Download Physics 102: Lecture 24 Heisenberg Uncertainty Principle Physics

Physics 102: Lecture 24
Heisenberg Uncertainty Principle
& Bohr Model of Atom
Physics 102: Lecture 24, Slide 1
Heisenberg Uncertainty Principle
Recall: Quantum Mechanics tells us outcomes of
individual measurements are uncertain
ΔpyΔy ≥
Uncertainty in
momentum
t
((along
l
y))
h
2
Uncertainty in
position
iti ((along
l
y))
Rough idea: if we know momentum very precisely,
we lose knowledge of location, and vice versa.
Thiss “uncertainty”
u ce a y iss fundamental:
u da e a : it arises
a ses because
quantum particles behave like waves!
Physics 102: Lecture 24, Slide 2
Electron diffraction
El t
Electron
beam
b
traveling
t
li through
th
h slit
lit will
ill diffract
diff t
Single slit diffraction pattern
Number of electrons
arriving at screen
w
θ
θ
electron
beam
Δpy = p sinθ
y
x
screen
Recall single-slit diffraction 1st minimum:
sinθ = λ/w
w = λ/sinθ = Δy
Δp y Δy = p sin θ
Physics 102: Lecture 24, Slide 3
λ
sin θ
= λp = h
Using de Broglie λ = h/p
Number of electrons
arriving at screen
py
w
electron
beam
Δp y ⋅ w = h
y
x
screen
Electron entered slit with momentum along
g x direction and no
momentum in the y direction. When it is diffracted it acquires a py
which can be as big as h/w.
The “Uncertainty in py” is
Δpy ≈ h/w.
An electron passed through the slit somewhere along the y
direction The “Uncertainty
direction.
Uncertainty in yy” is Δy ≈ w.
w
Physics 102: Lecture 24, Slide 4
Δp y ⋅ Δy ≈ h
Number of electrons
arriving at screen
py
w
electron
beam
Δp y ⋅ Δy ≈ h
y
x
screen
If we make the slit narrower (decrease w =Δy) the diffraction
peak gets broader (Δpy increases).
“If we know location very precisely, we lose knowledge of
momentum, and vice versa.”
Physics 102: Lecture 24, Slide 5
to be precise...
Δp y Δy ≥
h
2
Of course if we try to locate the position of the particle
alongg the x axis to Δx we will not know its x component
p
of
momentum better than Δpx, where
h
Δpx Δx ≥
2
and the same for z.
Checkpoint 1
According to the H.U.P., if we know the x-position of a particle, we
cannot know its:
(1)
y-position
y
position
((2))
x-momentum
x
momentum
(3)
Physics 102: Lecture 24, Slide 6
y-momentum
(4)
Energy
Atoms
•
•
•
•
Evidence for the nuclear atom
Today
Bohr model of the atom
Spectroscopy of atoms
Next lecture
Q
Quantum
t
atom
t
Physics 102: Lecture 24, Slide 7
Early Model for Atom
• Plum Pudding
– negative charges (electrons) in uniformly distributed
cloud
l d off positive
iti charge
h
lik
like plums
l
in
i pudding
ddi
+
+
+
- +
+
+
+
10 m across?
But how can you look inside an atom 10-10
Light (visible)
λ = 10-7 m
Electron (1 eV)
λ = 10-9 m
Helium atom
λ = 10-11 m
Physics 102: Lecture 24, Slide 8
Rutherford Scattering
1911: Scattering He++ (an “alpha particle”) atoms off of gold.
Mostly go through, some scattered back!
+
Plum pudding theory:
– electrons in “cloud” of
uniformly
if
l distributed
di t ib t d + charge
h
Æ electric field felt by alpha
never gets too large
To scatter at large angles, need
positive charge concentrated in small
region (the nucleus)
+
-
+
+ +
+
+
+
Atom is mostly empty space with a small (r = 10-15 m) positively
charged nucleus surrounded by “cloud” of electrons (r = 10-10 m)
Physics 102: Lecture 24, Slide 9
Nuclear Atom (Rutherford)
Large angle scattering
Nuclear atom
Classic nuclear atom is not stable!
Electrons orbit nucleus -> constant centripetal
acceleration
Accelerated charges radiate energy
Electrons will radiate and spiral into nucleus!
Early “quantum” model: Bohr
Physics 102: Lecture 24, Slide 10
Need
quantum
theory
Bohr Model is Science fiction
The Bohr model is complete nonsense.
Electrons do not circle the nucleus in little planetlike orbits.
The assumptions injected into the Bohr model
have no basis in physical reality
reality.
BUT the model does get some of the numbers
right for SIMPLE atoms…
atoms
Physics 102: Lecture 24, Slide 11
Hydrogen-Like Atoms
single electron with charge -e
nucleus with charge +Ze
Ze
(Z protons)
19 C
e=1
1.6
6 x 10-19
E H (Z
Ex:
(Z=1),
1) He
H (Z=2),
(Z 2) Li (Z=3),
(Z 3) etc
t
Physics 102: Lecture 24, Slide 12
The Bohr Model
Electrons circle the nucleus in orbits
Only certain orbits are allowed
2πr = nλ
n=1
-e
e
+Ze
Physics 102: Lecture 24, Slide 13
The Bohr Model
Electrons circle the nucleus in orbits
Only certain orbits are allowed
2πr = nλ
= nh/p
L = ppr = nh/2π = nħ
n=2
-e
e
+Ze
Angular momentum is quantized
Physics 102: Lecture 24, Slide 14
v is also quantized in the Bohr model!
An analogy: Particle in Hole
• Let’s define energy so that a free particle has
E=0
• Consider a particle trapped in a hole
• To free the particle, need to provide energy mgh
• Relative to the surface, energy = -mgh
mgh
– a particle that is “just free” has 0 energy
E 0
E=0
h
E=-mgh
Physics 102: Lecture 24, Slide 15
An analogy: Particle in Hole
• Quantized Energy: only fixed discrete
heights of particle allowed
• Lowest energy (deepest hole) state is called
the
h ““groundd state””
E 0
E=0
h
Physics 102: Lecture 24, Slide 16
“ground state”
For Hydrogen-like atoms:
Energy levels (relative to a “just free” E=0 electron):
mk 2 e 4 Z 2
13.6 ⋅ Z 2
En = −
≈−
eV ( where h ≡ h / 2π )
2
2
2
2h n
n
Radius of orbit:
2
2
n2
⎛ h ⎞ 1 n
rn = ⎜
= ( 0.0529
0 0529 nm )
⎟
2
Z
⎝ 2π ⎠ mke Z
Physics 102: Lecture 24, Slide 17
Checkpoint 2
h 2 1 n2
n2
rn = ( )
= (0.0529nm)
2
2π mke
k Z
Z
Bohr radius
If the electron in the hydrogen atom was 207 times
heavier (a muon), the Bohr radius would be
1) 207 Times Larger
h 2 1
Bohr Radius = ( )
2
π
2
mke
2)) Same Size
3) 207 Times Smaller
This “m”
m is electron
mass!
Physics 102: Lecture 24, Slide 18
ACT/Checkpoint 3
A single electron is orbiting around a nucleus
with charge +3. What is its ground state (n=1)
energy? (Recall for charge +1, E= -13.6 eV)
1)
2)
3)
E = 9 ((-13
13.6
6 eV)
E = 3 (-13.6 eV)
E = 1 (-13.6
( 13 6 eV)
V)
32/1 = 9
Z2
E n = −13.6eV 2
n
Note: This is LOWER energy since negative!
Physics 102: Lecture 24, Slide 19
ACT: What about the radius?
Z=3, n=1
1. larger than H atom
2 same as H atom
2.
3. smaller than H atom
h 2 1 n2
n2
rn = ( )
= (0.0529nm)
2
2π mke Z
Z
Physics 102: Lecture 24, Slide 20
Summary
• Bohr’s Model gives accurate values for electron
energy levels...
levels
• But Quantum Mechanics is needed to describe
electrons in atom.
atom
• Next time: electrons jump between states by
emitting or absorbing photons of the appropriate
energy.
Physics 102: Lecture 24, Slide 21
Some (more) numerology
• 1 eV = kinetic energy of an electron that has been
accelerated through a potential difference of 1 V
1 eV = qΔV = 1.6 x 10-19 J
• h (Planck’s constant) = 6.63 x 10-34 J·s
hc = 1240 eV·nm
31 kg
• m = mass of electron = 9.1 x 10-31
mc2 = 511,000 eV
• U = kke2/r,
/ so ke
k 2 has
h units
i eV·nm
V
(like
(lik hc)
h )
2πke2/(hc) = 1/137 (dimensionless)
“fi structure
“fine
t t
constant”
t t”
Physics 102: Lecture 24, Slide 22