chapter 8. quantum theory: techniques and applications...
TRANSCRIPT
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Chapter 8.
Quantum theory:Techniques and applications
-Translational (Vk=0)-Vibrational-Rotational-Techniques and approximation
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Translational motion
8.1 A particle in a box8.2 Motion in two and more dimensions8.3 Tunnelling
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8.1 A particle in a box
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What is “a particle in a box”?
A particle (mass: m) is confined,
Outside the wall: infinite potentialInside the wall: zero potential
Ex) -a gas molecule in 1-d container-Electronic structure of a metal or conjugated molecule
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Energy and wavefunction of a particle in a box
( ) sin cosk x C kx D kx
(0) ( ) 0k k L
2 2
2k
kE
m
With B.C.,
(quantized)1 2
2( ) sink
n xx
L L
2 2
2, 1,2,...
8n
n hE n
mL
How about momentum?
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2 2
2, 1,2,...
8n
n hE n
mL
Properties of the solutions (a particle in a box)
: Figure 8.2
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The average value of the linear momentum of a particle in a box: <p>=0
The average value of p2 of a particle in a box: <p2>=0
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Existence of zero point E:(for classical mechanics: lowest E is 0.)
1. Location is not completely indefinite.
Momentum is non-zero.
KE is non-zero.
2. ψ is curved. -> non-zero KE(0 at walls. But, smooth, continuous, and non-zero everywhere)
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Separation E,
For free particles, Translation E is not quantized any more.
-> the particles are not confined any more!
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When n becomes infinite,
-The particles are not bounded any more.-the probability becomes more uniform.-Classical mechanics emerges.
Correspondence principle
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Orthogonality and the bracket notation
Dirac bracket notation
Kronecker deltaOrthonormal
Wave functions corresponding to different energies are orthogonal
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Figure 8.3
The first five normalized wave function of a particle in a box
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Figure 8.4
The first two normalized wave function of a particle in a box and the corresponding probability distribution
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8.2 Motion in two and more dimensions
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a particle in a 2-d box Figure 8.5
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a particle in a 3-d box
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Figure 8.6
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Degeneracy in a 2-d box
Degenerate: the same Energy
Ex) for 2-d: square box (ψ1,2= ψ2,1)
Figure 8.7
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8.3 Tunnelling
Very important for light particles: electrons, muons, and moderate for protons
Ex. Isotope dep. Rxn ratep. 336
STM (p. 337)
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When E<V, the wave function does not decay to „0‟ abruptly!
No oscillation
Figure 8.8
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For, x<0, V=0
(0<x<L, E<V)
(since V-E>0…)
To the right of the barrier, x>L
(for x>L, V=0)
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Figure 8.9
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Figure 8.10
Continuity of Wavefunction and its slope at the boundary
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The wavefunction and its slope should be continuous,
(at x=0) (at x=L)
(at x=0) (at x=L)
• B‟=0after the barrier no particles travelling to the leftNote. We cannot say B≠0.
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Transmission probability
For high wide barriers,
2
2
'AT
A
E V
1L
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E<V: E>V
Classically T=0
1 2(2 )L mV
Classically T=1
Figure 8.11
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T exponentially decreases with and1 2mL
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A particle in a square well potential of finite depth
•Difference from an infinitely deep well case: finite number of bound states
Deeper and wider: greater the # of the states
a potential well of finite depthTwo lowest bound-state of wavefunctions for a particle in a potential well of finite depth
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Scanning Probe Microscopy
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Physical Chemistry Fundamentals: Figure 8.16
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Vibrational motion
8.4 The energy levels8.5 The wavefunctions
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8.4 The energy levels
Harmonic motion of a particle(w/ restoring force)
dVF kx V
dx
2
2
kxV
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Parabolic Potential E of a harmonic oscillator
2
2
kxV
• Displacement at equilibrium
• Narrowness dep. on k:
larger k, narrower the wall
-> stronger confinement
Figure 8.17
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• For the typical molecular oscillator:
E0~30 zJ (10-21J)
• Zero point E is non-zero.
-position is not completely uncertain
-momentum and KE is non-zero
-particle fluctuates around eq. position
Note) Classical Mechanics allows the particle to be perfectly still.Figure 8.18
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8.5 The wavefunctions
Difference betw. Particle in a box and Harmonic motion of a particle
1. In the oscillator, ψ →0 more slowly at large x.
(P.E. is proportional to x2.)
2. K.E. of the oscillator and curvature of the wave function depends on x in a more complex way
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The form of the wavefunctions
(Fig 19 and 20)
(Fig 21)
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Hermite polynomial
(orthogonal!!!
Not normalized yet.)
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(largest probability at eq)
Graph of Gaussian curve Waveftn and probability distribution for lowest E of a harmonic oscillaotr
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(Fig 21)
Waveftn and probability distribution for the first excited state of a harmonic oscillaotr
Figure 8.21
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The first 5 wavefunction
• The number of nodes are equal to ν.
• Even v : symmetrical
• Odd v : antisymmetrical
Figure 8.22
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The first 5 probability distribution
• Largest amplitude at high quantum numbers (see 20),
near the turning point of CM motion (Ek=0 or V=E).
In Classical Mechanics, the particle becomes slowest.
• At x=0, the particle is least likely to be found
(it travels most rapidly)
Figure 8.23
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The properties of the oscillators
Expectation value
Dirac‟s braket notation
' Or Matrix element
the same states
Ex. for harmonic oscillator,
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The mean potential E
Since total E is 1
2
KE and PE is equal… Special case of virial theorem
Virial theorem: if V=axb, then
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An oscillator can be found even at V>E!!!
Beyond its classical limit, p~0.079
(these tunnelling probabilities are indep. of mass and force const.
Macroscopic oscillator (such as pendulum) are in state with very high quantum number -> p(V>E)~0
But, for molecules are normally in their vib gnd state: p(V>E)~significant
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In classical mechanics,
(E=V=kx2/2, Ek=0)
Probability that an oscillator is stretched beyond its CM turning point,
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Rotational motion
8.6 Rotation in 2-d: the particle on a ring8.7 Rotation in 3-d: the particle on a sphere8.8 Spin
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8.6 Rotation in 2-d:
the particle on a ring
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: angular momentumJz
0, kV E E
z
hrJ pr
2 : moment of inertiaI mr2 2 2
22 2 2
z zp J JE
m mr I
Figure 8.24
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Qualitative origin of quantized rotation
Not acceptable: not single valuedand destructive
Acceptable:constructive
Figure 8.25
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Figure 8.26
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Cylindrical coordinate
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Cyclic boundary condition
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Shorter the wavelength,Angular momentum (z-axis)the larger in steps of h/2
Real parts of the waveftns of a particle on a ring
Figure 8.28
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Definition of angular momentum
Angular momentum operator
in cylindrical coordinate,
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Probability density of a particle in a definite state of angular momentum~uniform
Figure 8.30
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8.7 Rotation in 3-d
: e- in atomsRotating molecules
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• Two cyclic BC
• Two quantum numbers
• V=0
• r: constant
Waveftn of a particle on a surface
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The Schrodinger equation
Laplacianor del squaredor nabla squared
colatitude
azimuth
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Since r is constant,
Justification 8.6
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Same as 8.4 Associated Legendre ftn
w/ cyclic BC: l (quantum number) appearsml is restricted by l
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Orbital angular momentumquantum numberl : 0,1,2,3…
Magnetic quantum numberml : l, l -1, …,-l
Spherical harmonics
, ll mY
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Note: most probable location of particlesmigrates towards xy plane as increases
•ml doesn‟t affect E•2l+1 degeneracy
lm
Figure 8.34
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Angular momentum
Number of node w/ l
Higher momentum, Higher Ek
More nodal lines cut the equator (note: the curvature is greater in the direction)
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Space Quantization [1]
• For a given l,
Angular momentum for z-
axis: 2l +1
• Discrete range of orientations
• The orietation of rotating body is quantized
Figure 8.35
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Figure 8.36
Space Quantization [2]
• Beam of silver atom
• Magnetic field
• Classically expected
• Observation
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The vector model [1]
• No reference to x,y axis
• Once lz is known, lx and ly can‟t be known
(due to uncertainty principle)
• lz is known, lx and ly are complementary
(they are not commute)
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Physical Chemistry Fundamentals:
Figure 8.37
The vector model [2]
Schematic diagram
Better schematic diagram
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8.8 Spin
: not the actual spin motioninteract w/ magnetic field
Stern and Gelach observed two band(2l +1=2…l =1/2 ? … s instead l )
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Spin quantum number: sSpin magnetic quantum number: ms
for electron: s=1/2ms=1/2 ( ), -1/2 ( )
Figure 8.38
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• Electron, proton and neutron: spin ½ particles (s=1/2)
with the angular momentum of (3/4)1/2
Despite the mass difference, they have the same spin angular momentum
(In CM, proton and neutrons should spin slower.)
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• Photon:
Spin 1 particles (s=1)
with the angular momentum of (2)1/2
zero rest mass
zero charge
an energy hv
linear momentum hv/c
speed: c
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• Fermion: particles with half-integral spinsAll the elementary particles that constitutes matter
ex) electron and protons
• Boson: particles with integral spinsfundamental particles that are responsible for the
forces that binds fermions together
ex) photons (transmit EM forces that binds together electrically charged particles)
• Matter: is an assembly of fermions held together by bosons
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Table 8.3