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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