Absorption / Emission of Photonsand Conservation of Energy

Ef - Ei = hv Ei - Ef = 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 corresponding to m = -3, -2, -1, 0, 1, 2, 3

• 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

Photographic Film

Slit

White Light Source

Emission Spectra of Hydrogen

Prism

Photographic Film

Film

Slit

Low DensityGlowing

Hydrogen Gas

Discrete Emission Spectrum

Portion of the Absorption Spectrum of Hydrogen

Discrete Absorption Spectrum

Prism

Photographic Film

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 linesfrom the ground

state (n=1)

Atom de-excitation,Emission lines

from the excited states

Schrodinger equation for one electron atoms

Coulomb potential

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V (r) = −Ze2

(4πε0)r

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−h2

2m∇ 2 −

Ze2

(4πε0)r

⎢

⎣ ⎢

⎥

⎦ ⎥ψ (

r r ) = Eψ (

r r )

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ψ(r r )=ψ(r,θ,ϕ)=ψE,l,m(r,θ,ϕ)=RE,l(r)Υl,m(θ,ϕ)

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E = En = −Z 2e2

4πε0a0

1

2n2

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l = 0,1,...,n −1

m = −l.− l +1,..., l −1, l

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ψ(r r )=ψn,l,m(r,θ,ϕ)=Rn,l(r)Υl,m(θ,ϕ)

Radial and angular part

BORN POSTULATEThe probability of finding an electron in a certain region of space is proportional to ψ2, the square of the value of the wavefunction at that region.

ψ can be positive or negative. ψ2 is always positive

ψ2 is called the “electron density”

What is the physical meaning of the wave function?

E.g., the hydrogen ground state

1 1 3/2

ψ 1s = e -r/ao (ao: first Bohr radius=0.529 Å)

ao

1 1 3ψ2

1s = e -2r/ao

ao

ψ21s

r

Radial electron densitiesThe probability of finding an electron at a distance r from the

nucleus, regardless of direction

The radial electron density is proportional to r2ψ2

Surface = 4r2

r

Volume of shell = 4r2 r