Download Simple Harmonic Oscillator

Reminder: First exam Wednesday Oct. 7th in class
Announcement: No class this Friday Oct. 2nd!!!!!
Try practice problems instead!! Practice problems
will be posted to webpage after class
Homework due Wednesday Oct. 7th
Chapter 5: 12,15, 22, 23, 28
Simple Harmonic Oscillator
Simple harmonic
oscillators describe
many physical
situations: springs,
diatomic molecules
and atomic lattices.
Consider the Taylor expansion of an
arbitrary potential function:
1
V ( x)  V0  V1  [ x  x0 ]  V2  [ x  x0 ]2  ...
2
Near a minimum, V1[xx0] ≈ 0.
Simple Harmonic
Oscillator
Consider the second-order term
of the Taylor expansion of a
potential function:
Letting x0 = 0.
V ( x)  12  ( x  x0 ) 2  12  x 2
Substituting this into
Schrödinger’s equation:
 2 d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx

 m x 2 2mE 
d 2
2m   x 2
  2 
 E    2  2 
We have:
2
dx
 2



2mE
m
Let   2 and   2 , which yields:


2
d 2
2 2


x  
2
dx


The Parabolic
Potential Well
The wave function solutions
are:  n ( x)  H n ( x) exp( x 2 / 2)
where Hn(x) are Hermite
polynomials of order n.
n
|n |2
The Parabolic
Potential Well

Classically, the
probability of finding the
mass is greatest at the
ends of motion (because
its speed there is the
slowest) and smallest at
the center.


Classical
result

Contrary to the classical
one, the largest
probability for the lowest
energy states is for the
particle to be at (or near)
the center.
Correspondence Principle for the
Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase,
however, the solution approaches the classical result. This confirms the
Correspondence Principle for the quantum-mechanical simple
harmonic oscillator.
Classical
result
The Parabolic Potential Well
The energy levels are given by:
1
1
En  (n  )  / m  (n  )
2
2
The zero point
energy is
called the
Heisenberg
limit:
1
E 0  
2
CHAPTER 6
Structure of the Atom
The Atomic Models of Thomson and
Rutherford
Rutherford Scattering
The Classic Atomic Model
The Bohr Model of the Hydrogen Atom
Successes & Failures of the Bohr Model
Characteristic X-Ray Spectra and Atomic
Number
Atomic Excitation by Electrons
Niels Bohr (1885-1962)
The opposite of a correct statement is a false statement. But the opposite
of a profound truth may well be another profound truth.
An expert is a person who has made all the mistakes that can be made in
a very narrow field.
Never express yourself more clearly than you are able to think.
Prediction is very difficult, especially about the future.
- Niels Bohr
Structure of the Atom
Evidence in 1900 indicated that
the atom was not a fundamental unit:
1)
There seemed to be too many kinds
of atoms, each belonging to a distinct chemical
element (way more than earth, air, water, and fire!).
2)
Atoms and electromagnetic phenomena were intimately related
(magnetic materials; insulators vs. conductors; different emission
spectra).
3)
Elements combine with some elements but not with others, a
characteristic that hinted at an internal atomic structure
(valence).
4)
The discoveries of radioactivity, x-rays, and the electron (all
seemed to involve atoms breaking apart in some way).
Knowledge of atoms in 1900
Primitive view of an atom
Electrons (discovered in
1897) carried the negative
charge.
Electrons were very light,
even compared to the atom.
Protons had not yet been
discovered, but clearly
positive charge had to be
present to achieve charge
neutrality.
Thomson’s
Atomic Model
Thomson’s “plum-pudding”
model of the atom had the
positive charge spread
uniformly throughout a
sphere the size of the atom,
with electrons embedded in
the uniform background.
In Thomson’s view, when the atom was heated, the electrons could
vibrate about their equilibrium positions, thus producing
electromagnetic radiation.
Unfortunately, Thomson couldn’t explain spectra with this model.
Experiments of Geiger and Marsden
Rutherford, Geiger, and Marsden
conceived a new technique for
investigating the structure of
matter by scattering  particles
from atoms.
Experiments of Geiger and Marsden 2
Geiger showed that many  particles were scattered from thin
gold-leaf targets at backward angles greater than 90°.
Rutherford’s Atomic Model
Experimental observation
of many large angle
scattering events!
Experimental results were
not consistent with
Thomson’s atomic model.
Rutherford proposed that an
atom has a small positively
charged core (nucleus)
surrounded by the negative
electrons.
Geiger and Marsden
confirmed the idea in 1913.
Ernest Rutherford
(1871-1937)
The Classical Atomic Model
Consider an atom as a planetary system. Like gravity, the force on the
electron an inverse-square-law force. This is good.
Now, by Newton’s 2nd Law:
1 e2 mv 2
Fe 

2
4 0 r
r
where v is the tangential velocity of the electron:
v
e
4 0 mr
So:
K  12 mv 2 
Kinetic energy
1
2
e2
4 0 r
This is negative, so
the system is bound,
which is good.
e 2
The potential energy is: V 
4 0 r
The total
e2
e2
e 2
E  K V 


energy is then:
8 0 r 4 0 r 8 0 r
The planetary atom will emit light of a
particular frequency.
An electron in an orbit will emit a light wave at the orbital frequency.
Like planets in the solar system, the electron’s orbital frequency will
vary according to the electron’s distance from the nucleus.
So this model could, in principle, explain atoms’ discrete spectra.
But all frequencies seem possible…
The Planetary Model is Doomed!
Because an accelerated electric charge continually radiates energy
(electromagnetic radiation), the total energy must continually decrease.
So the electron radius must continually decrease!
The
electron
crashes
into the
nucleus!
Physics had reached a turning point in 1900 with Planck’s hypothesis
of the quantum behavior of radiation, so a radical solution would be
considered possible.