Lecture 17

Radiation from the Electron

According to the classical electromagnetic theory, the electron in a hydrogen atom produces electromagnetic radiation, just like an oscillating electric dipole, with the frequency of the radiation corresponding to that of the orbital motion f,

( )3/2

0

3/1240

00

2

21

421

ε

πεπ

femE

rme

rf

−=

=

Lecture 17

“Classical Dilemma”

•  The spectrum of the radiation should be continuous, with the frequency goes as

2/3−∝ rf

•  The atom is unstable, with the electron eventually crashes onto the proton.

Rutherford’s model implies an unstable atomic structure and is, therefore, wrong!

As the electron loses energy, its orbit shrinks and the frequency of radiation increases, until the electron crashes onto the proton.

Lecture 17

Spectrometer

Spectrometer is an instrument that is capable of measuring the wavelength (or frequency or energy) of each incident photon.

Lecture 17

Spectrum

A spectrum is simply a histogram of photons binned by their wavelengths (or frequencies or energies).

Lecture 17

Spectral Components

•  Continuum: smooth, featureless part of a spectrum

•  Emission line: a discrete feature above the continuum that is localized at a certain wavelength •  Absorption line: a discrete feature below the continuum that is localized at a certain wavelength

Observations show that each element has its own set of characteristic spectral lines and that the formation of the lines depends strongly on the physical conditions of the emission region such as temperature and density.

Lecture 17

Spectral Lines

Lecture 17

Hydrogen Lines

Balmer series:

,5,4,3,4

6.364 2

2=

−= mwherenm

mm

mλ

OR

⎟⎠⎞⎜

⎝⎛ −= 22

1211

mRH

mλ

where 15810.109677 −= cmRH is the Rydberg constant

Lecture 17

Hydrogen Lines Series

Rydberg-Ritz formula:

mnwheremn

RH <⎟⎠⎞⎜

⎝⎛ −= ,111

22λ

Balmer series is only a special case where n = 2

•  Lyman series: n=1 •  Paschen series: n=3 •  Brackett series: n=4 •  Pfund series: n=5

Other common series: Ritz Combination Principle: The difference of the frequencies of two lines in a line series is equal to the frequency of a spectral line which actually occur in another series from the same atomic spectrum.

Lecture 17

Bohr’s Postulates

•  The classical equations of motion are valid for electrons in atoms. However, only certain discrete orbits with the energies En are allowed. These are referred to as the energy levels of the atom.

Em − En = hf =hcλ

•  The motion of the electrons in these quantized orbits is radiationless. An electron can be transferred from an orbit with lower (negative) binding energy Em (i.e., larger r) to an orbit with higher (negative) binding energy En (smaller r), emitting photons in the process. The frequency (or wavelength) of the photons is given by

Lecture 17

Comparison with Experiments

To obtain the Rydberg-Ritz formula for hydrogen atoms, Bohr identified the energy terms as

En = − Rhcn2, Em = − Rhc

m2

•  With increasing orbital radius r, the laws of the quantum atomic physics approaches those of classical physics, i.e., Correspondence Principle.

where n and m are referred to as principal quantum numbers.

It is show to see, using the first postulate, that the orbital radius is proportional to the principle quantum number squared.

Lecture 17

Classical Limit

Considering the emission of photons according to the first two postulates for a transition between neighboring orbits, i.e., for m – n = 1, and for large n (or m). We get

hf = Rhc 1n2

− 1m2

⎛⎝

⎞⎠ = Rhc 1

n2− 1(n +1)2

⎛⎝⎜

⎞⎠⎟

= Rhc 2n +1n2 (n +1)2

For large n, we have f ≈ 2Rc

n3

Lecture 17

Comparison with Classical Physics

( )3/2

0

3/1240

21

εfemE −=

Applying the correspondence principle, for very large n (or orbital radius), we know that the frequency of the emitted photons is equal to that of the orbital motion.

Also, according to the classical theory, the total energy of an electron is given by

Plugging in the expression for f and comparing it to the formula in quantum theory (i.e., Bohr’s second postulate),

Lecture 17

Deriving Rydberg Constant

We have

Rhcn2

= 12m0e

4( )1/3ε 02/3

2Rcn3

⎛⎝

⎞⎠

2/3

finally,

R = m0e4

8ε 02h3c

= 109737.318 cm−1

RH = 109677.5810 cm−1

which is slightly different from