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)