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 Rotational motion

Harmonic Oscillator

Classical Harmonic Oscillator

2

2

( ) sin(2 )

F ma kx

d xm kx

dtx t A t b

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

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)

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

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

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)

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.

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

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.

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

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

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

The Rigid Rotor

1. Classical treatment2. Shrödinger equation3. Energy4. Wavefunctions: Spherical Harmonics5. Properties

The Rigid Rotor

2D (on a plane) circular motion with fixed radius.

3D: Rotational motion with fixed radius (spherical)

The Rigid Rotor

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

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

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

Plots of spherical harmonics and the corresponding square functions

From WolframMathWorld (just Google ‘Spherical harmonics’

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

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

Angular momentum and the Hydrogen Atom

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

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

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

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.

Hydrogen Atom

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)

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

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

• 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

Radial probability density (or radial distribution function)

See also Figure 10.5 Silbey

Node for 2s orbitalNodes for 3s orbital

Bohr radius, ao

Electron cloud picture1s 2s 3s

2p

Shapes of (orbitals)

NOTE:

This is not yet the 2.

Shapes of (orbitals)

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

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

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

More complicated systems…

MANY-ELETRON ATOMS

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

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

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

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

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

Going back to Pauli exclusion… implications…

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

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?

Linear combination of spin functions

1/ 2 (1) (2) (2) (1)

1/ 2 (1) (2) (2) (1)

SEATWORK TWO: Are these antisymmetric functions?

Therefore for the ground state He atom

1 (1)1 (2) 1/ 2 (1) (2) (2) (1) s s SPATIAL PART

SPIN PART

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?

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)

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)

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

Spin-Orbit coupling Coupling of the spin angular momentum S and orbital angular momentum L

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)