Absorption / Emission of Photons

Absorption / Emission of Photons

and Conservation of Energy

E_f - E_i = hv E_i - E_f = hv

hv

hv

Energy Levels of Hydrogen

Electron jumping to a higher energy level

E = 12.08 eV

Spectrum of Hydrogen, Emission lines

Bohr’s formula:

Hydrogen is therefore a fussy absorber / emitter of light

It only absorbs or emits photons with precisely the right energies dictated by energy conservation

Electron in a Hydrogen Atom

• The three quantum numbers:

– n = 1, 2, 3, … – l = 0, 1, …, n-1

– m = -l, -l+1, …, l-1, l

• For historical reasons, l = 0, 1, 2, 3 is also known

as s, p, d, f

1s Orbital

Density of the cloud gives

probability of where the electron

is located

2s and 2p Orbitals

Another diagram of 2p orbitals

Note that there are three different configurations corresponding to m = -1, 0, 1

3d Orbitals

Now there are five different configurations corresponding to m = -2, -1, 0, 1, 2

4f Orbitals

There are seven different configurations

• The excited atom usually de-excites in about 100 millionth of a second.

• The subsequent emitted radiation has an energy that matches that of the orbital change in the atom.

• This emitted radiation gives the characteristic colors of the element involved.

Emission Spectra

Continuous Emission Spectrum

Prism Slit

White Light Source

Emission Spectra of Hydrogen

Prism

Photographic Film

Film Slit

Low Density Glowing Hydrogen Gas

Discrete Emission Spectrum

Portion of the Absorption Spectrum of Hydrogen

Discrete Absorption Spectrum

Prism

Film Slit

White Light Source

Discrete Emission Spectrum

Hot

Hydrogen Gas

Absorption Spectra

• Frequencies of light that represent the correct energy jumps in the atom will be absorbed.

• When the atom de-excites, it may emit the same kinds of frequencies it absorbed.

• However, this emission can be in any direction.

Emission and Absorption

Continous Spectrum

Portion of the Emission Spectrum

Absorption Spectrum Hot Gas

Cold Gas

Absorption spectrum of

Sun

Emission spectra of

various

elements

Usually the Emission spectrum has more

“features” of the absorption spectrum

Atom excitation, Absorption lines from the ground

state (n=1)

Atom de-excitation, Emission lines

from the excited states

Schrodinger equation for one electron atoms

Coulomb potential

€

V (r) = − Ze² (4πε₀)r

€

−h²

2m ∇² − Ze² (4πε₀)r

⎢

⎣⎢ ⎥

⎦⎥ψ (r

r ) = Eψ (r r )

€

( ψ r r )

=ψ (r,

θ,ϕ )=

ψ

E,l

(r,

θ,ϕ )=

R (r)

Υ (θ,

ϕ)

E = E

= − Z

e

4πε

a

1 2n

€

l = 0,1,...,n −1

m = −l.− l +1,...,l −1,l

€

( ψ r r )

=ψ

n,l

(r,

θ,ϕ )=

R (r)

Υ (θ,

ϕ)

Radial and angular part

BORN POSTULATE

The probability of finding an electron in a certain region of space is proportional to ², the square of the value of the wavefunction at that region.

 can be positive or negative. ² is always positive

² is called the “electron density”

What is the physical meaning of the wave function?

E.g., the hydrogen ground state

 1 1 ^3/2

 _1s = e ^-r/ao (a_o: first Bohr radius=0.529 Å)

 a_o

 1 1 ³

²_1s = e ^-2r/ao  a_o

²_1s

r

Radial electron densities

The probability of finding an electron at a distance r from the nucleus, regardless of direction

The radial electron density is proportional to r²²

Surface = 4r²

r

Volume of shell = 4r² r