stanford qm
TRANSCRIPT
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Hermitian linear operator Real dynamical variable
represents observable quantity
Eigenvector A state associated with an
observable
Eigenvalue Value of observable associated
with a particular linear operator
and eigenvector
Copyright Michael D. Fayer, 2007
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Momentum of a Free Particle
Know momentum, p, and position, x, cancan predict exact location at any subsequent time.
Solve Newton's equations of motion.
p = mV
What is the quantum mechanical description?
Should be able to describe photons, electrons, and rocks.
Free particleparticle with no forces acting on it.
The simplest particle in classical and quantum mechanics.
classical
rock in interstellar space
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Momentum eigenvalue problem for Free Particle
P P P
momentum momentum momentum
operator eigenvalues eigenkets
Know operator want: eigenvectorseigenvalues
Schrdinger Representationmomentum operator
P ix
LaterDiracs Quantum Condition shows how to select operators
Different sets of operators form different representations of Q. M.
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Try solution
/i xP ce eigenvalue Eigenvalueobservable.
Proved that must be real number.
Also: If not real, function would blow up.
Function represents probability (amplitude)
of finding particle in a region of space.Probability cant be infinite anywhere. (More later.)
Function representing state of system in a particularrepresentation and coordinate system called wave function.
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Schrdinger Representation
momentum operator
P ix
/i xP ce Proposed solution
Check
/ /
/
/i x i x
i x
i ce i i cex
ce
Operating on function gives
identical function back
times a constant.
P P P
continuous range of eigenvalues
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/ipx ikx
pP ce ce momentum
eigenkets
momentum
wave functions
Have called eigenvalues =pp k k is the wave vector.
c is the normalization constant.
c doesnt influence eigenvalues
direction not length of vector matters.
Normalization
a b scalar product
a a scalar product of a vector with itself
1/ 2 1a a normalized Copyright Michael D. Fayer, 2007
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Normalization of the Momentum Eigenfunctionsthe Dirac Delta Function
b ab a d
scalar product of vector functions (linear algebra)
*
a ad .a a scalar product of vector function with itself
*1
0P P
if P PP P dx
if P P
normalization '
orthogonality '
P P
P P
2 ' 2 ( ')ik x ikx i k k xc e e dx c e dx
P kUsing
( ') cos( ') sin( ') .i k k xe dx k k x i k k x dx
Cant do this integral with normal methodsoscillates.
Copyright Michael D. Fayer, 2007
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Dirac d function
( ) 1x dx
Definition
( ) 0 if 0x x
For functionf(x) continuous atx = 0,
( ) ( ) (0)f x x dx f
or
( ) 0
( ) ( ) ( )
x a if x a
f x x a dx f a
d
d
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Physical illustration
x x=0
unit area
double height
half width
still unit area
Limit width 0, height area remains 1
Copyright Michael D. Fayer, 2007
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-40 -20 20 40
-0.2
0.2
0.4
0.6
0.8
1
Mathematical representation ofd function
sin( )g x
x
zeroth order spherical
Bessel function
g any real numberg =
Has valueg/
atx = 0.Decreasing oscillations for increasing |x|.
Has unit area independent of the choice ofg.
As g
dsin
( ) limg
gxx
xd
1. Has unit integral2. Only non-zero at pointx = 0
3. Oscillations infinitely fast, over
finite interval yield zero area.
Copyright Michael D. Fayer, 2007
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Use d(x) in normalization and orthogonality of momentum eigenfunctions.Want p p 1
p p' 0
2 '1 if '
0 if '
ik x ikxk k
c e e dxk k
Adjustc to make equal to 1.
( ') cos( ') sin( ')i k k xe dx k k x i k k x dx
Evaluate
g cos( ') sin( ')limg
g
k k x i k k x dx
Rewrite
g
sin( ') cos( ')lim
( ') '
g
g
k k x i k k x
k k k k
Copyright Michael D. Fayer, 2007
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g
sin( ') cos( ')lim
( ') '
g
g
k k x i k k x
k k k k
2 sin ( ')lim
( ')g
g k k
k k
2 ( ')k kd d function multiplied by 2.
2 ( ') 0 if 'k k k kd The momentum eigenfunctions areorthogonal. We knew this already.Proved eigenkets belonging to differenteigenvalues are orthogonal.function
To find out what happens fork = k' must evaluate .2 ( ')k kd But, whenever you have a continuous range in the variable of a vector function
(Hilbert Space), can't define function at a point. Must do integral about point.'
'
( ') ' 1 if
k k
k k
k k dk k k
d
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if
if2
2 '1'
0 '
P PP P
c P P
Therefore
P are orthogonal and the normalization constant is
2 1
21
2
c
c
The momentum eigenfunction are
1( )
2
ikx
p x e k p momentum eigenvalue
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Wave Packets
P P p P
momentum momentum momentum
operator eigenvalue eigenket
Copyright Michael D. Fayer, 2007
Free particle wavefunction in the Schrdinger Representation
1 1( ) cos sin2 2
ikx
pP x e kx i kx k p
State of definite momentum
2k
wavelength
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1 1( ) cos sin2 2
ikx
pP x e kx i kx k p
What is the particle's location in space?
x
real imaginary
Not localized Spread out over all space.
Equal probability of finding particle from .
Know momentum exactly No knowledge of position.
to
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Not Like Classical Particle!
Plane wave spread out over all space.
Perfectly defined momentum
no knowledge of position.
What does a superposition of states of definite momentum look like?
( ) ( ) ( )p p
p
x c p x dp indicates that wavefunction is composed
of a superposition of momentum eigenkets
coefficienthow much of each
eigenket is in the superposition
must integrate because continuous
range of momentum eigenkets
Copyright Michael D. Fayer, 2007
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0
0
k k
ikx
k
k k
e dk
Then Superposition ofmomentum eigenfunctions.
Same as integral for normalization but
integrate overk aboutk0
instead of overx aboutx = 0.
02sin( )
ik x
k
k xx
xe
Rapid oscillations in envelope ofslow, decaying oscillations.
Work with wave vectors - /k p
k0k k0 k0 + kk
Wave vectors in the range
k0
k to k0 +
k.In this range, equal amplitude of each state.
Out side this range, amplitude = 0.
0k k
2sin( )At 0 have lim 0 2
k xx x k
x
First Example
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0 0 02sin( ) cos sink kx k x i k x k kx
Writing out the 0
ik xe
X
k
This is the envelope. It is filled in
by the high frequency real and imaginary
oscillations
Wave Packet
The wave packet is now localized in space.
As | | large, small.kx
Greater k more localized.Large uncertainty inp small uncertainty inx.
Copyright Michael D. Fayer, 2007
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Born Interpretation of Wavefunction
Schrdinger Concept of Wavefunction
Solution to Schrdinger Eq.
Eigenvalue problem (see later).
Thought represented
Matter Waves
real entities.
Born put forward
Like E field of light amplitude of classical E & M wavefunction
proportional toE field, .
Absolute value square ofE field Intensity.
E
2 *| |E E E IBorn Interpretationsquare of Q. M. wavefunction proportional to probability.
Probability of finding particle in the regionx + x given by
*Prob( , ) ( ) ( )
x x
x
x x x x x dx
Q. M. Wavefunction Probability Amplitude.
Wavefunctions complex.
Probabilities always real.
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Localization caused by regions of constructive and destructive interference.
5 waves Cos( ) 1.2, 1.1, 1.0, 0.9, 0.8ax a
-30 -20 -10 10 20 30
-4
-2
2
4
Sum of the 5 waves
Copyright Michael D. Fayer, 2007
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Wave Packetregion of constructive interference
Combining different ,
free particle momentum eigenstates
wave packet.
P
Concentrateprobabilityof finding particle in some region of space.
Get wave packet from
0
0
( )( )k
ik x
f k k
x
x e dk
Wave packet centered around
x = x0,
made up ofk-states centered aroundk = k0.
0 weighting fu c n) o( n tif k k Tells how much of eachk-state is in superposition.
Nicely behaved dies out rapidly away fromk0
.
k
f(k-k0)
k0 Copyright Michael D. Fayer, 2007
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0 weighting fu c n) o( n tif k k Only havek-states neark = k0. 0 0
1ik x x
e e At x = x0
Allk-states in superposition in phase.
Interfere constructively add up.
All contributions to integral add.
For large (x
x0) 0 0 0cos sinik x xe k x x i k x x Oscillates wildly with changingk.
Contributions from differentk-states destructive interference.
0| | 0kx x Probability of finding particle 02
k wavelength
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Gaussian Wave Packet
Gaussian distribution ofk-states
2
0
22
( )1( ) exp
2( )2
k kf k
kk
Gaussian in k, with
normalizedk
2 2
02
1( ) exp[ ( ) / 2 ]
standard deviat
2
ion
G y y y
Gaussian Function
2o
2o
( )
( )2( )
2
1
( ) 2
k k
ik x xk
k x e e dkk
Gaussian Wave Packet
0
0( )
ik x xef k k
Written in terms ofxx0 Packet centered aroundx0.
Copyright Michael D. Fayer, 2007
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Gaussian Wave Packet
Looks like momentum eigenket atk = k0
times Gaussian in real space.
Width of Gaussian (standard deviation)
1
k
The bigger k, the narrower the wave packet.Whenx = x0, exp = 1.
For large |x - x0|, ifk large, real exp term
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x = x0
1
k
k smallpacket wide.
Gaussian Wave Packets
1
k
x = x0
k largepacket narrow.
To get narrow packet (more well defined position) must superimpose
broad distribution ofk states. Copyright Michael D. Fayer, 2007
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Brief Introduction to Uncertainty Relation
Probability of finding particle
betweenx &x + x*Prob( ) ( ) ( )
x x
x
x x x dx
Written approximately as
( ) ( )x xFor Gaussian Wave Packet
* 2 2
0exp ( ) ( )k k x x k (noteProbabilities are real.)This becomes small when
2 2
0( ) ( ) 1x x k 0Call , then
1x
x x x
k
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1x k
1
pk
x p
x p
Superposition of states leads to uncertainty relationship.
Large uncertainty inx, small uncertainty inp, and vis versa.
(See later that
differences from choice of 1/e point instead of)/2x p
spread or "uncertainty" in momentum
momentumk p
k p
Copyright Michael D. Fayer, 2007
Ob i Ph t W P k t
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-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1850 2175 2500 2825 3150
frequency (cm-1
)
In
tensity(Arb.
Units)
368 cm-1 FWHM
Gaussian Fit
/p h
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
-0.2 -0.1 0 0.1 0.2
Delay Time (ps)
Intensity(Arb.
Units)
40 fs FWHMGaussian fit
collinearautocorrelation
Observing Photon Wave Packets
Ultrashort mid-infrared optical pulses
1
29
28
0
/ /
2.4 10 kg-m/s (14%)
1.7 10 kg-m/s
p h p h
p
p
-15
5
40 fs = 40 10 s
1.2 10 m 12 m
t
x c t
x
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CRT - cathode ray tube
-
cathode
filament
controlgrid
screenelectronsboil off
electrons aimed electronshit screen
accelerationgrid
+
e-
Old style TV or computer monitor
Electron beam in CRT.
Electron wave packets like bullets.
Hit specific points on screen to give colors.
Measurement localizes wave packet.
Copyright Michael D. Fayer, 2007
El t diff ti Electron beam impinges
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Peaks line up.
Constructive interference.
in coming
electron
wave"
out goingwavescrystal
surface
Electron diffraction
many grating
different directionsdifferent spacings
Low Energy Electron Diffraction (LEED)
from crystal surface
Electron beam impingeson surface of a crystallow energy electron diffractiondiffracts from surface.Acts as wave.Measurementdetermines
momentum eigenstates.
Copyright Michael D. Fayer, 2007
G V l iti
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Group Velocities
Time independent momentum eigenket
1
2pikxP x e p k
Direction and normalization ket still not completely defined.
Can multiply by phase factor realie
Ket still normalized.
Direction unchanged.
Copyright Michael D. Fayer, 2007
F ti i d d t E
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For time independent Energy
In the Schrdinger Representation (will prove later)
Time dependence of the wavefunction is given by
/i E t i te e E Time dependent phase factor.
1,i t i t
e e Still normalized
The time dependent eigenfunctions of the momentum operator are
0( ) ( )
,k
i k x x k tx t e
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Time dependent Wave Packet
0, ( )exp ( )k x t f k i k x x k t dk
time dependent phase factor
Photon Wave Packet in a vacuum
Superposition of photon plane waves.
Need c c 2k
c
2
Copyright Michael D. Fayer, 2007
U i k
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Using ck 0( , ) ( )exp ( )k x t f k ik x x ct dk
Have factored outk.Att = 0,
Packet centered atx = x0.
Argument of exp. = 0. Max constructive interference.
At later times,
Peak atx = x0 + ct.
Point where argument of exp = 0 Max constructive interference.
Packet moves with speed of light.
Each plane wave (k-state) moves at same rate,c.
Packet maintains shape.
Motion due to changing regions of constructive and destructive
interference of delocalized planes waves
Localization and motion due to superposition
of momentum eigenstates.Copyright Michael D. Fayer, 2007
Photons in a Dispersive Medium
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Photons in a Dispersive Medium
Photon enters glass, liquid, crystal, etc.
Velocity of light depends on wavelength.
n() Index of refraction
wavelength dependent.
n()
bluered
glassmiddle of visible
n ~ 1.5
2 c / n( ) 2 / c
n( ) 2
V k
V
Dispersion, no longer linear ink.( )k
Copyright Michael D. Fayer, 2007
W P k t
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Wave Packet
, ( )exp ( ) ( ) )k ox t f k i k x x k t dk
Still looks like wave packet.
0At 0,t x xMoves, but not with velocity of light.
Different states move with different velocities
blue waves move slower than red waves.
Velocity of a single state in superposition
Phase Velocity.
Copyright Michael D. Fayer, 2007
V l it l it f t f k t
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Velocityvelocity of center of packet.
00,t x x max constructive interferenceArgument of exp. zero for allk-states at 0 , 0.x x t The argument of the exp. is
0( ) ( )k x x k t (k) does not change linearly withk.
Argument will never again be zero
for allk in packet simultaneously.
Most constructive interference when argument changes withk
as slowly as possible.
This produces slow oscillations.
k-states add constructively, but not perfectly.
Copyright Michael D. Fayer, 2007
No longer perfect constructive interference at peak
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-3 -2 -1 1 2 3
-1
-0.5
0.5
1
Red waves get ahead.
Blue waves fall behind.
No longer perfect constructive interference at peak.
The argument of the exp. is 0( ) ( )k x x k t When changes slowly as a function ofk within range ofk determined byf(k)
Constructive interference.
Find point of max constructive interference. changes as slowly as possible withk. 0
00
k
x x tk k
0
0
k
x x tk
max of packetat time, t. Copyright Michael D. Fayer, 2007
Distance Packet has moved at time t
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Distance Packet has moved at time t
0
0( )
k
x x tk
d Vt
Point of maximum constructive interference (packet peak) moves at
0
Group VelocitygVk
k
Vg speed of photon in dispersive medium.
Vp phase velocity = /konly same when (k) = ck vacuum (approx. true in
low pressure gasses)
Copyright Michael D. Fayer, 2007
Electron Wave Packet
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Electron Wave Packet
de Broglie wavelength
/p h k (1922 Ph.D. thesis)A wavelength associated with a particle
unifies theories of light and matter.
2 2
( ) /2 2
p kk E
m m E
Non-relativistic free particleenergy divided by .
p kUsed
Copyright Michael D. Fayer, 2007
V Free non zero rest mass particle
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VgFree non-zero rest mass particle
2( )
/2
g
k k kV p m
k m k m
/ /p m mV m VGroup velocity of wave packet same as Classical Velocity.
However, description very different. Localization and motion due tochanging regions of constructive and destructive interference.
Can explain electron motion and electron diffraction.
Correspondence PrincipleQ. M. gives same results as classical mechanics
when in classical regime.
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Q. M. Particle Superposition of Momentum Eigenstates
Partially localized Wave Packet /2x p
PhotonElectron
Photon wave packet description of light same as
wave packet description of electron.
Therefore, Electron and Photon can act like wavesdiffract
or act like particleshit target.
WaveParticle duality of both light and matter.