氫原子光譜之分析

氫原子的 Bohr 理論模型

利用分光儀分析光譜

Importance of the Hydrogen Atom

• A structural model can also be used to describe a very small-scale system, the atom

(利用結構模型描述原子系統 )• The hydrogen atom is the only atomic system

that can be solved exactly ( 氫原子為原子系統中唯一有正確解的原子 )• Much of what was learned about the hydrogen

atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+

( 氫原子模型的理論計算可推廣至擁有單一電子的離子 )

Light From an Atom

• The electromagnetic waves emitted from the atom can be used to investigate its structure and properties– Our eyes are sensitive to visible light– We can use the simplification model of a wave

to describe these emissions

物質 (原子 )的電子能階

激發態不穩定

Emission Spectra Examples

The Bohr Theory of Hydrogen

• In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory

• His model includes both classical and non-classical ideas

• He applied Planck’s ideas of quantized energy levels to orbiting electrons

E h f

Niels Bohr

• 1885 – 1962• An active participant in

the early development of quantum mechanics

• Headed the Institute for Advanced Studies in Copenhagen

• Awarded the 1922 Nobel Prize in physics– For structure of atoms

and the radiation emanating from them

Bohr’s Assumptions for Hydrogen• The electron moves in circular orbits around the

proton under the electric force of attraction. • ( 電子對原子核做圓周運動 )

• Only certain electron orbits are stable and these are the only orbits in which the electron is found.

• ( 電子在特定的軌道才能穩定的運動 )

• Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit.

• ( 電子在特定軌道之間躍遷時，吸收或發出光子 )

• The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum.

Quantization of angular momentum

v

( )

eL m r

n

��������������

特定的軌道大小由電子的角動量決定

Bohr’s Assumptions for Hydrogen• ( 電子對原子核做圓周運動 )

• ( 電子在特定的軌道才能穩定的運動 )

• ( 電子在特定軌道之間躍遷時，吸收或發出光子 )

Quantization of angular momentum

v

( )

eL m r

n

��������������

特定的軌道大小由電子的角動量決定

22

0

1 1v

2 4

eE K U m

r

2 2

20

v 1

4

emr r

i fE E h f

Wave Properties of Particles

Louis de Broglie• 1892 – 1987• Originally studied

history• Was awarded the

Nobel Prize in 1929 for his prediction of the wave nature of electrons

Wave Properties of Particles• Louis de Broglie postulated that because

photons have both wave and particle characteristics, perhaps all forms of matter have both properties

• The de Broglie wavelength of a particle is

h h

p mv

de Broglie wavelength of a particle

(1) Baseball m=1kg ; v=10 m/s

(2) Electron

34356.63 10

6.63 1010 /

h J sm

p kg m s

2 191. v 100 100 1.6 10

2K E m eV J

34

31 19

10

6.63 10

2 ( . ) 2 9.11 10 100 1.6 10

1.2 10

h h J s

p m K E kg J

m

wave nature of electrons

L

L = 3λ

2

(1)2

de Broglie wavelength

v (2)v

v

v

2

where n=1,2,3,2

r n

nr

h h hm

p m

L r p

r m

m r

h n

nhn

������������������������������������������

Quantization of angular momentum

v

( )

eL pr m r

n

��������������

Mathematics of Bohr’s Assumptions and Results

• Electron’s orbital angular momentum

where n = 1, 2, 3,… -------- (2)• The total energy of the atom is

• The centripetal force

• The total energy

vm r n

22

0

1 1v

2 4

eE K U m

r

2 2

20

------(11

4)

v emr r

2

0

1 1

2 4

eE

r

Bohr Radius

v vn

m r nmr

2

2

2 2 2

20 0

1 1

4 4

v

n

me em mr rr

r

r

220

2 whe 1,2,3,4...re

.

.

nn

hr

m e

n

The radii of the Bohr orbits are quantized

When n = 1, the orbit has the smallest radius, called

the Bohr radius,

n is called a quantum number

20

0 2 0.0529 nm h

am e

2

020

2

2

n

hr a

m e

nn

0

2

4

2

0

2 2

1 1

2 4

8

wher 1,2,3,4.....( quantum number )e

nn

n

eE

r

meE

h n

n

Radii and Energy of Orbits

2

13.606nE eV

n

n=1

n=2

n=3

Wavelength of Emitted Photons

• The wavelengths are found by

• RH = Rydberg constant

0

4

2 2 22

1 1

8i ff i

h c meE E h f

h n n

0

2 23 2

4

22

1 1 1 1 1

8 f i f i

me

h n n n nc

HR

RH = 1.097373 2 x 107 m-1

ni

nf

Energy Level Diagram

Balmer Series

– H is red, = 656.3 nm

– H is green, = 486.1 nm

– H is blue, = 434.1 nm

– H is violet, = 410.2 nm

0

4

2 38

me

h c HR

0

2 23 2

4

22

1 1 1 1 1

8 f i f i

me

h n n n nc

HR

計算 RH 與 h

• 利用 ，先算 RH 。

• 再利用 0

43

28

meh

c

HR

0

43

2:

8

meh

c

-99 -10210 10提示 先整理 或算出HR

計算 h

Extension to Other Atoms

• Bohr extended his model for hydrogen to other elements in which all but one electron (He+ and Li2+) had been removed

0

1 ; Reduced

4 massn e

n e

M m

M m

21

v2

2eE K U

Z

r

2 2

20

v 1

4

Ze

r r

0

2 22

8E

nh

2 4

n

Z e

charge of nucleusZe =

v r n

分光儀

光柵分光儀Diffraction Grating Spectrometer

sin ; 0, 1...d m m

sind

m

瞄準器 望遠鏡

透鏡

物鏡目鏡

Emission Spectra Examples

• End of LectureEnd of Lecture

雙狹縫干涉

Young’s Double Slit Experiment, Schematic

• Thomas Young first demonstrated interference in light waves from two sources in 1801

• The narrow slits, S1 and S2 act as sources of waves

• The waves emerging from the slits originate from the same wave front and therefore are always in phase

雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉

1sin ( ) ; 0, 1, 2...

2d n n

dd

1 1 0 1

2 2 0 2

1 2

2

0

( , ) sin( )

( , ) sin( )

1

2

E l t E kl t

E l t E kl t

E E E

I c E

合

合 振幅

0

20 0

( , ) sin( )

2 ; 2 ;

1

2

E x t E kx t

k f c fk

I c E

0 1 0 2sin( ) sin( )E E kl wt E kl wt 合

0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合

2 1( ) 2

sin 2

sin

2

; 0, 1, 2...

k l l lk

n

n

d n

d n

建設性干涉

0 1

2

max 0 0

2 sin( )

12

2

E E kl wt

I c E

合

2 1( ) (2 1)

sin (2

; 0, 1,

2 1)

2..1

sin ( )2

.

k l l l n

d

d n

n

n

k

破壞性干涉

0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合

雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉

1sin ( ) ; 0, 1, 2...

2d n n

dd

Diffraction Grating

• The condition for maxima is

• The integer n is the order number of the diffraction pattern

• If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

Bright

sin ; 0, 1, 2...d n n

• 講義結束

分光儀

Diffraction Grating Spectrometer

• The collimated beam is incident on the grating

• The diffracted light leaves the gratings and the telescope is used to view the image

• The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders

sin ; 0, 1, 2...d m m

Emission Spectra Examples

雙狹縫干涉

Young’s Double Slit Experiment, Schematic

• Thomas Young first demonstrated interference in light waves from two sources in 1801

• The narrow slits, S1 and S2 act as sources of waves

• The waves emerging from the slits originate from the same wave front and therefore are always in phase

1 1 0 1

2 2 0 2

1 2

2

0

( , ) sin( )

( , ) sin( )

1

2

E l t E kl t

E l t E kl t

E E E

I c E

合

合 振幅

0

20 0

( , ) sin( )

2 ; 2 ;

1

2

E x t E kx t

k f c fk

I c E

0 1 0 2sin( ) sin( )E E kl wt E kl wt 合

0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合

2 1( ) 2

sin 2

sin

2

; 0, 1, 2...

k l l lk

n

n

d n

d n

建設性干涉

0 1

2

max 0 0

2 sin( )

12

2

E E kl wt

I c E

合

2 1( ) (2 1)

sin (2

; 0, 1,

2 1)

2..1

sin ( )2

.

k l l l n

d

d n

n

n

k

破壞性干涉

0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合

雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉

1sin ( ) ; 0, 1, 2...

2d n n

dd

Diffraction Grating

• The condition for maxima is

• The integer n is the order number of the diffraction pattern

• If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

Bright

sin ; 0, 1, 2...d n n

Diffraction Grating Spectrometer

• The collimated beam is incident on the grating

• The diffracted light leaves the gratings and the telescope is used to view the image

• The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders

sin ; 0, 1, 2...d m m

Emission Spectra Examples

Emission Spectra Examples

• 講義結束