Today: Bohr Model More Details
1. Model for discrete electron orbits in atoms.
2. Prediction of allowed radii from new
assumptions, esp. quantized angular
momentum.
3. Discrete electronic energies calculated.
What deeper physical meaning comes from
the Bohr Model’s successes??
• HWK 8 due Wed. 10AM.
• Week 8 online participation available until Tuesday
• Reading for Monday.: TZ&D Chap. 6.1-6.4.
Bohr model background
1. 1/λnm = R (1/m2 - 1/n2) – Big experimental target
2. Gravity -1/r2 force gives orbits. Planetary resonances
Coulomb -ke2/r2 force between electron and proton,
So might expect orbits.
3. Classical EM says electron going in circle should
radiate energy, and spiral in. (accelerating charge
radiates)
proton
+
-
Bohr’s additional hypothesisa. Fixed orbits are stable (quant.)
and at fixed energies
b. But WHY??
Review Bohr Model – see book 5.6
Bohr started with 3 basic ideas:
Ordinary
Classical
Mechanics
1. Mech. energy: E = KE + PE = ½mv2 - ke2/r
2. Centripetal Force: Fcent = mv2/r = ke2/r2
3. Angular Momentum Quantization L = n Totally new idea:
Derived from
Correspondence
Principle
Classical Angular Momentum
An object rotates about an axis. The
classical angular momentum of the object
is given by:
A. L = pr
B. L = p ir
C. L = r × p
D. Angular momentum depends only on
torque, not momentum or position
E. None of the above
Classical Angular Momentum
L=r×p
r
p
Angular momentum can
always be calculated from:
Angular momentum can be
changed with net torque:
For a circular orbit:
An object rotates about
an axis at some position
from the axis and with
some momentum.
L=r×p
τ ≡ r × FNET
dL
=
dt
L = p r = mvr
Review Bohr Model – see book 5.6
Bohr started with 3 basic ideas:
Ordinary
Classical
Mechanics
1. Energy Cons.: E = KE + PE = ½mv2 - ke2/r
2. Centripetal Force: Fcent = mv2/r = ke2/r2
3. Angular Momentum Quantization L = n Totally new idea:
Derived from
Correspondence
Principle
Solve 3 for v ⇒ mvr = n ⇒ v = n /mr
Sub 3 into 2, solve for r to get rn = n2 2/mke2 = n2aB
Hydrogen orbital radii
Sub 2 into 1 to get E = -ke2/2r
Hydrogen
Sub rn into E to get En = -mk2e4/2 2n2 = E1/n2 energies
where E1 = -13.6eV = ground state energy of H
& aB= 2/mke2 = Bohr radius = size of H in gnd state.
Note: k =1/4πε0 (textbook)
Successes of Bohr Model
• Explains source of Balmer formula and predicts
empirical constant from fundamental constants:
1/λ12 = R(1/n22 - 1/n12) ⇔ Ephoton = E1(1/n22 - 1/n12)
R = 1/(91.2nm) = mk2e4/4πc 3
• Explains variations in R for different single
electron atoms.
• Predicts approximate size of hydrogen atom
• Explains (sort of) why atoms emit discrete
spectral lines
• Explains (sort of) why electron doesn’t spiral into
nucleus
What things CAN’T the Bohr model explain?
• WHY is angular momentum quantized?
• WHY don’t electrons radiate when they
are in fixed orbitals?
• How does electron know which level to
jump to? (i.e. how to predict intensities of
spectral lines)
…and other issues, but these are biggies.
Ideas for how to resolve these
problems?
L
Waves
λ1=2L
• Physicists at this time
may have been
confused about atoms,
but they understood
waves really well.
• They understood that for
standing waves,
boundary conditions
mean that waves only
have discrete modes.
• E.g. guitar strings
= node = fixed point
that doesn’t move.
f1=c/2L
λ2=L
f2=c/L
λ3=2L/3
f3=3c/2L
λ4=L/2
f4=2c/L
λ5=2L/5
f5=5c/2L
λn=2L/n
f5=nc/2L
…
Standing Waves on a Ring
Just like standing wave
on a string, but now the
two ends of the string
are joined.
What are the restrictions on the wavelength?
A. r = λ
B. r = nλ
n = 1, 2, 3, …
C. πr = nλ
D. 2πr = nλ
E. 2πr = λ/n
Standing Waves on a Ring
• Answer: D. 2πr = nλ
• Circumference = 2πr
• To get standing wave on ring:
Circumference = nλ
Must have integer number of wavelengths to
get constructive, not destructive, interference.
• n = number of wavelengths
deBroglie Waves
• deBroglie (French grad student)
suggested: maybe electrons are actually
little waves going around the nucleus.
• This seems plausible because…
– Standing waves have quantized frequencies,
might be related to quantized energies.
– Einstein had shown that light, typically thought
of as waves, have particle properties. Might
not electrons, typically thought of as particles,
have wave properties?
deBroglie Waves
What is n for electron wave in this picture?
4
3
A.
B.
C.
D.
E.
1
1
5
10
10
20
Cannot determine from picture
Answer: C. 10
5
2
6
7
9
8
deBroglie Waves
n=1
n=2
n=3
…n=10
= node = fixed point
that doesn’t move.
deBroglie Waves
• If electron orbits are standing waves, there is a
relationship between orbital radius and
wavelength.
• But what is the wavelength of an electron?!
• For photons, it was known that photons have
(momentum)
momentum E= pc= hc/ λ
p
⇒ p=h/λ ⇒ λ=h/p
λ
• deBroglie proposed that this
(wavelength)
is also true for massive particles (particles w/mass)!
• λ=h/p=“deBroglie wavelength”
Critical New Idea: Particles have a wavelength!
deBroglie Waves
Given the deBroglie wavelength (λ=h/p)
and the condition for standing waves on a ring
(2πr = nλ), what can you say about the angular
momentum L of an electron if it is a deBroglie
wave?
A.
B.
C.
D.
E.
L = n /r
L=n
L = n /2
L = 2n /r
L = n /2r
(Recall:
= h/2π)
L = angular momentum = pr
p = (linear) momentum = mv
deBroglie Waves
• Substituting the deBroglie wavelength (λ=h/p)
into the condition for standing waves (2πr = nλ),
gives:
2πr = nh/p
• Or, rearranging:
pr = nh/2π
L=n
• deBroglie EXPLAINS quantization of angular
momentum, and therefore EXPLAINS
quantization of energy!
deBroglie Waves
• This is a great story.
• But is it true? If electrons are waves, then
shouldn’t there be other observed effects?
• If so, why no observations of electron
waves?
• What would you need to see to believe
that electrons are described by WAVES?
• Next: Electron interference!
Models of the Atom
• Thomson – Plum Pudding
–
–
–
–
–
– Why? Known that negative charges can be removed from atom.
– Problem: just a random guess
• Rutherford – Solar System
– Why? Scattering showed hard core.
– Problem: electrons should spiral into nucleus in ~10-11 sec.
+
• Bohr – fixed energy levels
– Why? Explains spectral lines.
– Problem: No reason for fixed energy levels
+
• deBroglie – electron standing waves
– Why? Explains fixed energy levels
– Problem: still only works for Hydrogen.
• Schrodinger – quantum wave functions
– Why? Explains everything!
– Problem: None (except that it’s hard to understand)
+
–