De Broglie Waves, Bohr's Quantization, and Electron Scattering in Physics
undefined
1
PHYS 3313 – Section 001
Lecture #16
Monday, Mar. 24, 2014
Dr.
Jae
hoon
Yu
•
De Broglie Waves
•
Bohr’s Quantization Conditions
•
Electron Scattering
•
Wave Packets and Packet Envelops
•
Superposition of Waves
•
Electron Double Slit Experiment
•
Wave-Particle Duality
Monday, Mar. 24, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Monday, Mar. 24, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
Announcements
•
Research paper template has been uploaded onto
the class web page link to research
•
Special colloquium on April 2, triple extra credit
•
Colloquium this Wednesday at 4pm in SH101
De Broglie Waves
•
Prince Louis V. de Broglie suggested that mass particles
should have wave properties similar to electromagnetic
radiation
•
Thus the wavelength of a matter wave is called the de
Broglie wavelength:
•
This can be considered as the probing beam length scale
•
Since for a photon,
E = pc
and
E = hf
, the
energy can be
written as
Monday, Mar. 24, 2014
3
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
•
What is
the formula for De Broglie Wavelength?
•
(a) for a tennis ball, m=0.057kg.
•
(b) for electron with 50eV KE, since KE is small, we can use non-relativistic
expression of electron momentum!
•
What are the wavelengths of you running at the speed of 2m/s? What about your
car of 2 metric tons at 100mph? How about the proton with 14TeV kinetic energy?
•
What is the momentum of the photon from a green laser?
Calculate the De Broglie wavelength of (a) a tennis ball of mass 57g traveling 25m/s
(about 56mph) and (b) an electron with kinetic energy 50eV.
Ex 5.2: De Broglie Wavelength
Monday, Mar. 24, 2014
4
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Bohr’
s Quantization Condition
•
One of Bohr’
s assumptions concerning his hydrogen atom
model was that the angular momentum of the electron-nucleus
system in a stationary state is an integral multiple of
h
/2
π
.
•
The electron is a standing wave in an orbit around the proton.
This standing wave will have nodes and be an integral number
of wavelengths.
•
The angular momentum becomes:
Monday, Mar. 24, 2014
5
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Electron Scattering
•
Davisson and Germer experimentally observed that electrons were diffracted
much like x rays in nickel crystals.
•
George P. Thomson (1892–1975), son of J. J. Thomson,
reported seeing the effects of electron diffraction in
transmission experiments. The first target was celluloid,
and soon after that gold, aluminum, and platinum were
used. The randomly oriented polycrystalline sample of
SnO
2
produces rings as shown in the figure at right.
Monday, Mar. 24, 2014
6
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
•
Photons, which we thought were waves, act particle
like (eg Photoelectric effect or Compton Scattering)
•
Electrons, which we thought were particles, act
particle like (eg electron scattering)
•
De Broglie: All matter has intrinsic wavelength.
–
Wave length inversely proportional to momentum
–
The more massive… the smaller the wavelength… the
harder to observe the wavelike properties
–
So while photons appear mostly wavelike, electrons
(next lightest particle!) appear mostly particle like.
•
How can we reconcile the wave/particle views?
Monday, Mar. 24, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
7
De Broglie matter waves suggest a further description.
The displacement of a wave is
This is a solution to the wave equation
Define the wave number
k
and the angular frequency
as:
The wave function is now:
Wave Motion
Monday, Mar. 24, 2014
8
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Wave Properties
•
The phase velocity is the velocity of a point on the
wave that has a given phase (for example, the crest)
and is given by
•
A phase constant
shifts the wave:
.
Monday, Mar. 24, 2014
9
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
(When
=
/2)
Principle of Superposition
•
When two or more waves traverse the same region, they act
independently of each other.
•
Combining two waves yields:
•
The combined wave oscillates within an envelope that
denotes the maximum displacement of the combined
waves.
•
W
h
e
n
c
o
m
b
i
n
i
n
g
m
a
n
y
w
a
v
e
s
w
i
t
h
d
i
f
f
e
r
e
n
t
a
m
p
l
i
t
u
d
e
s
a
n
d
f
r
e
q
u
e
n
c
i
e
s
,
a
p
u
l
s
e
,
o
r
w
a
v
e
p
a
c
k
e
t
,
c
a
n
b
e
f
o
r
m
e
d
,
w
h
i
c
h
c
a
n
m
o
v
e
a
t
a
g
r
o
u
p
v
e
l
o
c
i
t
y
:
Monday, Mar. 24, 2014
10
10
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Fourier Series
•
Adding 2 waves isn’t localized in space… but
adding lots of waves can be.
•
T
h
e
s
u
m
o
f
m
a
n
y
w
a
v
e
s
t
h
a
t
f
o
r
m
a
w
a
v
e
p
a
c
k
e
t
i
s
c
a
l
l
e
d
a
F
o
u
r
i
e
r
s
e
r
i
e
s
:
•
Summing an infinite number of waves yields the
Fourier integral:
Monday, Mar. 24, 2014
11
11
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Wave Packet Envelope
•
The superposition of two waves yields a wave number and angular
frequency of the wave packet envelope.
•
The range of wave numbers and angular frequencies that produce the
wave packet have the following relations:
•
A
G
a
u
s
s
i
a
n
w
a
v
e
p
a
c
k
e
t
h
a
s
s
i
m
i
l
a
r
r
e
l
a
t
i
o
n
s
:
•
The localization of the wave packet over a small region to describe a
particle requires a large range of wave numbers. Conversely, a small
range of wave numbers cannot produce a wave packet localized within
a small distance.
Monday, Mar. 24, 2014
12
12
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
•
A Gaussian wave packet describes the envelope of a
pulse wave.
•
The group velocity is
Gaussian Function
Monday, Mar. 24, 2014
13
13
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Dispersion
•
Considering the group velocity of a de Broglie wave packet
yields:
•
The relationship between the phase velocity and the group
velocity is
•
H
e
n
c
e
t
h
e
g
r
o
u
p
v
e
l
o
c
i
t
y
m
a
y
b
e
g
r
e
a
t
e
r
o
r
l
e
s
s
t
h
a
n
t
h
e
p
h
a
s
e
v
e
l
o
c
i
t
y
.
A
m
e
d
i
u
m
i
s
c
a
l
l
e
d
n
o
n
d
i
s
p
e
r
s
i
v
e
w
h
e
n
t
h
e
p
h
a
s
e
v
e
l
o
c
i
t
y
i
s
t
h
e
s
a
m
e
f
o
r
a
l
l
f
r
e
q
u
e
n
c
i
e
s
a
n
d
e
q
u
a
l
t
o
t
h
e
g
r
o
u
p
v
e
l
o
c
i
t
y
.
Monday, Mar. 24, 2014
14
14
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Waves or Particles?
•
Young’s double-slit diffraction
experiment demonstrates the wave
property of light.
•
However, dimming the light results
in single flashes on the screen
representative of particles.
Monday, Mar. 24, 2014
15
15
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Electron Double-Slit Experiment
•
C. Jönsson of Tübingen,
Germany, succeeded in 1961
in showing double-slit
interference effects for
electrons by constructing very
narrow slits and using
relatively large distances
between the slits and the
observation screen.
•
This experiment demonstrated
that precisely the same
behavior occurs for both light
(waves) and electrons
(particles).
Monday, Mar. 24, 2014
16
16
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Which slit?
•
To determine which slit the electron went through: We set up a light
shining on the double slit and use a powerful microscope to look at
the region. After the electron passes through one of the slits, light
bounces off the electron; we observe the reflected light, so we know
which slit the electron came through.
•
Use a subscript
“
ph
”
to denote variables for light (photon). Therefore
the momentum of the photon is
•
The momentum of the electrons will be on the order of
.
•
The difficulty is that the momentum of the photons used to determine
which slit the electron went through is sufficiently great to strongly
modify the momentum of the electron itself, thus changing the
direction of the electron! The attempt to identify which slit the electron
is passing through will in itself change the interference pattern.
Monday, Mar. 24, 2014
17
17
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Wave particle duality solution
•
The solution to the wave particle duality of an event
is given by the following principle.
•
B
o
h
r
’
s
p
r
i
n
c
i
p
l
e
o
f
c
o
m
p
l
e
m
e
n
t
a
r
i
t
y
:
I
t
i
s
n
o
t
p
o
s
s
i
b
l
e
t
o
d
e
s
c
r
i
b
e
p
h
y
s
i
c
a
l
o
b
s
e
r
v
a
b
l
e
s
s
i
m
u
l
t
a
n
e
o
u
s
l
y
i
n
t
e
r
m
s
o
f
b
o
t
h
p
a
r
t
i
c
l
e
s
a
n
d
w
a
v
e
s
.
•
P
h
y
s
i
c
a
l
o
b
s
e
r
v
a
b
l
e
s
a
r
e
t
h
e
q
u
a
n
t
i
t
i
e
s
s
u
c
h
a
s
p
o
s
i
t
i
o
n
,
v
e
l
o
c
i
t
y
,
m
o
m
e
n
t
u
m
,
a
n
d
e
n
e
r
g
y
t
h
a
t
c
a
n
b
e
e
x
p
e
r
i
m
e
n
t
a
l
l
y
m
e
a
s
u
r
e
d
.
I
n
a
n
y
g
i
v
e
n
i
n
s
t
a
n
c
e
w
e
m
u
s
t
u
s
e
e
i
t
h
e
r
t
h
e
p
a
r
t
i
c
l
e
d
e
s
c
r
i
p
t
i
o
n
o
r
t
h
e
w
a
v
e
d
e
s
c
r
i
p
t
i
o
n
.
Monday, Mar. 24, 2014
18
18
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Uncertainty Principle
•
It is impossible to measure simultaneously, with no
uncertainty, the precise values of
k
and
x
for the same
particle. The wave number
k
may be rewritten as
•
For the case of a Gaussian wave packet we have
T
h
u
s
f
o
r
a
s
i
n
g
l
e
p
a
r
t
i
c
l
e
w
e
h
a
v
e
H
e
i
s
e
n
b
e
r
g
’
s
u
n
c
e
r
t
a
i
n
t
y
p
r
i
n
c
i
p
l
e
:
Monday, Mar. 24, 2014
19
19
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Energy Uncertainty
•
If we are uncertain as to the exact position of a particle, for
example an electron somewhere inside an atom, the particle
can’
t have zero kinetic energy.
•
The energy uncertainty of a Gaussian wave packet is
combined with the angular frequency relation
•
Energy-Time Uncertainty Principle:
.
Monday, Mar. 24, 2014
20
20
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Probability, Wave Functions, and the Copenhagen
Interpretation
•
The wave function determines the likelihood (or probability)
of finding a particle at a particular position in space at a
given time.
•
The total probability of finding the electron is 1. Forcing this
condition on the wave function is called normalization.
Monday, Mar. 24, 2014
21
21
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
The Copenhagen Interpretation
•
Bohr’s interpretation of the wave function consisted
of 3 principles:
•
The uncertainty principle of Heisenberg
•
The complementarity principle of Bohr
•
The statistical interpretation of Born, based on probabilities
determined by the wave function
•
Together these three concepts form a logical
interpretation of the physical meaning of quantum
theory. According to the Copenhagen interpretation,
physics depends on the outcomes of measurement.
Monday, Mar. 24, 2014
22
22
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Particle in a Box
•
A particle of mass
m
is trapped in a one-dimensional box of width
l
.
•
The particle is treated as a wave.
•
The box puts boundary conditions on the wave. The wave function must be zero at the walls
of the box and on the outside.
•
In order for the probability to vanish at the walls, we must have an integral number of half
wavelengths in the box.
•
The energy of the particle is
.
•
The possible wavelengths are quantized which yields the energy:
•
The possible energies of the particle are quantized.
Monday, Mar. 24, 2014
23
23
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
Discover the fascinating concepts of De Broglie waves, Bohr's quantization conditions, and electron scattering in physics. Delve into the wave-particle duality, electron double-slit experiments, and the groundbreaking observations by Davisson and Germer. Uncover the implications of mass particles having wave properties and the intriguing behavior of electrons in diffraction experiments. Dive into the world of quantum physics with this intriguing lecture series.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
PHYS 3313 Section 001 Lecture #16 Monday, Mar. 24, 2014 Dr. Jae Jaehoon Yu De Broglie Waves Bohr s Quantization Conditions Electron Scattering Wave Packets and Packet Envelops Superposition of Waves Electron Double Slit Experiment Wave-Particle Duality Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Yu 1 Dr. Jaehoon Yu
Announcements Research paper template has been uploaded onto the class web page link to research Special colloquium on April 2, triple extra credit Colloquium this Wednesday at 4pm in SH101 Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 2
De Broglie Waves Prince Louis V. de Broglie suggested that mass particles should have wave properties similar to electromagnetic radiation many experiments supported this! Thus the wavelength of a matter wave is called the de Broglie wavelength: l = h p This can be considered as the probing beam length scale Since for a photon, E = pc and E = hf, the energy can be written as E =hf = pc = pl f Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 3
Ex 5.2: De Broglie Wavelength Calculate the De Broglie wavelength of (a) a tennis ball of mass 57g traveling 25m/s (about 56mph) and (b) an electron with kinetic energy 50eV. l =h What is the formula for De Broglie Wavelength? (a) for a tennis ball, m=0.057kg. l =h p= 0.057 25 p 6.63 10-34 = 4.7 10-34m (b) for electron with 50eV KE, since KE is small, we can use non-relativistic expression of electron momentum! 1240eV nm 2 0.511MeV 50eV hc l =h h = p= = = 0.17nm 2meK 2mec2K What are the wavelengths of you running at the speed of 2m/s? What about your car of 2 metric tons at 100mph? How about the proton with 14TeV kinetic energy? What is the momentum of the photon from a green laser? Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 4
Bohrs Quantization Condition One of Bohr s assumptions concerning his hydrogen atom model was that the angular momentum of the electron-nucleus system in a stationary state is an integral multiple of h/2 . The electron is a standing wave in an orbit around the proton. This standing wave will have nodes and be an integral number of wavelengths. nl = nh 2pr = p The angular momentum becomes: nh 2p pp = nh L = rp = 2p=n Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 5
Electron Scattering Davisson and Germer experimentally observed that electrons were diffracted much like x rays in nickel crystals. direct proof of De Broglie wave! l =Dsinf n George P. Thomson (1892 1975), son of J. J. Thomson, reported seeing the effects of electron diffraction in transmission experiments. The first target was celluloid, and soon after that gold, aluminum, and platinum were used. The randomly oriented polycrystalline sample of SnO2 produces rings as shown in the figure at right. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 6
Photons, which we thought were waves, act particle like (eg Photoelectric effect or Compton Scattering) Electrons, which we thought were particles, act particle like (eg electron scattering) De Broglie: All matter has intrinsic wavelength. Wave length inversely proportional to momentum The more massive the smaller the wavelength the harder to observe the wavelike properties So while photons appear mostly wavelike, electrons (next lightest particle!) appear mostly particle like. How can we reconcile the wave/particle views? Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 7
Wave Motion De Broglie matter waves suggest a further description. The displacement of a wave is 2p l Y x,t ( )= Asin ( ) x-vt This is a solution to the wave equation 2Y x2=1 2Y t2 v2 Define the wave number k and the angular frequency as: k 2p l Y x,t w =2p l = vT and T [ ] ( )= Asin kx-wt The wave function is now: Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 8
Wave Properties The phase velocity is the velocity of a point on the wave that has a given phase (for example, the crest) and is given by vph=l T= l 2p 2p T =w k A phase constant shifts the wave: Y x,t ( )= Asin kx-wt +f [ = Acos kx-wt [ (When = /2) ] . ] Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 9
Principle of Superposition When two or more waves traverse the same region, they act independently of each other. Combining two waves yields: ( )= Y1x,t Dk 2 x-Dw cos kavx-wavt ( )+Y2x,t ( )= 2Acos ( ) Y x,t t 2 The combined wave oscillates within an envelope that denotes the maximum displacement of the combined waves. When combining many waves with different amplitudes and frequencies, a pulse, or wave packet, can be formed, which can move at a group velocity: ugr=Dw Dk Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 10
Fourier Series Adding 2 waves isn t localized in space but adding lots of waves can be. The sum of many waves that form a wave packet is called a Fourier series: ( Aisin kix-wit [ ] i )= Y x,t Summing an infinite number of waves yields the Fourier integral: ( [ ] )= A k ( )cos kx-wt Y x,t dk Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 11
Wave Packet Envelope The superposition of two waves yields a wave number and angular frequency of the wave packet envelope. Dk 2 x-Dw 2 The range of wave numbers and angular frequencies that produce the wave packet have the following relations: DkDx = 2p DwDt = 2p A Gaussian wave packet has similar relations: DkDx =1 DwDt =1 2 2 The localization of the wave packet over a small region to describe a particle requires a large range of wave numbers. Conversely, a small range of wave numbers cannot produce a wave packet localized within a small distance. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 12
Gaussian Function A Gaussian wave packet describes the envelope of a pulse wave. Y x,0 ( ( ) )= Y x ( )= Ae-Dk2x2cos k0x ugr=dw The group velocity is dk Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 13
Dispersion Considering the group velocity of a de Broglie wave packet yields: ugr=dE dp=pc2 E The relationship between the phase velocity and the group velocity is ugr=dw dk= dk )=vph+kdvph ( d vphk dk Hence the group velocity may be greater or less than the phase velocity. A medium is called nondispersive when the phase velocity is the same for all frequencies and equal to the group velocity. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 14
Waves or Particles? Young s double-slit diffraction experiment demonstrates the wave property of light. However, dimming the light results in single flashes on the screen representative of particles. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 15
Electron Double-Slit Experiment C. J nsson of T bingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen. This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles). Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 16
Which slit? To determine which slit the electron went through: We set up a light shining on the double slit and use a powerful microscope to look at the region. After the electron passes through one of the slits, light bounces off the electron; we observe the reflected light, so we know which slit the electron came through. Use a subscript ph to denote variables for light (photon). Therefore the momentum of the photon is h >h Pph= lph d Pe=h ~h The momentum of the electrons will be on the order of . le d The difficulty is that the momentum of the photons used to determine which slit the electron went through is sufficiently great to strongly modify the momentum of the electron itself, thus changing the direction of the electron! The attempt to identify which slit the electron is passing through will in itself change the interference pattern. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 17
Wave particle duality solution The solution to the wave particle duality of an event is given by the following principle. Bohr s principle of complementarity: It is not possible to describe physical observables simultaneously in terms of both particles and waves. Physical observables are the quantities such as position, velocity, momentum, and energy that can be experimentally measured. In any given instance we must use either the particle description or the wave description. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 18
Uncertainty Principle It is impossible to measure simultaneously, with no uncertainty, the precise values of k and x for the same particle. The wave number k may be rewritten as k =2p l =2p h p= p2p =p h For the case of a Gaussian wave packet we have DkDx =DpDx =1 2 Thus for a single particle we have Heisenberg s uncertainty principle: DpxDx 2 Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 19
Energy Uncertainty If we are uncertain as to the exact position of a particle, for example an electron somewhere inside an atom, the particle can t have zero kinetic energy. ( ) 2 Dp 2m 2 2 Kmin=pmin 2m 2ml2 The energy uncertainty of a Gaussian wave packet is DE = hDf = hDw = Dw 2p combined with the angular frequency relation DwDt =DEDt =1 2 Energy-Time Uncertainty Principle: . DEDt 2 Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 20
Probability, Wave Functions, and the Copenhagen Interpretation The wave function determines the likelihood (or probability) of finding a particle at a particular position in space at a given time. P y ( )dy = Y y,t ( ) 2dy The total probability of finding the electron is 1. Forcing this condition on the wave function is called normalization. P y ( )dy - - + + Y y,t ( ) 2dy = =1 Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 21
The Copenhagen Interpretation Bohr s interpretation of the wave function consisted of 3 principles: The uncertainty principle of Heisenberg The complementarity principle of Bohr The statistical interpretation of Born, based on probabilities determined by the wave function Together these three concepts form a logical interpretation of the physical meaning of quantum theory. According to the Copenhagen interpretation, physics depends on the outcomes of measurement. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 22
Particle in a Box A particle of mass m is trapped in a one-dimensional box of widthl. The particle is treated as a wave. The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside. In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box. The energy of the particle is . The possible wavelengths are quantized which yields the energy: The possible energies of the particle are quantized. Monday, Mar. 24, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu 23
