Quantum mechanics review

• Reading for week of 1/28-2/1– Chapters 1, 2, and 3.1,3.2

• Reading for week of 2/4-2/8– Chapter 4

Schrodinger Equation (Time-independent)

where

The solutions incorporate boundary conditions and are a family of eigenvalues with increasing energy and corresponding eigenvectors with an increasing number of nodes.

The solutions are orthonormal.

nn EH

nmmn d *

VmH

VTH

22

2

Physical properties: Expectation values

nAnA

or

dAA nn

*

Dirac notation or bra-ket notation

Physical properties: Hermitian Operators

mAnAnAmAnmmn

Real Physical Properties are Associated with Hermitian Operators

Hermitian operators obey the following:

The value <A>mn is also known as a matrix element, associated with solving the problem of the expectation value for A as the eigenvalues of a matrix indexed by m and n

Zero order models:

Particle-in-a-box: atoms, bonds, conjugated alkenes, nano-particles

Harmonic oscillator: vibrations of atoms

Rigid-Rotor: molecular rotation; internal rotation of methyl groups, motion within van der waals molecules

Hydrogen atom: electronic structure

Hydrogenic Radial Wavefunctions

Particle-in-a-3d-Box

x

a

V(x)

V(x) =0; 0<x<a

V(x) =∞; x>a; x <0

b y ; c z

c

zn

b

yn

a

xn

abczyx

nnn zyx

sinsinsin

8nx,y,z = 1,2,3, ...

V

zyxmVmVTH2

2

2

2

2

2222

22

Particle-in-a-3d-Box

x

a

V(x)

V(x) =0; 0<x<a

V(x) =∞; x>a; x <0

b y ; c z

2

2

2

2

2

22

8 c

n

b

n

a

n

m

hE zyx

nnn zyx

0111

8 2

2

2

2

2

22

111

cbam

hE zyx

Zero point energy/Uncertainty Principle

In this case since V=0 inside the box E = K.E.

If E = 0 the p = 0 , which would violate the uncertainty principle.

2

px

Zero point energy/Uncertainty Principle

More generally

Variance or rms:

If the system is an eigenfunction of then is precisely determined and there is no variance.

A

A

22

AAA

2,

pxixppxpx

Zero point energy/Uncertainty Principle

BABA ,2

1

If the commutator is non-zero then the two properties cannot be precisely defined simultaneously. If it is zero they can be.

Harmonic Oscillator 1-d

F=-k(x-x0) Internal coordinates; Set x0=0

22

22

21;2 kxVV

dx

dH

Hermite polynomials

535

424

33

22

1

0

v

3216120

164812

812

42

2

1

)1(22

v

qqqqH

qqqH

qqqH

qqH

qqH

qH

edq

deqH q

n

nq

Harmonic Oscillator Wavefunctions

2

1

vv !v2

N

V = quantum number = 0,1,2,3

/

Hv = Hermite polynomials Nv = Normalization Constant

25.0vvv

xeHN x

25.011 2 xexN

kE )(v 21

v

25.000

xeN

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1

Raising and lowering operators:

Recursion relations used to define new members in a family of solutions to D.E.

lowering2

raising2

0

0

0

0

ˆ

ˆ

piX

piX

a

a

1

11

VVVa

VVVa

Rotation: Rigid Rotor

01

22

2

VprLRII

LH

0,2

2222

i

zyx

LL

LLLL

Rotation: Rigid Rotor

Wavefunctions are the spherical harmonics

imml

mm

lm

ePml

mll

Y

)(cos!

!

4

12)1(

,

lmlm

lmlmz

YllYL

YmYL

)1(2

Operators L2 ansd Lz

,lmlm Y

Degeneracy

Angular Momemtum operators the spherical harmonics

Operators L2 ansd Lz

llmm

llmmz

lmlm

lmlmz

lllmLml

mlmLml

YllYL

YmYL

''2

''

2

)1(''

''

)1(

Rotation: Rigid Rotor

01

22

2

VprLRII

LH

Eigenvalues are thus:

llmmI

lllm

I

Lml ''

2

2

)1(

2''

l = 0,1,2,3,…

Lots of quantum mechanical and spectroscopic problems have solutions that can be usefully expressed as sums of spherical harmonics.

e.g. coupling of two or more angular momentumplane wavesreciprocal distance between two points in space

Also many operators can be expressed as spherical harmonics:

lmYml LM''The properties of the matrix element above are well known and are zero unless

-m’+M+m = 0l’+L+l is even

Can define raising and lowering operators for these wavefunctions too.

The hydrogen atom

r

emH

222

2

Set up problem in spherical polar coordinates. Hamiltonian is separable into radial and angular components

,lmnlnlm YrR

n

the principal quantum number, determines energy

l

the orbital angular momentum quantum number

l= n-1, n-2, …,0

m

the magnetic quantum number -l, -l+1, …, +l

molekJeVRn

R

n

eEn /13126.13;

2 222

4