4.3 the free-particle schrodinger eq. chapter. 4 wave
TRANSCRIPT
PHYS-3301
Sep. 28, 2017
Lecture 10
AnnouncementCourse webpage
http://www.phys.ttu.edu/~slee/3301/
Textbook
HW4 (due 10/5)
Chapter 413, 15, 20, 31, 36, 41, 48, 53, 63, 66
Exam 1 (10/5) Chapters 2, 3, & 4
Chapter. 4Wave & Particles II
Outline:
• A Double-Slit Experiment (watch “video”)• Properties of Matter Waves• The Free-Particle Schrödinger Equation• Uncertainty Principle• The Bohr Model of the Atom• Mathematical Basis of the Uncertainty Principle –The Fourier Transform
“Matter” behaving as “Waves”
4.3 The Free-Particle Schrodinger Eq.[Q] How do we determine the ψ(x,t) of a matter wave?[A] Matter Waves
-- wave eq. obeyed by matter waves è Schrödinger eq.-- for free particle, in the absence of external forces,
[Q] Schrodinger eq. is “complex”. Matter wave is not “real”?[A] Reason for “i” :: NOT matter waves are “unreal”
BUT they can’t be represented by a single real function !!
e.g. like E&M waves, 2 parts (E & B) and a complex function,carrying twice the info. of a real one.
===> In E&M, we could treat E & B as a single complex unit by including “i” w/o making either field “unreal” ===> Combined WF, G = E + iCB
(Wave Function)2 = Probability Density
real now!!
The Free-Particle
Schrodinger Wave
Equation
),( tx!
2*|),(|),(),( txtxtx !!! =
Probability Wave Function
Probability
Density
a complex function),( tx!
4.3 The Free-Particle Schrödinger Eq.***** Probability Density
--- Probability of detecting the particle ~ (wave’s Amplitute)2
[Q] What does this mean if wave has 2 parts; E & B or real & imaginary part of ψ(x,t)?
Probability density = |ψ(x,t)|2
per unit lengthper unit volumeprobability of finding the particle in certain region!!
),(),(),( 21 txitxtx !!! +"
Probability Density
),(),(),( 21
*txitxtx !!! #"
1#=i
== ),(),(|),(|*2
txtxtx !!!
=+##=12212211
!!!!!!!! iiii
2211!!!! +=
),(),(|),(|2
2
2
1
2txtxtx !!! +=
),(),(),( 21 txitxtx !!! +"
Probability Density = ?
),(),(),( 21
*txitxtx !!! #"
Complex Conjugate
1#"i
12 #=i
4.3 The Free-Particle Schrödinger Eq.The Plane Wave[Solution] -- complex exponential
The Plane Wave
Ue!Is the Plane Wave
a solution of the Schrodinger Equation?
Is the Plane Wave
a solution of the Schrodinger Equation?
Is the Plane Wave
a solution of the Schrodinger Equation?
Taking the partial derivatives on both sides..
Is the Plane Wave
a solution of the Schrodinger Equation?
Is the Plane Wave
a solution of the Schrodinger Equation?
22
)(22
mv
m
mvE ==
YES
Let’s see how the Schrödinger wave eq. relates to the classical physics of particle?
So. Our answer is “YES”.
It’s a solution of Schrödinger eq.
Is the Plane Wave
a solution of the Schrodinger Equation?
22
)(22
mv
m
mvE ==
YES
Mean: Particle’s KE = Total ETrue, classically!!since a free particle has no PE
Is the Plane Wave
a solution of the Schrodinger Equation?
22
)(22
mv
m
mvE ==
YES
The Schrödinger equation is related to a classical accounting of energy!!
Mean: Particle’s KE = Total ETrue, classically!!since a free particle has no PE
The Magnitude of a Plane Wave
Constant in space and time!
The Magnitude of a Plane Wave
Constant in space and time!
Constant Probability Density
= constant (x,t)
Kinetic
energy
Plane
wave
Uncertainty
Principle
Gaussian Wave form
!2
1=!! xpx = minimum uncertainty
The Uncertainty Relations
in 3 Dimensions
The Uncertainty Relations and the
Fourier Transform
Any wave may be expressed mathematically as a
superposition of plane waves of different
wavelengths and amplitudes
Fourier Transform
dxexkikx!
"
"!
!= )(2
1)(~ #
$#
dkekxikx
)(~
2
1)( !
"
"!
= #$
#
Fourier Transform
Fourier Transform
dxexkikx!
"
"!
!= )(2
1)(~ #
$#
dkekxikx
)(~
2
1)( !
"
"!
= #$
#
Spectral
Contentxikx
eAex 02)2/()( !" #
=
Gaussian Wave Packet
2)/(22|)(| !" xeAx#
=
Probability Density:
?)(~ =k!xikxeAex 02)2/(
)("! #
=
Gaussian Wave Packet
Find the
“Spectral Content”:
?
dxexkikx#
$
$#
!= )(2
1)(~ !
%!
xkp ˆ0!
"=
?)(~ =k!xikxeAex 02)2/(
)("! #
=
Gaussian Wave Packet
Find the
“Spectral Content”:
?
dxexkikx#
$
$#
!= )(2
1)(~ !
%!
xkp ˆ0!
"=
!
!!
!"
"+"
!
!"
""
=
=
dxeA
k
dxeeAek
xkkix
xikxikx
)()4/1(
)2/(
022
02
2)(~
)(2
1)(~
#
#
$%
$%
)(
4/1
0
2
4/22
kkib
a
aedze
abbzaz
"=
=
=!!
!"
+"
#
$
20
2
20
2
022
)(
2)(
)()4/1(
)(~
42
)(~
2)(~
kk
kk
xkkix
eA
k
eA
k
dxeA
k
!!
!!
"
"!
!+!
=
=
= !
#
#
#
$
#%
$#$
%
$%
G“Spectral Content” is
a Gaussian function !
Not a complex
function
20
2)(
)(~kk
eA
k!!
="
#
"$xikx
eAex 02)2/(
)("$ !
=2
02
)()(~
kke
Ak
!!=
"
#
"$xikx
eAex 02)2/(
)("$ !
=
22)(
!!! ==%%=%%
""kxpx
Gaussian Wave Packet
Minimal Uncertainty
1 standard deviation
A Single-Slit A Single-Slit
Spectral Content