 1.6 x 10-19 J
Spectroscopy
Spectrum of “white” light
is essentially continuous
continuous.
Spectrum
p
of hydrogen
y g
gas is clearly discrete.
What’s going on here?
Light and Energy
“cooler”
“hotter”
Diff
Different
t colors
l
off li
light
ht → different
diff
t photon
h t energies
i
E photon = p photon c =
h
λ photon
c
the wavelength λ determines
color of light
Planck’s constant: h = 6.6x10-34 J·s
Energy Quantization in Atoms
Consider a hydrogen atom
((1 p
proton and 1 electron))
It turns out that the
electron may only
assume certain orbits.
N=1
Then U + Kelectron can be
only certain values.
N=2
N=3
Bohr Model of the Atom
Energy Quantization in Atoms
−13.6 eV
EN ≡ K e + U e =
N2
N = 1, 2,3, etc
electronic energy levels of hydrogen atom
(no other atom has these levels!)
CLICKER QUESTION 1
Suppose that these are the
quantized energy levels (K+U) for
an atom. Initially the atom is in its
[ ] (symbolized by a dot).
ground state
An electron with kinetic energy
6 eV collides with the atom and
excites it. What is the remaining
kinetic energy of the electron?
A) 9 eV
B) 6 eV
C) 5 eV
D) 3 eV
E) 2 eV
Only possible excitation: -9 eV → - 5 eV.
Not enough
g K in electron for any
y other excitation.
System = atom + electron: ΔEatom + ΔEelectron = W + Q = 0
4 eV
ΔEatom = [(-5
[( 5 eV) – (-9
( 9 eV)] = 4 → ΔEelectron = -4
Kf,electron = 2 eV
(no change in rest energies, etc.)
Quantum Mechanics
…
In this course we won’t touch most of quantum mechanics.
It’s a very interesting story, however . . .
Emission and Absorption of Photons
emitted
photon
absorbed
photon
How Do We Determine Energy Levels?
We look at light emitted from some
gas of atoms,
atoms and we see photons
with energies
1 eV,, 2 eV,, 3 eV,, 6 eV,, 8 eV,, 9 eV
Play with the numbers for a while
while.
The following energy levels are
consistent with this data:
-10 eV,, -9 eV,, -7 eV,, -1 eV
(or -11, -10, -8, -2 etc.)
CLICKER QUESTION 2
Suppose that these are
the quantized energy
levels (K+U) for an atom.
If the atom is excited to
the second excited state
(marked by a dot), what
are the possible energies
photons it might
g emit?
of p
A) 2, 5, and 9 eV
B) 3, 4, and 7 eV
C) 3 or 7 eV
D) 5 or 9 eV
E) 2 eV
Possible atomic transitions:
• -2 → -9 gives ΔEatom =-7 eV which gives Ephoton = 7 eV
OR
• -2 → -5 gives Ephoton = 3 eV, followed by -5→-9 gives Ephoton = 4 eV
CLICKER QUESTION 3
Light consisting of photons
with a range of energies from
1 to 7.5 eV passes through
this collection of objects.
j
A collection of these
atoms is kept very
cold, so that all are in
the ground state.
A)
B)
C)
D)
E)
2 eV, 5 eV, 9 eV
3 eV,
V 4 eV
V
0.5 eV, 3 eV, 4 eV
4 eV, 7 eV
3 eV, 4 eV, 7 eV
What photon energies will be
absorbed from the light
g beam
(“dark lines”)?
NOTE: Excited states fall back to the ground state so quickly that we’ll never
see “double transitions” like -9 → -5→ -2.
Joseph von Fraunhofer
Solar Spectrum
Quantizing Two Interacting Atoms
U for two atoms
If atoms don’t move too far from equilibrium,
q
, U looks like Uspring.
Thus, energy levels should correspond to a quantized spring . . .
Quantized Vibrational Energy Levels
Classical harmonic oscillator:
2
E = 12 mv 2 + 12 ks 2 = 12 kAmax
A value
Any
l off A is
i allowed
ll
d → any E is
i possible.
ibl
Quantum harmonic oscillator:
EN = N hω0 + E0
where
h
N = 00, 11, 22, . . .
ω0 =
Only certain values of E are possible.
Note that levels are evenly spaced:
ΔE = hω0
ks
m
Quantized Vibrational Energy Levels
ffar away from
f
equilibrium,
ilib i
atomic
t i b
bond
d
doesn’t behave as quantum spring
(levels not evenly spaced)
Nearly uniform spacing:
ks
ΔE = h ω 0 = h
m
equilibrium
CLICKER QUESTION 4
Pb: ks ~ 5 N/m Al: ks ~ 16 N/m
Which vibrational energy level diagram
represents Pb, and which is Al?
A) A is Pb and B is Al
B) A is Al and B is Pb
C) A is both Pb and Al
D) B is both Pb and Al
ks
ΔE = hω0 = h
m
ks,Al
, > ks,Pb
,
mAl < mPb
ΔEAl > ΔEPb
ω0,Al > ω0,Pb
CLICKER QUESTION 5 (if time)
Two atoms joined by a chemical bond
can be modeled as two masses
connected
t db
by a spring.
i
In one such molecule, it takes
0.05 eV to raise the molecule from its
vibrational ground state to the first
f
excited vibrational energy state.
How much energy is required to
raise the molecule from its first
excited
it d state
t t tto th
the second
d excited
it d
vibrational state?
A) 0.0125 eV
B) 0.025 eV
C) 0.05 eV
D) 00.10
10 eV
E) 0.20 eV
CLICKER QUESTION 6 (if time)
Molecule A: 2 atoms of mass MA
Molecule B: 2 atoms of mass 4MA
Stiffness of interatomic bond is
approximately the same for both.
ks
ΔE = hω0 = h
m
Which molecule has vibrational
energy levels spaced closer
together?
A) Molecule A
B) Molecule B
C) the spacing is the same
m⇑
→ ΔE⇓
CLICKER QUESTION 7 (if time)
Suppose the atoms in diatomic
molecules C and D had approximately
the same masses, but . . .
Which molecule has vibrational
energy levels spaced closer
together?
Stiffness of bond in C is 3 times as large C) Molecule C
D) Molecule D
as stiffness of bond in D.
D
E) the spacing is the same
ks
ΔE = hω0 = h
m
k⇑
→ ΔE ⇑