PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 32:
PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101 ...
Transcript of PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101 ...
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 1
PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101
Plan for Lecture 23 (Chapter 40-42):
Some topics in Quantum Theory
1. Particle behaviors of electromagnetic waves
2. Wave behaviors of particles
3. Quantized energies
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 3
www.wfu.edu/physics/sps/spszone52012conf/welcome.html
Part of SPS zone 5 conference April 20-21, 2012
Offer 1 point extra credit for attendance*
*After the lecture, email me that you attended. In the following email exchange you will be asked to answer one question about the lecture.
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 4
Webassign hint:
NdN
md1 ,grooves/cm 4160For
sin
==
= λθ
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 5
Webassign hint:
λλθ == md sin2:spotbright for Condition
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 6
If you have not already done so – please reply to my email concerning your intentions regarding Exam 4.
The material you have learned up to now in PHY 113 & 114 was known in 1900 and is basically still true. Some details (such as at high energy, short times, etc. ) have been modified with Einstein’s theory of relativity, and with the development of quantum theory.
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 7
Which of the following technologies do not need quantum mechanics. A. X-ray diffraction B. Neutron diffraction C. Electron microscope D. MRI (Magnetic Resonance Imaging) E. Lasers
Which of the following technologies do not need quantum mechanics. A. Scanning tunneling microscopy B. Atomic force microscopy C. Data storage devices D. Microwave ovens E. LED lighting
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 8
From Physical Review Letters March 20, 2000 -- Volume 84, Issue 12, pp. 2642-2645
Image of Si atoms on a nearly perfect surface at T=7 K.
Image made using atomic force microscopy.
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 9
Quantum physics –
Electromagnetic waves sometimes behave like particles
one “photon” has a quantum of energy E=hf
momentum p=h/λ=hf/c
Particles sometimes behave like waves “wavelength” of particle related to momentum:
λ=h/p
quantum particles can “tunnel” to places classically “forbidden”
Stationary quantum states have quantized energies
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 10
Classical physics
Wave equation for electric field in Maxwell’s equations (plane wave boundary conditions):
Equation for particle trajectory r(t) in conservative potential U(r) and total energy E
( )( )ctxkEtxx
ct
−=∂∂
=∂∂ sinˆ),( :examplefor max2
22
2
2
jEEE
( )
200
2
ˆ21)( :examplefor
21
tgtt
EUdtdm
kvrr
rr
−+=
=+
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 11
Particle properties Wave properties
Position as a function of time is known -- r(t) Particle is spatially confined when E≤U(r). Particles are independent.
Phonomenon is spread out over many positions at an instant of time. Notion of spatial confinement non-trivial. Interference effects.
Particle wave properties in classical physics
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 12
Mathematical representation of particle and wave behaviors.
Consider a superposition of periodic waves at t=0:
( )∑=i
i xkEtxE sin),( max
single wave (one value of k)
superposed wave (many values of k)
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 13
[ ] ( )2
max2 sin)0,(
∑=
ii xkExE
∆x
∆x
∆k = 10
∆k = 1
∆x ∆k ≈ 2π
∆x smaller more particle like
∆k smaller more wave like
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 14
∆x ∆k ≈ 2π Heisenberg’s uncertainty principle
khhp ===π2λ/π2/
λ
De Broglie’s particle moment – wavelength relation:
Heisenberg’s hypotheses: 2
≥∆∆ px
2
≥∆∆ Et
h = 6.6×10-34 Js = 4.14×10-15 eVs
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 15
Wave equations
Electromagnetic waves:
Matter waves: (Schrödinger equation)
2
22
2
2
xc
t ∂∂
=∂∂ EE
( ) ( )txxUxm
txt
i ,)(2
, 2
22
Ψ
+
∂∂
−=Ψ∂∂
−
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 16
Electromagnetic waves Matter waves
Vector – E or B fields Second order t dependence Examples:
Scalar – probability amplitude First order t dependence Examples:
( )
( )tkxc
EtxB
tkxEtxE
z
y
ωsin),(
ωsin),(
max
max
−=
−=/
0 )sin(),( iEtekxtx −Ψ=Ψ
Comparison of different wave equations
00
2
0
//
30
8
1),( 00
aeE
eea
tr tiEar
πε
π
−=
=Ψ −−
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 17
What is the meaning of the matter wave function Ψ(x,t)?
Ψ(x,t) is not directly measurable
|Ψ(x,t)|2 is measurable – represents the density of particles at position x at time t.
For a single particle system – represents the probability of measuring particle at position x at time t.
For many systems of interest, the wave function can be written in the form Ψ(x,t) = ψ(x)e-iEt/ |Ψ(x,t)|2 = | ψ(x)|2
1),( 2 =∫ Ψ∞
∞−dxtx
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 18
Wave-like properties of particles
Louis de Broglie suggested that a wavelength could be associated with a particle’s momentum
xi
xhihp
∂∂
−≡∂∂
−⇒= π2λ
“Wave” equation for particles – Schrödinger equation
( ) ( )txt
hitxxUxm
,π2
,)(2 2
22
Ψ∂∂
−=Ψ
+
∂∂
−
( ) ( )
( ) ( )
,,)(2
ψ, :ions wavefunctstate-Stationary
2
22
/
txEtxxUxm
et iEt
Ψ=Ψ
+
∂∂
−
=Ψ −
rr
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 19
( ) ( )
( ) ( )
( )
2
or 2
2
2
sin
,,2
ψ, :0)( -- particle free -- Example
2
2
220
2
22
/
λλ
λπ
mhE
mEhk
mkE
(kx)eΨx,t Ψ
txEtxxm
etrU
-iEt/
iEt
==⇒=
=
=
Ψ=Ψ
∂∂
−
=Ψ= −
rr
Example: Suppose we want to create a beam of electrons (m=9.1x10-31kg) for diffraction with λ=1x10-10m. What is the energy E of the beam?
( )( ) 15010421010192
10662
1721031
234
2
2
eV J. mkg.
J.mhE -
-=×=
⋅×⋅
×==
−
−
λ
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 20
Electron microscope Typically E=120,000-200,000 eV for high resolution EM
From Microscopy Today article May 2009
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 21
Electrons in an infinite box:
( ) ( )
( )
3,2,1 πsinψψ
0for ψ2
ψ
0
2
22
=
=
≤≤
∂∂
−=
nL
xnx
Lxxxm
xE
mnEn 2
222π=
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 22
Electrons in a finite box:
finite probability of electron existing outside of classical region
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 23
Why would it be interesting to study electrons in a finite box?
A. It isn’t B. It is the mathematically most simple example of
quantum system C. Quantum well systems can be manufactured to
design new devices
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 24
Tunneling of electrons through a barrier
surface region
vacuum
tip
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 25
How a scanning tunneling microscope works:
Developed at IBM Zurich by Gerd Binnig and Heinrick Rohrer who received Nobel prize in 1986.
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 26
Visualization of | ψ(x)|2
A surface if a nearly perfect Si crystal
Physical Review Letters -- March 20, 2000 -- Volume 84, Issue 12, pp. 2642-2645
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 27
The physics of atoms –
Features are described by solutions to the matter wave equation – Schrödinger equation:
( ) ( )tUm
tt
i ,)(2
, 2
22
rrr
r Ψ
+
∂∂
−=Ψ∂∂
−
“reduced” mass of electron and proton
rZe
0
2
πε4−
( ) ( )
nm 0529.0πε4
eV 6.131πε8
:Solutions
ψ, :ions wavefunctstate-Stationary
2
20
0
2
2
200
22
/
==
−=−=
=Ψ −
mea
nZ
naeZE
et
n
iEt
rr
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 28
( )22 ψπ4 rr
Form of probability density for ground state (n = 1)
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 29
Angular degrees of freedom
-- since the force between the electron and nucleus depends only on distance and not on angle, angular momentum L ≡ r x p is conserved. Quantum numbers associated with angular momentum:
( ) ( )states 12 of total
1,....2,1,0 122
+≤≤−=−=+=
mmLn
z
L
Notation: dps ⇒⇒⇒= 2 ,1 ,0
s p
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 30
eV 6.13 2
2
nZEn −=
Summary of results for H-atom:
n = 1
n = 4 n = 3
n = 2
Balmer series
spectra
degeneracy associated with each n: 2n2
4/19/2012 PHY 114 A Spring 2012 -- Lecture 23 31
Atomic states of atoms throughout periodic table:
( ) ( )rrr
r ψ)(2
ψ 2
22
+
∂∂
−= Um
E
effective potential for an electron in atom