39. More About Matter Wavesphome.postech.ac.kr/user/genphys/download/chap39,40_p.pdf1913, Bohr’s...
Transcript of 39. More About Matter Wavesphome.postech.ac.kr/user/genphys/download/chap39,40_p.pdf1913, Bohr’s...
39. More About Matter Waves39. More About Matter Waves• Quantum mechanics and Schrödinger Equation
39-2. String Waves and Matter Waves
tiezyxtzyx ωψ −=Ψ ),,(),,,(Wave function : Describe the state of a particleSchrödinger equation:
[ ] 0)(82
2
2
2
=−+ ψπψ xUEh
mdxd
Probability density: |ψ|2
- Traveling wave
- Standing waves as in strings : confined in a finite space
Energy is quantized: discrete states : confinement principle
Particles like electrons: Matter waves
39-3. Energies of a Trapped Electron • One-Dimensional Traps
A particle: confined in 0 ≤ x ≤ LThe wave function is as in a standing wave.
Wave length of standing waves
..... 3, 2, 1, for 2 and 2
=== nnLnL λλ
)2(L
nk πλπ==
n : a quantum numberWavefunction:
⎟⎠⎞
⎜⎝⎛= x
LnAxnπψ sin)(
• Quantized Energies in an infinite potential well
Length WaveBroglie de : ph
=λ
LhnhmKp 2//2 === λ ) 0( EKU =⇒=
22
22
82n
mLh
mpEn ⎟⎟
⎠
⎞⎜⎜⎝
⎛== for n = 1, 2, 3, …..
Quantized Energies
• Energy Changes
lowhigh EEE −=Δ
An electron makes a quantum jump (transition) only if the received energy = ΔE(excited from the lower-energy state to the higher-energy state)
(i) Excitation by the absorption of light:lowhigh EEEhf −=Δ=
(ii) The excited electron becomes quickly de-excited and emits light (a photon) with an energy lowhigh EEhf −=
(i) (ii)
39-4. Wave Functions of a Trapped ElectronWavefunction:
⎟⎠⎞
⎜⎝⎛= x
LnAxnπψ sin)(
• Probability of DetectionProbability p(x) of
detection at position xwith a width dx
for 0 ≤ x ≤ L
= Probability density |ψn(x)|2 at position x (width dx)
)(sin)()( 222
LnAdxxxp nπψ ==
Probability of detection between x1 and x2
= dxL
nAxpx
x
x
x ∫∫ = 2
1
2
1
)(sin)( 22 π
n becomes larger: probability density becomes more uniform
• Normalization condition
1)(2 =∫+∞
∞−dxxnψ LA /2 =⇒
• Zero-point Energy
Lowest energy ∞→→⎟⎟
⎠
⎞⎜⎜⎝
⎛= as 0
8 2
2
1 LmLhE
39-5. An Electron in a Finite WellSchrödinger equation:
[ ] 0)(82
2
2
2
=−+ ψπψ xUEh
mdxd
if U0 = 450 eV and L = 100pm
For 0 < x < L : U = 0
For x < 0 or x > L : U = U0
082
2
2
2
=+ ψπψh
mEdxd
2
22 8
hmEk π
= )/2(
Wave equation
λπ=k
if E < U0 E – U <0
0||82
2
2
2
=−− ψπψ EUh
mdxd
ikxex ±~)(ψ
||802
22 EU
hm
−=πκ xex κψ −~)(
)(~)( Lxex −−κψ
for x < 0
for x > L
For x < 0 or x > L : U = U0
if E > U0 E – U >0
0)(802
2
2
2
=−+ ψπψ UEh
mdxd
)(802
22 UE
hmq −=
π iqxex ±~)(ψ
Wave equation
Energy is not quantized (continuous).
• Solving differential equation
- General Solution.
LxBeAex ikxikxII ≤≤+= − 0 )(ψ
0 )( ≤= xCex xI
κψ
LxDex LxIII ≥= −− )( )(κψ
- Continuity conditions (boundary conditions)
0at )0()0( and )0()0( === xdx
ddx
d IIIIII
ψψψψ
Lxdx
Lddx
LdLL IIIIIIIIII === at )()( and )()( ψψψψ
39-8. The Bohr Model of the Hydrogen Atom• Hydrogen Atom : one proton and one electron
Observed emission spectrum (Balmer)
.. 5, 4, 3, for ,1211
22 =⎟⎠⎞
⎜⎝⎛ −= n
nR
λ
1913, Bohr’s semiclassical theory Quantization of Angular Momentum in the Bohr model
de Broglie ; a dual nature of MatterBohr’s theory : Semiclassical theory
r λ
A standing wave form
rn πλ 2= L,3,2,1=n
vmh
ph
e
==λ
rvm
nh
e
π2= hnhnvrmL e ===π2
,
• The Orbital Radius is Quantized in the Bohr Model
rvm
re
rqq
kF2
2
2
02
21
41
===πε
hnrmvL ==rmnv h
=⇒
.... 3, 2, 1, for ,20
22
02
=== nnanme
hrπε
radiusBohr : pm 92.52m 10291.5 112
02
0 ≈×== −
mehaπε
• Orbital Energy is Quantized
Total Energy :
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=+=
remvUKE
2
0
221
41πε r
e2
081πε
−=
..... 3, 2, 1, for ,eV 6.13J10180.218 22
18
2220
4
=−=×
−=−=−
nnnnh
meEn ε
• Energy Changes
lowhigh EEEhf −=Δ=222
0
4 18 nhmeEn ε
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2232
0
4 118
1
lowhigh nnchmeελ
⎟⎟⎠
⎞⎜⎜⎝
⎛−= 22
111
highlow nnR
λ
1732
0
4
m 10097.18
-
chmeR ×==ε
Balmer’s equation :nlow = 2, and nhigh = 3, 4, 5, …
39-9. Schrödinger’s Equation and the Hydrogen Atom
rerU
0
2
4)(
πε−
=Electrical potential energy :
• Energy Levels and Spectra of the Hydrogen Atom
Red Blue Violet Near ultravioletλ (nm) 65
6.3
486.
1
434.
1
410.
239
7.0
388.
9
364.
6
Spectrum: absorption or emission lines
• Quantun Numbers for the Hydrogen Atom
Each set of quantum numbers (n, l, ml)n : principle quantum number : n = 1, 2, 3, …..l : orbital quantum number : l = 0, 1, 2, …, n − 1ml: orbital magnetic quantum number : ml = − l, − (l − 1), …, + (l − 1), + l
• The Wave Function of the Hydrogen Atom’s Ground State
0
30
11 a/r
s ea
−
π=ψ
0/230
21
1 ars e
a−=
πψ : Probability density
drrea
dVdrrP ars
2/230
21 41)( 0 π
πψ −==Radial
Probability
0/2230
4)( arera
rP −=
1)(0
=∫∞
drrP
ψ2(r)
• Hydrogen Atom States with n =2
(n, l, ml) = (2, 0, 0), (2, 1, -1), (2, 1, 0), (2, 1, 1)
(2, 0, 0) state ψ2(r)
(2, 1, 0) state ψ2(r) (2, 1, ±1) state ψ2(r)
n, l = n -1 (n >>1) state P(r)
40. All About Atoms40. All About Atoms
40-2. Some Properties of Atoms- Atoms are stable.- Atoms combine with each other.
• Ionization energy
Inert Noble gasHe, Ne, Ar, Kr, Xe
Alkali metal;A highly reactiveLi, Na, K, Rb, Cs
Six periods2, 8, 8, 18, 18, 32
• Atoms Emit and Absorb Light:
• Atoms Have Angular Momentum and Magnetism
• The Eistein-de Haas Experiment
lowhigh EEhf −=
orborb Lme rr
2−=μ
: and orborb Lrrμ
both perpendicular to the plane of the orbit.
Show that angular momentum and magnetic moment are coupled.
As a magnetic field applied, the cylinder begins to rotate.
netrot L- Lrr
=
40-3. Electron Spin• spin angular momentum, spin S
an intrinsic quantum number (a spin quantum number)
r
n shell : afford 2n2 statesl subshell : 2(2l+1) states
a spin magnetic quantum number ms = 21±
Quantum number Symbol Allowed Values Related toPrincipal n 1, 2, 3, …. Distance from the nucleusOrbital l 0, 1, 2, …, (n − 1) Orbital angular momentumOrbital magnetic ml 0, ±1, ±2, …., ±l Orbital angular momentum (z component)Spin s ½ Spin angular momentumSpin magnetic ms ± ½ Spin angular momentum (z component)
n : 1, 2, 3, ······ : K, L, M, ······ shelll : 0, 1,···, n-1 : s, p, d, ······ subshell
or 1s, 2s, 2p, 3s, …, ns, np, nd,…
Quantum numbers
40-4. Angular Momenta and Magnetic Dipole MomentsOrbital Angular Momentum and Magnetism
Classical : L = mvr for a circular motionSemi-Classical (Bohr Model) : hnmvrL == (n = 1, 2, 3, ···)
Quantum Mechanics : ( )hr
1|| +== llLL
Lowest L = 0 : Spherically Symmetry
(l = 0, 1, 2, ···, n-1)
Lme
orb
rr
2−=μOrbital magnetic dipole moment
h)1(2
+= llme
orbμ
Neither nor can be measured.orbμr Lr
Bllzorb mmm
e μμ −=−=2,h
Bohr magneton J/T 10274.92
24−×==m
eB
hμ
hlz mLz component : =
Semi-classical angle θ :LLz=θcos
(ml = 0, ±1, ± 2, ···, ± l)
measured quantity
• Spin Angular Momentum and Spin Magnetic Dipole MomentMagnitude S of the spin angular momentum S
r
( ) hhh 866.0)1(1 21
21 =+=+= ssS
Sme
s
rr−=μspin magnetic dipole moment
h)1( += ssme
sμ
Neither nor can be measured.sμr
Sr
hh
szBsszs mSmmme
=−=−= ,2, μμ
measured quantities
• Orbital and Spin Angular Momenta Combined
total angular momentum Jr
for more than one electron
)()( 2121 zz SSSLLLJrrrrrrr
+⋅⋅⋅+++⋅⋅⋅++=
effective magnetic dipole momenteffμr
40-5. Stern-Gerlach Experiment
Ag
BBU zμμ −=⋅−=
Magnetic moment of Ag atom parallelor antiparallel to the magnetic filed
Magnetic potential energy: rr
Force along z-axis:
dzdB
dzdUF zz μ−=−= gradient of
magnetic field
μz = −μ and μ : reflected oppositely
40-7. The Pauli Exclusion PrincipleNo two electrons in an atom can ever be in the same quantum state.
: No same set of quantum numbers (n, l, ml, ms)