Modern Physics (II)

Chapter 9: Atomic Structure

Chapter 10: Statistical Physics

Chapter 11: Molecular Structure

Chapter 12-1: The Solid StateChapter 12-2: Superconductivity

Serway, Moses, Moyer: Modern PhysicsTipler, Llewellyn: Modern Physics

Modern Physics I

Chap 3: The Quantum Theory of Light Blackbody radiation, photoelectric effect, Compton effect

Chap 4: The Particle Nature of Matter Rutherford’s model of the nucleus, the Bohr atom

Chap 5: Matter Waves de Broglie’s matter waves, Heisenberg uncertainty principle

Chap 6: Quantum Mechanics in One Dimension The Born interpretation, the Schrodinger equation, potential wells

Chap 7: Tunneling Phenomena (potential barriers)

Chap 8: Quantum Mechanics in Three Dimensions Hydrogen atoms, quantization of angular momentums

de Broglie’s intriguing idea of “matter wave” (1924)

Extend notation of “wave-particle duality” from light to matter

For photons,

E hf h

Pc c

Suggests for matter,

h

P de Broglie wavelength

E

fh

de Broglie frequency

P: relativistic momentum

E: total relativistic energy

The wavelength is detectable only for microscopic objects

Chapter 5: Matter Waves

(x,t ) contains within it all the information that can be known about the particle

Normalization: 2, 1x t dx

,x t

Finite, single-valued, and continuous on x and t

Properties of wavefunction

,x t

x

must be “smooth” and continuous where U(x) has a finite value

The particle can be found somewhere with certainty

(x, t ) is an infinite set of numbers corresponding to the wavefunction value at every point x at time t

The one-dimensional Schrödinger wave equation

2 2

2

,, , ,

2

x tx t U x t x t i

m x t

Time-independent Schrödinger equation U(x,t ) = U(x), independent of time

2 2

2

( )

2

d xU x x E x

m dx

, E

i tx t x e

, ,x x t x t x xP

Solution:

Probability density at any given position x (independent of time)

stationary states

E: total energy of the particle

Time-independent Schrödinger equation

Separation of variables : (x,t ) = (x)·(t )

2 2

2

( )

2

x tt x x t x

d dU i

m dx dt

2 2

2

1 ( ) 1

2

d td xU x i

m x dx t dt

Independent of t Independent of x

= E = constant

2 2

2

( )

2

xx x x

tt

dU E

m dxd

i Edt

spatial

temporal

tE

i te

E

A particle in a finite square well

elsewhere ,U

Lx0 0,U(x)

o

U(x)

0 L

U0

I II III

x

Region II

Region I, III

Need to solve Schrodinger wave equation Need to solve Schrodinger wave equation in regions I, II, and IIIin regions I, II, and III

2

2 2

( )( ) ( )

2[ ] 0

xx x

d mU E

dx

1( ) xI x C e Region I:

2( ) xIII x D e Region III:

Region II: 1 2( ) sin cosII x B kx B kx

> 0

Consider: E < U0

The wavefunctions look very similar to those for the infinite square well, except the particle has a finite probability of “leaking out” of the well

0 L

U0

I II III

x

n=1

(x)U0

Finite square Well

n = 2

U(x)

0 L

U0

I II III

x

U0

n = 3

U(x)

0

I II III

x

Penetration depth

2 om U E

No classical analogy !!

Example: A particle in an infinite square well of width L

2 2 2 2

2

2

2 2

2 2

nn

n

n n

nn

n nL n k

k L

P k

P nE

m mL

Momentum is quantized. Energy is quantized !

The notion of quantum number: n

The Square Barrier Potential

where 0( )

0 elsewhere oU x L

U x

Ux

x0

Uo

L

I II III

Resonance transmission at certain energies E > U0

2 2 2

22o

nE U

mL

A finite transmission through the barrier at E < U0 if the barrier is made sufficiently thin

Expectation valuesFor a given wavefunction (x,t ), there are two types of measurable quantities: eigenvalues, expectation values

Observables (可觀測量 ) and Operators (算符 )An “observable” is any particle property that can be measured

, , x t x tQ Q dx

Expectation value Q predicts the average value for Q

The Schrödinger wave equation: H Ψ=[E]Ψ

, , x t x tQ q (x,t ) is the “eigenfunction” and q is the “eigenvalue”

q = constant

Examples of eigenvalues and eigenfunctions

, , l lm mz l l lL Y m Y

2 2, ,( 1) l lm m

l lL Y l l Y

U = central forces

[ ] ( )i kx tP A ik e ki x i

[ ] ( )i kx tE i i A i et

U = 0, a free particle i kx tAe

Three-dimensional Schrödinger equation

2 2 2 2

2 2 2, , , ( , , , ) , , , , , ,

2x y z t x y z t x y z t x y z t

hU ih

m x y z t

2

2, , , ( , , , ) , , , , , ,

2x y z t x y z t x y z t x y z t

hU ih

m t

2

2

2r r r rU E

m

Time-independent Schrödinger equation:

Particle in a system with central forces

x

y

z

nucleus

electron

r

( )( )( )

4 o

Ze eU r U r

r

Require use of spherical coordinates

( , ) ( , , , ) ( , , , )r t x y z t r t

a central force !!

2

2, , , , , ,

2r r r rU E

m

Time-independent Schrödinger equation

22

20

dm

d

2

2

1sin 1 0

sin sin

d d m

d d

22 2 2

11 2 0r

d dR mr E U R

r dr dr r

R r

, ,r r R r

ml

l

n

principal quantum number orbital quantum number magnetic quantum number

For any central force U(r ), angular momentum is quantized by the rules

and

Since |L| and Lz are quantized differently, L cannot orient in the z-axis direction. |L| > Lz

( 1) L l l

z lL m ml = 0, 1, 2, …

= 1, 2, 3, … (n-1)

2 2

28no o

e ZE

a n

n = 1, 2, 3,…

Degeneracy for a given n

21

0

12 nn

2, , ,n n nP r dr r R r R r dr

Probability of finding electron of a hydrogen-like atom in the spherical shell between r and r + dr from the nucleus

l = 0

0.52 Å

The first excited state: n = 2 fourfold degenerate

200 2s state, is spherically symmetric

121211210 , , 2p states, is not spherically symmetric

Excited States of Hydrogen-like Atoms

2

0212

121

(2/23/2009, 2h)

0/,

Zr nanR r e