Download Particle in a box

Hydrogen atom energies
Quantized energy levels:
Each corresponds to
different
 Orbit radius
 Velocity
 Particle wavefunction
 Energy
Each described by a
quantum number n


Prof. Clint Sprott
takes us on a tour
of fractals.

Zero energy
n=4
n=3
n=2
E2 = "
13.6
eV
22
E1 = "
13.6
eV
12
!
13.6
eV
n2
Thu. Nov. 29 2007
13.6
eV
32
!
n=1
En = "
E3 = "
Energy
Friday Honors lecture
!
Physics 208, Lecture 25
2
!
Quantum ‘Particle in a box’
Classical vs Quantum
Classical: particle bounces back and forth.

Particle confined to a fixed region of space
e.g. ball in a tube- ball moves only along length L
Sometimes velocity is to left, sometimes to right

L

L
Classically, ball bounces back and forth in tube.



This is a ‘classical state’ of the ball.

Identify each state by speed,
momentum=(mass)x(speed), or kinetic energy.
Quantum mechanics:


Classical: any momentum, energy is possible.
Quantum: momenta, energy are quantized

Particle represented by wave: p = mv = h / λ
Different motions: waves traveling left and right
Quantum wave function:

Thu. Nov. 29 2007
Physics 208, Lecture 25
3
Quantum version
Quantum state is both velocities at the same time
" = 2L

!

momentum
h h
p= =
" 2L


" = 2L
Wave traveling right ( p = +h/λ )
!
Wave traveling left ( p = - h/λ )
Determined by standing wave condition L=n(λ/2) :
"( x) =
2 % 2# (
sin'
x*
L & $ )
Thu. Nov. 29 2007
One halfwavelength
!
Quantum wave function:
superposition of both motions.
Physics 208, Lecture 25
Different speeds correspond to different λ
subject to standing wave condition
integer number of half-wavelengths fit in the tube.
Wavefunction: " ( x ) =
Ground state is a standing wave, made equally of

!
L
4
p = mv = h / λ

One halfwavelength
Physics 208, Lecture 25
Different quantum states


superposition of both at same time
Thu. Nov. 29 2007
"=L
Two halfwavelengths
5
!
Thu. Nov. 29 2007
2 % 2# (
sin'
x*
L & $ )
momentum
n=1
!
p=
n=2
!
Physics 208, Lecture 25
h h
=
# po
" 2L
momentum
h h
p = = = 2 po
" L
6
!
1
Particle in box question
Particle in box energy levels
Quantized momentum
p=

A. are equally spaced everywhere
B. get farther apart at higher energy
C. get closer together at higher energy.
!

Energy = kinetic
2
p 2 ( npo )
E=
=
= n2Eo
2m
2m
Or Quantized Energy
!
Thu. Nov. 29 2007
Physics 208, Lecture 25
7
h
h
=n
= npo
"
2L
n=5
Energy

A particle in a box has a mass m.
Its energy is all kinetic = p 2 /2m.
Just saw that momentum in state n is npo.
It’s energy levels
n=4
En = n2Eo
n=3
n=quantum number
n=2
n=1
Thu. Nov. 29 2007
Physics 208, Lecture 25
8
!
Quantum dot: particle in 3D box
Question
CdSe quantum dots
dispersed in hexane
(Bawendi group, MIT)
A particle is in a particular quantum state in a box of length L.
The box is now squeezed to a shorter length, L/2.
The particle remains in the same quantum state.
The energy of the particle is now
Color from photon
absorption


Energy level spacing increases
as particle size decreases.
i.e
E n +1 " E n =
Thu. Nov. 29 2007
Determined by energylevel spacing
Decreasing particle size
A. 2 times bigger
B. 2 times smaller
C. 4 times bigger
D. 4 times smaller
E. unchanged
Physics 208, Lecture 25
9
(n + 1)
2
8mL2
Thu. Nov. 29 2007
h2
"
n 2h 2
8mL2
Physics 208, Lecture 25
10
!
Interpreting the wavefunction

Higher energy wave functions
Probabilistic interpretation
The square magnitude of the wavefunction |Ψ|2 gives the
probability of finding the particle at a particular spatial
location
Wavefunction
L
n
p
n=3
h
3
2L
E
Probability
h2
3
8mL2
2
Probability = (Wavefunction)2
! n=2 ! 2 h
2L
! n=1
Thu. Nov. 29 2007
Wavefunction
Physics 208, Lecture 25
11
! h
2L
Thu. Nov. 29 2007
!
22
h2
8mL2
h2
8mL2
Physics 208, Lecture 25
12
!
2
Probability of finding electron


Quantum Corral
Classically, equally likely to find particle anywhere
QM - true on average for high n
D. Eigler (IBM)
Zeroes in the probability!
Purely quantum, interference effect
Thu. Nov. 29 2007
Physics 208, Lecture 25
13


48 Iron atoms assembled into a circular ring.
The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
Thu. Nov. 29 2007
Physics 208, Lecture 25
14
Particle in a box, again
Scanning Tunneling Microscopy
L
Particle contained entirely
within closed tube.
Tip
Wavefunction
Open top: particle can escape if
we shake hard enough.
Sample


But at low energies, particle
stays entirely within box.
Over the last 20 yrs, technology developed to controllably
position tip and sample 1-2 nm apart.
Is a very useful microscope!
Thu. Nov. 29 2007
Physics 208, Lecture 25
15
Like an electron in metal
(remember photoelectric effect)
Thu. Nov. 29 2007
Physics 208, Lecture 25
16
Two neighboring boxes
Quantum mechanics says
something different!

Low energy
Classical state

Low energy
Quantum state
Probability =
(Wavefunction) 2
Quantum Mechanics:
some probability of the
particle penetrating
walls of box!
When another box is brought nearby, the
electron may disappear from one well, and
appear in the other!
The reverse then happens, and the electron
oscillates back an forth, without ‘traversing’ the
intervening distance.
Nonzero probability of being outside the box.
Thu. Nov. 29 2007
Physics 208, Lecture 25
17
Thu. Nov. 29 2007
Physics 208, Lecture 25
18
3
Question
Suppose separation between boxes increases by a factor of two.
The tunneling probability
Example:
Ammonia molecule
N
H
A. Increases by 2
H
H
B. Decreases by 2
C. Decreases by <2


‘high’ probability
D. Decreases by >2

E. Stays same


Ammonia molecule: NH3
Nitrogen (N) has two equivalent
‘stable’ positions.
Quantum-mechanically tunnels
2.4x1011 times per second (24 GHz)
Known as ‘inversion line’
Basis of first ‘atomic’ clock (1949)
‘low’ probability
Thu. Nov. 29 2007
Physics 208, Lecture 25
19
Thu. Nov. 29 2007
20
Tunneling between conductors
Atomic clock question

Suppose we changed the ammonia molecule so
that the distance between the two stable positions
of the nitrogen atom INCREASED.
The clock would
A. slow down.
B. speed up.
C. stay the same.
Physics 208, Lecture 25


Make one well deeper:
particle tunnels, then stays in other well.
Well made deeper by applying electric field.
This is the principle of scanning tunneling microscope.
N
H
H
H
Thu. Nov. 29 2007
Physics 208, Lecture 25
21
Scanning Tunneling Microscopy
Thu. Nov. 29 2007
Physics 208, Lecture 25
22
Surface steps on Si
Tip, sample are quantum
‘boxes’
Tip
Potential difference induces
tunneling
Tunneling extremely sensitive
to tip-sample spacing
Sample


Over the last 20 yrs, technology developed to controllably
position tip and sample 1-2 nm apart.
Is a very useful microscope!
Thu. Nov. 29 2007
Physics 208, Lecture 25
23
Images courtesy
M. Lagally,
Univ. Wisconsin
Thu. Nov. 29 2007
Physics 208, Lecture 25
24
4
Manipulation of atoms



Quantum Corral
Take advantage of tip-atom interactions to
physically move atoms around on the surface
This shows the assembly
of a circular ‘corral’ by
moving individual Iron
atoms on the surface of
Copper (111).
The (111) orientation
supports an electron
surface state which can
be ‘trapped’ in the corral
Thu. Nov. 29 2007
Physics 208, Lecture 25
D. Eigler (IBM)


D. Eigler (IBM)
25
The Stadium Corral
48 Iron atoms assembled into a circular ring.
The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
Thu. Nov. 29 2007
Physics 208, Lecture 25
26
Some fun!
D. Eigler (IBM)
Again Iron on copper. This was assembled to investigate quantum chaos.

Kanji for atom (lit. original child)
Iron on copper (111)
The electron wavefunction leaked out beyond the stadium too much to to observe
expected effects.
Thu. Nov. 29 2007
Physics 208, Lecture 25
27
Thu. Nov. 29 2007
Carbon Monoxide man
Carbon Monoxide on Pt (111)
Physics 208, Lecture 25
D. Eigler (IBM)
28
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