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Physical Chemistry III (728342)
The Schrödinger EquationPiti TreesukolKasetsart UniversityKamphaeng Saen Campus
http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon
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Solving Schrödinger Equations Eigenvalue problem
Simple cases• The free particle • The particle in a box • The finite potential well • The particle in a ring • The particle in a spherically
symmetric potential • The quantum harmonic oscillator • The hydrogen atom or hydrogen-like
atom • The particle in a one-dimensional
lattice (periodic potential)
VEmdxd
ExVdxd
m
22
2
2
22
2
)(2
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Solutions for a Particle in a 1-D Box Solve for possible r and s in
that satisfy the Schrödinger equation.
Not all solutions are acceptable. The wave function needs to be continuous in all regions
sxBrxA cossin
a0for )2(cos)2(sin
)2(12/112/1
12/1
xxmEBxmEA
mEsr
II
x
0I 0III conditionsboundary 0)(0)0(
aII
II
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Using the boundary conditions
• Only these allowed energy can make the wavefunction well-behaved.
Solutions for a Particle in a 1-D Box
0 00cos0sin)0( BBAII
...3,2,1 ;8
)2( 0)2(sin
0)2(sin)(
2
22
2/112/1
12/1
nma
hnE
namEamE
amEAaII
n=1n=2
n=3
n=4En
ergy
E1
E2
E3
E4
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Solutions for a Particle in a 1-D Box Normalizability of the
wavefunction
The lowest energy is when n=1 For Macroscopic system,
n is very large
axa
xna
aiA
cxdxcx
dxa
xnAdxdx
a
aa
II
0for sin2
2
41
2sin
sin||||||1
2/1
2/1
0
2
0
22
0
22
Normalization constant
2
2
8mahE Zero-point
energy
groundstate
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Solutions for a Particle in a 3-D Box For the 3-D box (axbxc), the
potential outside the box is infinity and is zero inside the box
Edzd
dyd
dxd
m
2
2
2
2
2
22
2
)()()( zZyYxX
EzZzZ
myYyY
mxXxX
m
)()("
22
)()("
22
)()("
22
2
222/1
2
222/1
2
222/1
8 ;sin2)(
8 ;sin2)(
8 ;sin2)(
mchnE
czn
czZ
mbhn
Eb
ynb
yY
mahnE
axn
axX
zz
z
yy
y
xx
x
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Solutions for a Particle in a 3-D Box
• Inside the box:• Outside the box:
• Total energy
• The ground state is when nx=1 ny=1 nz=1
0
sinsinsin8 2/1
c
znb
yna
xnabc
zyx
2
2
2
2
2
22
8 cn
bn
an
mhE zyx
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Degeneracy A Particle in a 3-D box with
a=b=c
• States have the same energy.
An energy level corresponding to more than one states is said to be degenerate.
The number of different state belonging to the level is the degree of degeneracy.
sinsinsin2 2/3
czn
byn
axn
azyx
2222
28 zyx nnnmahE
, , 112121211,, zyx nnn
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Orthogonality and the Bracket Notation Two wavefunctions are
orthogonal if the integral of their product vanishes
Dirac Bracket Notation 0* dmn
mnmn
mn
nm
nm
0 1
Kronecker Delta
)( 0 mnmnmn
1 nnnn
m
n
m
n
* bra
ket
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A Finite Depth Potential Box A potential well with a finite
depth Classically
• Particles with higher energy can get out of the box
• Particles with lower energy is trapped inside the box
Ener
gy
x
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A Finite Depth Potential Box If the potential energy does not
rise to infinity at the wall and E < V, the wavefunction does not decay abruptly to zero at the wall• X < 0 V = 0• X > 0 V = constant• X > L V = 0
Ener
gy
x=0
Particle’s energy
Barrier Height
x=L
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The Tunneling Effect
2/12 0 mEkBeAex ikxikx
2/1
2
2
2 2
0
VEmDeCe
EVdxd
m x
xixi
2/12 mEkFeEeLx ikxikx
Ener
gy
x=0 x=L
Incident wave
Reflected waveTransmitted wave
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The Tunneling Effect Boundary conditions (x=0, x=L)
• F = 0 because there is no particle traveling to the left on the right of the barrier
iklikLLL FeEeDeCeLx
DCBAx
0
xbc
xbc
bcbc
ji
ji
)()(
)()(
FunctionFirst Derivative
Function
First Derivative
iklikLLL ikFeikEeDeCeLx
DCikBikAx
0
2
2
)
)
EProb(
AProb(
dtransmitte
incident
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The Tunneling Effect Transmission
Probability
• For high, wide barriers , the probability is simplified to
• The transmission probability decrease exponentially with the thickness of the barrier and with m1/2.
The leakage by penetration through classically forbidden zones is called tunneling.
VE
eeA
ET
LL
incident
dtransmitte
/ when
1161
12
2
2
2
2
LeT 2116
1L
Tran
smiss
ion
Coeffi
cient
E/V00.0 1.0 2.0 3.0 4.0
1.0
0.5
0.0
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The Harmonic Oscillator (Vibration) A Harmonic Oscillator and
Hooke’s Law• Harmonic motion: Force is
proportional to its displacement: F= –kx when k is the force constant.
l0
l
l0
)(
)(
2
2
02
2
0
kxdt
xdm
llkdt
ldm
kxllkf
21
cossin)( 21
mk
tctctx
2
21)(
)(
kxxV
dxdVxf
Harmonic Potential
x
V
0Equilibrium
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The 1-D Harmonic Oscillator Symmetrical well
Schrödinger equation with harmonic potential:
Ekxdxd
m
kxxV
22
22
2
21
2
21)(
Solving the equation by using boundary conditions that .
The permitted energy levels are
The zero-point energy of a harmonic oscillator is
... 3 2, 1, ,0 2/1
21
v
mkvEv
0
21
0 EDisplacement,
x
Pote
ntia
l En
ergy
, V
0
1
2
3
4
5
6
7
2)( axxV
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Solutions for 1-D Harmonic Oscillator Wavefunctions for a harmonic
oscillator
• Hermite polynomials;
4/12
2/)/( 2
)(
.)(
mkexHNx
fnGaussianxinpolynomialNx
x
12072048064)(
12016032)(
124816)(
128
24
21
2466
355
244
33
22
1
0
yyyyH
yyyyH
yyyH
yyyH
yyH
yyHyH
yH
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2/)/(00
2
)( xeNx
2/)/(11
22)(
xexNx
2/)/(2
2
22
2
24)(
xexNx
2/)/(3
3
33
2
128)(
xexxNx
Wavefunctions
2/1
2/1 !21
N
Normalizing Factors 2/1
mk
0
1
2
3
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At high quantum number (>>) harmonic oscillator has their highest amplitudes near the turning points of the classical motion (V=E)The properties of
oscillators• Observables
• Mean displacement 0
1
2
3
4
5
6
7
n
ˆˆ* dx
0
)()(
)()(
2
2
22
2
2
2
2
121
122
22
2/2/22
222*
dyeHHHN
dyeyHHN
dyeHyeHN
dxeHxeHNdxxx
y
y
yy
xx
' if !2' if 0
2/1'
2
dyeHH y
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8%
• Mean square displacement
• Mean potential energy
• Mean kinetic energy
The tunneling probability decreases quickly with increasing .
Macroscopic oscillators are in states with very high quantum number.
2/1212
)(mkx
Emk
kkxV
21
21
21
2/121
212
21
)(
EVEEK 21
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Rigid Rotor (Rotation) 2-D Rotation
• A particle of mass m constrained to move in a circular path of radius r in the xy plane.E = EK+VV = 0EK = p2/2m
• Angular momentum Jz= pr• Moment of inertia I = mr2
x
y
z
r
IJE
mrIprJm
pE
z
z
2
;;2
2
22
Not all the values of the angular momentum are permitted!
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• Using de Broglie relation, the angular momentum about the z-axis is
• A particle is restricted to the circular path thus cannot take arbitrary value, otherwise it would violate the requirements for satisfied wavefunction.
hrJ
hpprJ z
z
;
lmr 2
Allowed wavelengths
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• The angular momentum is limited to the values
• The possible energy levels are
,2,1,0 22
l
lll
z
m
mhmr
hrmhrJ
ml > 0
ml < 0
Im
IJE lz
22
222
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Solutions for 2-D rotation Hamiltonian of 2-D rotation
• The radius of the path is fixed then
The Schrödinger equation is
2
2
22
22
2
2
2
22
112
,
2,
rrrrmrH
yxmyxH
2
22
2
2
2
2
22 dd
Idd
mrH
22
2 2IE
dd
2/1
2/1
2)2(
)( IEmel
im
m
l
l
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• Cyclic boundary condition
• The probability density is independent of
l
l
lll
l
mim
imimim
m
e
eee
2
2/1
2
2/1
2
)(
)2()2()2(
)2()(
1ie
,2,1,0
1
)(1)2(2
2
l
m
mm
m
m
l
l
l
l
positive be must
21
22 2/12/1*
ll
ll
imim
mmee
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Spherical Coordinates Coordinates
defined by r, , *
www.mathworld.wolfram.com
0 20
0
angle Polar
angle Azimuthal
Radius r
rzxy
zyxr
1
1
222
cos
tan
cossinsinsincos
rzryrx
θ̂sin1φ̂1r̂
sinV
r̂sina
θ̂sinφ̂r̂s
2
2
rrr
drddrd
drd
drrddrd
sinsin
1sin
1122
2
222
22
rrrr
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3-D Rotation A particle of mass m that
free to move anywhere on the surface of a sphere radius r.• The Schrödinger equation
rV
mH 2
2
2
fixed is
travel to free is it whenever
0r
V
)()(),(),,(2
22
r
Em
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Using spherical coordinate
• Discard terms that involve differentiation wrt. r
sinsin
1sin
1
11
2
2
22
222
22
rrrr
sinsin
1sin
1112
2
222
22
rr
2
22
22
;2
2
21
mrIIE
mEr
mEr
Legendrian
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Plug the separable wavefunction into the Schrödinger equation
The equation can be separated into two equations
22
2
2
2
2
2
2
2
2
sinsinsin1
sinsinsin
sinsin
1sin
1
dd
dd
dd
dd
dd
dd
222
2
sinsinsin1
dd
ddm
dd
l
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The normalized wavefunctions are denoted , which depend on two quantum numbers, l and ml, and are called the spherical harmonics.
Solutions for 3-D Rotation),(,
lmlY
0 0
10
1
2
0
1
2
),(, lmlYl lm
2/1
41
cos43 2/1
ie
sin
83 2/1
1cos316
5 22/1
ie
sincos
85 2/1
ie 222/1
sin32
5
llllml l ,,2,1,2,1,0
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• For a given l, the most probable location of the particle migrates towards the xy-plane as the value of |ml| increases
• The energy of the particle is restricted to the values
Energy is quantized.Energy is independent of ml values.A level with quantum number l is (2l+1) degenerate.
,2,1,02
)1( lI
llE
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Angular Momentum & Space Quantization Magnitude of angular
momentum z-component of angular
momentum The orientation of a rotating
body is quantizedz
ml = 0
ml = +1
ml = -1
ml = +2
ml = -2
... 2, 1, ,01 2/1 lll
lllmm ll ,...,1 ,
+2
+1
0
–1
–2
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Spin The intrinsic angular
momentum is called “spin”• Spin quantum number; s = ½
• Spin magnetic quantum number; ms = s, s–1, … –s
Element particles may have different s values• half-integral spin: fermions (electron, proton)
• integral spin: boson (photon)
ms = +½
ms = –½
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Key Ideas Wavefunction
• Acceptable• Corresponding to boundary conditions
Modes of motion (Functions to explain the motion)• Translation Particle in a box• Vibration Harmonic oscillator• Rotation Rigid rotor
Tunneling effect