Transcript uncertainty, atom

PH 103
Dr. Cecilia Vogel
Lecture 19
Review
Matter Waves
Outline
Uncertainty Principle
Tunneling
Atomic model
Nucleus and electrons
The quantum model
quantum numbers
Position Uncertainty
A wave is not at one
place.
Dx = uncertainty in
position
 = spread in positions
where the wave is.
Dx
Momentum Uncertainty
A wave is not moving in just one
way.
Dp = uncertainty in
momentum
 = spread in ways the wave
moves.
Dp
Heisenberg Uncertainty Principle
h
DxDp x 
2
What it means:
You cannot know position and momentum
both very precisely at the same time
If you measure momentum, you disturb the
position, so you no longer know the
position accurately -- and vice versa
This disturbance is random, indeterminate
(unlike letting a little air out when you
measure the tire pressure)
Heisenberg Uncertainty Principle
h
DxDp x 
2
Heisenberg Uncertainty Principle
h
DxDp x 
2
Zero-point motion:
Any confined particle cannot have a
definite momentum
in particular, it cannot have zero
momentum
any confined particle will have some
kinetic energy -- some “zero-point
motion”
Heisenberg Uncertainty Principle
h
DxDp x 
2
What it does not mean:
It does not mean you can’t measure
position (or momentum) very precisely.
It does not mean you need better
measuring instruments.
It does NOT just a matter of not knowing:
If Dx is large enough, an electron will pass
thru both of two slits and interfere with itself
Another Uncertainty Principle
h
DEDt 
2
What it means
If you only have a small time Dt to measure
energy, you can’t accurately measure energy.
If a particle only lives for a short time Dt, you
can’t accurately measure its energy.
 Since E=mc2, you can’t accurately measure its
mass!
Unstable particles have uncertain mass.
Another Uncertainty Principle
For a short enough period of time Dt, you
can violate conservation of energy by DE.
h means you can measure DE in time Dt
Dt 
2DE for these times, energy conservation
cannot be violated
h means you can’t measure DE in time Dt
Dt 
2DE so the universe can violate energy
conservation for shorter times
and “get away with it”
Classically, potential energy cannot
be greater than the total energy
Otherwise the kinetic energy would be
negative! K = E - U
Places where U>E are classically
forbidden
Tunneling
Waves can tunnel into regions where they
“shouldn’t” be -- if region is small
enough.
Light waves tunnel through region,
even when they “should” have totally
reflected,
if region is very narrow.
Matter waves tunnel through
“classically forbidden regions”
Tunneling
Wait, did you say a particle can
tunnel into classically forbidden
region
where the kinetic energy would be
negative?!!?
YUP
Another example of violating
conservation of energy for
short enough time - HUP
Examples of Quantum
Tunneling
One type of Scanning
Tunneling Microscope = STM
A small, metal needle passes very near a
material.
Electrons from the needle can tunnel through
the small gap and into the material.
The smaller the gap, the more likely the
tunneling.
The more tunneling happens, the stronger the
current of electrons.
As the needle scans across the surface
the tunneling current gives an outline of
the material.
Early Atomic Models
You’ve learned about many physics models
(theories) that are “wrong.”
So far, these models have been useful.
F=ma & K=½mv2 are good when v<<c.
The ray model of light is good for short
wavelength.
etc
WARNING:
The early atomic models are not useful,
except to see how we disprove theories.
Nuclear Model of Atom
a tiny, massive, dense nucleus
at the center of the atom
surrounded by electrons
very little of the mass of the atom is
electrons
most of the volume of the atom is
electrons
Orbits
Where are the electrons?
Electrons do NOT orbit the nucleus, like
planets orbit Sun
Although it seems reasonable, since
the electric force and the gravitational
force are very similar:
but…
1
F 2
r
Two Problems with Orbits
1) An orbiting electron is an accelerating
charge, and
accelerating charges give off EM
radiation (like an antenna),
thus giving off energy.
The electron would gradually lose all
its energy.
That doesn’t happen -- atoms are
stable.
Second Problem with Orbits
2) Quantization
A planet can be in any size orbit with
any orbital energy,
but electrons in atoms have only
certain -- quantized -- energy levels.
Orbit model can’t explain why.
Current Model of Orbit
Electron “cloud” is wavefunction
describes the probability of electron being
at various points around the nucleus.
Electron wave behavior based on
Schroedinger equation.
The electron states are quantized
3 quantum numbers for spatial state:
n, ℓ, mℓ.
http://www.falstad.com/qmatom/directions.html
Principle Quantum
Number
Principle quantum number, n,
n = 1, 2, 3, 4, 5, ....
tells what “shell” the electron is in.
n=1 is called the K-shell,
n=2 is the L-shell, etc
tells a lot about the electron’s energy
for hydrogen atom, it determines the
electron’s energy

13
.
6
eV
for hydrogen atom: E 
n
n
2