quantum mechanical model systems
DESCRIPTION
Quantum Mechanical Model Systems. Erwin P. Enriquez, Ph. D. Ateneo de Manila University CH 47. Based on mode of motion. Translational motion: Particle in a Box Infinite potential energy barrier: 1D, 2D, 3D Finite Potential energy barrier Free particle Harmonic Oscillator - PowerPoint PPT PresentationTRANSCRIPT
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Quantum Mechanical Model Systems
Erwin P. Enriquez, Ph. D.Ateneo de Manila University
CH 47
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Based on mode of motion
Translational motion: Particle in a Box Infinite potential energy barrier: 1D, 2D, 3D Finite Potential energy barrier Free particle
Harmonic Oscillator Rotational motion
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Harmonic Oscillator
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Classical Harmonic Oscillator
2
2
( ) sin(2 )
F ma kx
d xm kx
dtx t A t b
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Quantum Harmonic Oscillator (H.O.)
2 2 2
2
ˆ
2 2
H E
d kxE
m dx
5 103
0
V x( )
100100 x
2
( )2
kxV x
Schrödinger Equation Potential energy
v = 0,1, 2, 3, …v
1(v )
2E hv
SOLUTION:Allowed energy levels
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Solving the H.O. differential equationPower series methodTrial solution:
Substituting in H. O. differential equation:
Rearranging and changing summation indices:
Mathematically, this is true for all values of x iff the sum of the coefficients of xn
is equal to zero. Thus,
rearranging:
2 2
2 2
0
( )qx qx
nn
n
x e f x e c x
2"( ) 2 '( ) 2 ( ) 0f x qf x mE q f x
22
0 0 0
( 2)( 1) 2 2 0n n nn n n
n n n
n n c x q nc x mE q c x
22( 2)( 1) 2 2 0n n nn n c qnc mE q c
2
2
2 2
( 2)( 1)n n
qn mE qc c
n n
2-TERM RECURSION RELATION FOR COEFFICIENTS: Two arbitrary constants co (even) and c1
(odd)
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General solution
• Becomes infinite for very large x as x ∞.
• This is resolved by ‘breaking off’ the power series after a finite number of ters, e.g., when n = v Thus, our recursion relation becomes:
2
2
0
qxn
nn
x e c x
2
2
v 2 v
2 v 20
(v 2)(v 1)
2 v 2 0
1(v )
2
q mE qc c
q mE q
E hv
When n > v, coefficient is zero (truncated series, zero higher terms)
v = 0,1, 2, … also, QUANTIZED E levels
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Quantum Harmonic Oscillator
2
2v v v( ) ( )
q
x N H q e
2
kq x
1/ 2
v v
1
2 v!
kN
SOLUTIONv=1
v=2
v=3
v=4
0
1
v 1 v v-1
( ) 1
( ) 2
( ) 2 ( ) 2v ( )
H q
H q q
H q qH q H q
Hermite Polynomialsgenerated through recursion formula
Nomalization constant
Example:
What is
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General properties of H.O. solutions
Equally spaced E levels Ground state = Eo = ½ h (zero-point energy) The particle ‘tunnels’ through classically
forbidden regions The distribution of the particle approaches the
classically predicted average distribution as v becomes large (Bohr correspondence)
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Molecular vibration
• Often modeled using simple harmonic oscillator
• For a diatomic molecule:
1 2 1 2
2 2 2 2 21 2
1 2 1 22 21 1 2 2
1 2
ˆ ( , ) ( , )
( )( , ) ( , )
2 2 2
H x x E x x
d d k x xx x E x x
m dx m dx
r x x
In Cartesian system, the differential equation is non-separable. This can be solved by transforming the coordinate system to the Center-of-Mass coordinate and reduced mass coordinates.
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2 2 2 2 21 2
1 2 1 22 21 1 2 2
1 2
1 21 2
1 2
2 2 2 2 2
2 2
( )( , ) ( , )
2 2 2
( , ) ( , )2 2 2
d d k x xx x E x x
m dx m dx
r x x
m mR x x
M MM m m
d d krr R E r R
dr M dR
Reduced mass-CM coordinate system
1 2
1 2
m m
m m
Separable differential equation
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2 2 2 2 2
2 2
2 2 2
2
2 2
2
( , ) ( ) ( )
1 ( ) 1 ( )( )
( ) 2 2 ( ) 2
1 ( )( )
( ) 2 2
1 ( )
( ) 2
r R
r Rr T
r R
rr r
r
RR
R
r R r R
d r kr d Rr E
r dr R M dR
d r krr E
r dr
d RE
R M dR
Separation of variables (DE)
Particle of reduced mass 'motion' (just like Harmonic oscillator case)
Center-of-mass motion, just like Translational motion case
The motion of the diatomic molecule was ‘separated’ into translational motion of center of mass, and
Vibrational motion of a hypothetical reduced mass particle.
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H. O. model for vibration of molecule
• E depends on reduced mass,
• Note: particle of reduced mass is only a hypothetical particle describing the vibration of the entire molecule
1(v )
2
1
2
E hv
kv
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AnharmonicityVibrational motion does not follow the parabolic potential especially at high energies.
CORRECTION:
21 1(v ) (v )
2 2 eE hv hv
e is the anharmonicity constant
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Selection rules in spectroscopy
• For excitation of vibrational motions, not all changes in state are ‘allowed’.
• It should follow so-called SELECTION RULES
• For vibration, change of state must corrspond to v= ± 1.
• These are the ‘allowed transitions’.
• Therefore, for harmonic oscillator:
v 1 vE E E hv
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The Rigid Rotor
1. Classical treatment2. Shrödinger equation3. Energy4. Wavefunctions: Spherical Harmonics5. Properties
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The Rigid Rotor
2D (on a plane) circular motion with fixed radius.
3D: Rotational motion with fixed radius (spherical)
The Rigid Rotor
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2
22 2
v2
v
1 1v
2 2 2
drv
dt r
I mr
L m r pr I
LT m I
I
Classical treatment
Motion defined in terms of
• Angular velocity
• Moment of inertia
• Angular momentum
• Kinetic energy
The Rigid Rotor
Linear velocity Linear
frequency
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Quantum mechanical treatmentShrödinger equation
Laplacian operator in Spherical Coordinate System
2
2
22
22 2
2 2 2 2 2
ˆ ( , , ) ( , , )
ˆ ˆ ( , , ) ( , , )
( , , ) ( , , )2
( , , ) ( , , )2
1 1 1sin
sin sin
H x y z E x y z
T V x y z E x r z
x y z E x y zm
r E rm
RR R R R R
The Rigid Rotor
In spherical coordinate system
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2 22
2 2 2 2 2
1 1 1sin ( , , ) ( , , )
2 sin sinR R E R
m R R R R R
2 2
2 2 2
2
( , , ) ( ) ( , )
1 1sin ( , ) ( , )
2 sin sin
ˆ ( , ) ( , )
( 1)
20,1,2,...
0, 1,...,
m ml l
R R r Y
Y EYmR
HY EY
l lE
Il
m l
Substituting into Schrodinger equation:
Since R is fixed and by separation of variable:
SOLUTION:
SPHERICAL HARMONICS
(Table 9.2: Silbey)
l = azimuthal quantum number
Degeneracy = 2l+1
The Rigid Rotor
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Plots of spherical harmonics and the corresponding square functions
From WolframMathWorld (just Google ‘Spherical harmonics’
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Notes:
• E is zero (lowest energy) because, there is maximum uncertainty for first state given by
• We do not know where exactly is the particle (anywhere on the surface of the ‘sphere’)
00
1
4Y
The Rigid Rotor
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For a two-particle rigid rotor
• The two coordinate system can be Center of Mass and Reduced Mass
• since radius is fixed, the distance between the two particles R is also fixed
• The kinetic energy for rotational motion is:
• The result is the same: Spherical Harmonics as wavefunctions (but using reduced mass)
2 2
22 2
L LT
I R
The Rigid Rotor
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Angular momentum and the Hydrogen Atom
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Angular Momentum
• This is a physical observable (for rotational motion)
• A vector (just like linear momentum)
• Recall: right-hand rule
• L2 =L∙ L=scalar
L
x y z
x z y
y x z
z y x
i j k
L r p x y z
p p p
L yp zp
L zp xp
L xp yp
The Rigid Rotor
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Angular momentum operators
2 2 2 2
22 2
2 2
ˆ
ˆ
ˆ
ˆ ˆ ˆ ˆ
1 1ˆ sinsin sin
x
y
z
x y z
L i y zz y
L i z xx z
L i x yy x
L L L L
L
NOTE: SAME AS FOR RIGID ROTOR CASE
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Angular momentum eigenfunctions
Are the spherical harmonics:
l =0,1,2,…
m=0, ±1,…, ±l
The z-component is also
solved (Lx and Ly are
Uncertain)
2 2
2
ˆ ( , ) ( , )
= ( 1) ( , )
m ml l
ml
L Y L Y
l l Y
ˆ ( , ) ( , )m mz l lL Y m Y
REMINDER: SKETCH ON THE BOARD. FIGURE 9.9 and 9.10 SILBEY
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RECALL: HCl rotational energies (l is called J)
Angular momentum and rotational kinetic energy
RECALL 2 2
22 2
L LT
I R
2
2
ˆˆ
2
LT
R
2
ˆ ( , ) ( , )
( 1)
2
m ml lHY EY
l lE
I
The spherical harmonics are eigenfunctions of both Hamiltonian and Angular Momemtum Square operators.
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Hydrogen Atom
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To be solved to get the wavefunction
for the electron
H-atom: A two-body problem: electron and nucleus
2
2 2 22 2
22
ˆ ( , , , , , ) ( , , , , , )
ˆ ( ) ( ) ( ) ( )
( )4
1 1( ) ( ) ( )
( ) 2 4 ( ) 2
1( )
( ) 2
1
(
e e e N N N e e e N N N
N N N N
o
CM CM To CM CM
CM CM CMCM CM
H x y z x y z E x y z x y z
H q q E q q
ZeV r
r
Zeq q q E
q r q M
q Eq M
q
2 22 ( )
) 2 4 o
e N e Ne
e N N
Zeq E
r
m m m mm
m m m
Note that the reduced mass is approx. mass of e-. Thus
e
describes translational motion of entire atom (center of mass motion)
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Shrödinger equation for electron in H-atom
2 2 22
2 2 2 2 2
1 1 1sin ( , , ) ( , , )
2 sin sin 4 e ee o
Zer r E r
m r r r r r r
24 2
2 3 2 2
( , , ) ( ) ( , )
1,2,...
0,1,..., 1
0, 1,...,
4 (4 )
me nl l
ydn
o
r R r Y
n
l n
m l
R Ze ZE
c n n
RADIAL FUNCTIONS (depends on quantum numbers n and l
SPHERICAL HARMONICS
Quantized energy (as predicted by Bohr as well), Ryd = Rydberg constant.
E depends only on n
Degenerary = 2n2E1s = -13.6 eV
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Hydrogen atom wavefunctions
• Are called atomic orbitals• Technically atomic orbital is a wavefunction = • Given short-cut names nl:
• When l = 0, s orbitall = 1 p l = 2, d
3
2
1
11 o
Zr
as
o
Zs e
a
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• Probability density =
• Radial probability density (r part only)= gives probability density of finding electron at given distances from the nucleus
Probability =
• The spherical harmonics squared gives ‘orientational dependence’ of the probability density for the electron:
Plotting the H-atom wavefunction
2 2 ( )nl
r R r dr
22 2 ( ) ( , )nl
me lR r Y
2( , )m
lY
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Radial probability density (or radial distribution function)
See also Figure 10.5 Silbey
Node for 2s orbitalNodes for 3s orbital
Bohr radius, ao
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Electron cloud picture1s 2s 3s
2p
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Shapes of (orbitals)
NOTE:
This is not yet the 2.
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Shapes of (orbitals)
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Properties for Hydrogen-like atom
• H, He+, Li2+
• Energy depends only on n
• Degeneracy: 2n2 degenerate state (including spin)
• The energies of states of different l values are split in a magnetic field (Zeeman effect) due to differences in orbital angular momentum
• The atom acts like a small magnet: eL
Magnetic dipole moment
Magnetogyric ratio of the electron
Orbital angular momentum
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Electron spin
• Spin is purely a relativistic quantum phenomenon (no classical counterpart)
• Shown by Dirac in 1928 as a relativistic effect and observed by Goudsmit and Uhlenbeck in 1920 to explain the splitting (fine structure) of spectroscopic lines
• There is a intrinsic SPIN ANGULAR MOMENTUM, S for the electron which also generates a spin magnetic moment:
• The spin state is given by s=1/2 for the electron, and with two possible spin orientations given by ms = +1/2 or -1/2 (spin up or spin down)
• Spin has no classical observable counterpart, thus, the operators are postulated, and follows closely that of the angular momentum (Table 10.2 of Silbey)
2e
e
g eS
m
ge = 2.002322, electron g factor
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Pauli Exclusion Principle
• The wavefunction of any system of electrons must be antisymmetric with respect to the interchange of any two electrons
• The complete wavefunction including spin must be antisymmetric:
• In other words, each hydrogen-like state can be multiplied by a spin state of up or down
• Thus, “No two electrons can occupy the same state = otherwise, each must have different spins” or “no two electrons in an atom can have the same 4 quantum numbers n, l, ml, ms.”
( , , ) ( )sr g m Spatial part Spin part
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More complicated systems…
MANY-ELETRON ATOMS
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He atom
• Three-body problem (non-reducible)
• Not solved exactly!
• Use VARIATIONAL THEOREM to find approximate solutions
2 2 2 2
2 21 2
1 2 12
1ˆ2 4e o
Ze Ze eH
m r r r
+2
-
-
r12r1
r2
ˆ| |gsE f H f
Kinetic energy of e-s
Electrostatic repulsion between e’s and attraction of each to nucleus
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Variational Theorem (or principle)
• One of the approximation methods in quantum mechanics• States that the expectation value for energy generated for any
function is greater than or equal to the ground state energy
• Any function f is a “Trial Function” that can be used, and can be parameterized (f = f(a)) wherein a can be adjusted so that the lowest E is obtained.
ˆ| | gsf H f E
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He-atom approximation• As a first approximation, neglect the e-e repulsion
part. 2 2 2
2 21 2
1 2
2 2 2 22 21 2
1 2
1 2
1 1 1 1
2 2 2 2
1 2
1 2 1 1 1 2
1ˆ 2 4
1 1ˆ 2 4 2 4
ˆ ˆ ˆ
ˆ
ˆ
( ) ( )
e o
e o e o
He s s
Ze ZeH
m r r
Ze ZeH
m r m r
H H H
H E
H E
E E E
r r
2
12
e
r
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Applying variational principle
• Calculating the ‘expectation value from this trial function’ yields:
• 2E1s=8(-13.6) eV=-108.8 eV
• Subtracting repulsive energy of two electrons by evaluating:
• Total energy is -74.8 eV versus experimental -79.0 eV.
1 1 1 2 1 1 1 2ˆ( ) ( ) | | ( ) ( )calculated s s s sE r r H r r
2*
1 212
34.04 o
ed d eV
r
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Parameterization of the trial function
• The trial wavefunction may be ‘improved’ by parameterization
• For the He atom, the ‘effective nuclear charge’ Z is introduced in the trial wavefunction and its value is adjusted to get the lowest variational energy:
1 2 1 2ˆ( , ; ) | | ( , ; )
0calcd q q Z H q q Z dE
dZ dZ
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Going back to Pauli exclusion… implications…
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Permutation operator
• Permutation operator
• Permutation operator squared
• Eigenvalues
• f is symmetric function
• f is antisymmetric
12
212
212
12
12
ˆ ( (1,2)) (2,1)
ˆ ( (1,2)) 1 (1,2)
1
ˆ ( (1,2)) 1 (1,2)
ˆ ( (1,2)) 1 (1,2)
P f f
P f f
P
P f f
P f f
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Including spin states
( 1/ 2)
( 1/ 2)
g
g
( , , ) ( )sr g m
(1) (2) (1) (2) (1) (2) (2) (1)
SINGLE ELECTRON SYSTEM
TWO- ELECTRON SYSTEM (e.g., He atom:
SEATWORK 1: Which of the functions above are antisymmetric, symmetric?
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Linear combination of spin functions
1/ 2 (1) (2) (2) (1)
1/ 2 (1) (2) (2) (1)
SEATWORK TWO: Are these antisymmetric functions?
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Therefore for the ground state He atom
1 (1)1 (2) 1/ 2 (1) (2) (2) (1) s s SPATIAL PART
SPIN PART
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Fermions and Bosons Quantum particles of half-integral spins are called
FERMIONSS = ½, 3/2, etc. (two spin states, plus or minus)
-requires antisymmetric functions-follows Fermi-Dirac statistics
Quantum particles with integral spins are called BOSONS
S = 1, 2, etc.-requires symmetric functions-follows Bose-Einstein statistics
SEATWORK 2b: What are electrons? Protons?
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Slater determinants and STO
• Slater in 1929 proposed using determinants of spin functions for the spin part
1 (1) (1) 1 (1) (1)1/ 2
1 (2) (2) 1 (2) (2)
s s
s s
Atomic wavefunctions that use hydrogenic functions in a Slater determinant are called Slater-type Orbitals (STO)
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First excited state of He• Triply degenerate because of the spin states
1
2
3
1/ 2 1 (1)2 (2) 1 (2)2 (1) (1) (2)
1/ 2 1 (1)2 (2) 1 (2)2 (1) (1) (2) (2) (1)
1/ 2 1 (1)2 (2) 1 (2)2 (1) (1) (2)
s s s s
s s s s
s s s s
Antisymmetric spatial part This time, symmetric spin part
MULTIPLICITY = 2S +1S = total spin angular momentaTRIPLET M = 3 parallel spins for 2 e’sSINGLET M =1 opposite spins for 2 e’s
SEATWORK 3: What is that principle called, when the lower energy state consists of parallel spins in separate degenerate orbitals? (No erasure please… on your answer)
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Hartree-Fock Self-Consistent Field (HF-SCF) Method
Variational methodTrial function for electronic wavefunction:
V is a ‘smeared’ out potential due to all the electrons
1 1 1 1 2 2 2 3 n n
22
=g (r , , )g (r , , )...g (r , , )
( ) ( , )
( )2
i
i
n n
mi i i l i i
i i i i ii
g R r Y
V r g gm
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Spin-Orbit coupling Coupling of the spin angular momentum S and orbital angular momentum L
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Atomic units
• Short-cut way to write Shrodinger equation is to not include constants… values obtained are generic ‘atomic units’ which can be converted back…
• Table 10.7 Silbey
• E.g., 1a .u. of length is = 1 Bohr radius = 0.052 Å
21
2
ZE
r
SEATWORK 4: What is the length (in Angstroms) equivalent to 3.5 atomic unit of length?
-13.6 eV is how much in a. u.? (look up in Silbey)