The world of Atoms

Quantum MechanicsTheory that describes the physical properties of smallest particles (atoms, protons, electrons, photons)

"A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it."

Max Planck

Erwin Schrödinger"I don't like it and I'm sorry I ever had anything to do with it."

"An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them"

Werner Heisenberg

"It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom."

Max Born

The hydrogen atom

Niels Bohr (1885-1962)

- electron orbits around the nucleus like a wave

- orbit is described by wavefunction

- wavefunction is discrete solution of wave equation

- only certain orbits are allowed

- orbits correspond to energy levels of atom

The hydrogen atomIn the Bohr model of the atom, the hydrogen atom is like a planetary system with the electron in certain allowed circular orbits.

The Bohr model does not work for more complicated systems!

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Ψ6(r)

Quantum numbersEach orbital is characterized by a set of quantum numbers.

Principal quantum number (n): integral values (1,2,3). Related to the size and energy of the orbital.

Angular momentum quantum number (l): integral values from 0 to (n-1) for each value of n.

Magnetic quantum number (ml): integral values from - l to l for each value of n.

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Ψn,l ,m l(r)

Quantum numbersHow many orbitals are there for each principle quantum number n = 2 and n = 3?

For each n, there are n different l-levels and (2l+1) different ml levels for each l.

n=2: n = 2 different l-levels

(2l+1) = 2 x 0 + 1 = 1 ml-levels for l = 0

l = 0, 1

(2l+1) = 2 x 1 + 1 = 3 ml-levels for l = 1Total: 1 + 3 = 4 levels for n = 2

Quantum numbersHow many orbitals are there for each principle quantum number n = 2 and n = 3?

For each n, there are n different l-levels and (2l+1) different ml levels for each l.

n=3: n = 3 different l-levels

(2l+1) = 2 x 0 + 1 = 1 ml-levels for l = 0

l = 0, 1,2

(2l+1) = 2 x 1 + 1 = 3 ml-levels for l = 1

Total: 1 + 3 + 5 = 9 levels for n = 3(2l+1) = 2 x 2 + 1 = 5 ml-levels for l = 2

The total number of levels for each n is n2

Quantum numbers

Names of atomic orbitals are derived from value of l :

Quantum numbersQuantum numbers for the first four levels in the hydrogen atom.

What is the meaning of ?

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Ψ

Wavefunction itself is not an observable!

Square of wavefunction is proportional to probability density

“I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality - that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model. This seems to me the permanent upshot of Heisenberg's principle of uncertainty. (Albert Einstein, on Quantum Theory, 1934”

Wavefunction and probability

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Ψn,l ,m l

‘function’

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Ψn,l ,m l

r

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Ψn,l,m l

2

‘probability’

Quantum numbersA subshell is a set of orbitals with the same value of l. They have a number for n and a letter indicating the value of l.

l = 0 (s) l = 1 (p) l = 2 (d)

l = 3 (f) l = 4 (g)

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Ψn,l,m l

2

Orbital Shapes

Heisenberg uncertainty principle

Life is uncertain!

Where’s the electron?

Werner Heisenberg

That’s quite uncertain!

Heisenberg uncertainty principleIt is not possible to know both the position and momentum of an electron at the same time with infinite precision.

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Δx ⋅Δ mv( ) ≥ h4π

Δx is the uncertainty in position.

Δ(mv) is the uncertainty in momentum.

h is Planck’s constant.

Heisenberg

The s orbitals in hydrogen

The higher energy orbitals have nodes, or regions of zero electron density.

orbital surfaces

probability distributions

s-orbitals have n-1 nodes.

The 1s orbital is the ground state for hydrogen.

The orbital is defined as the surface that contains 90% or the total electron probability ( ).

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Ψn,l,m l

2

Pauli exclusion principleHow many electrons fit into 1 orbital?

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Ψ2,1,02

ms = +1/2

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Ψ2,1,02

ms = -1/2

Only 2 electrons fit into 1 orbital: 1 spin up1 spin down

Pauli exclusion principle

As the temperature is lowered, bosons pack much closer together, while fermions remain spread out.

Electrons are fermions. There are also bosons

Energy Levels

n =1

n =2

n =3n =4n =5

n =∞

E

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E = −RHZ 2

n2

⎛

⎝ ⎜

⎞

⎠ ⎟

RH = 2.178 x 10-18 J

Z = atomic number

n = energy level

Energy Transitions

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ΔE = −RHZ 2

n f2

− Z2

ni2

⎛

⎝ ⎜

⎞

⎠ ⎟

For the energy change when moving from one level to another:

n =1

n =2

n =3n =4n =5

n =∞

E transition

Lines and Colors

Change in energy corresponds to a photon of a certain wavelength:

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ΔE = hυ = hcλ

Change in energy Frequency of

emitted light

Wavelength of light emitted

Lines and Colors

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ΔE = −RHZ 2

n f2

− Z2

ni2

⎛

⎝ ⎜

⎞

⎠ ⎟

What is the wavelength of the photon that is emitted when the hydrogen atom falls from n=3 into n=2?

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ΔE = − 2.178 ×10−18J( )12

22 − 12

32

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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=3.03 ×10−19J

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3.03×10−19J = hcλ

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λ =6.626 ×10−34 ⋅3.00 ×108

3.03×10−19

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=656 nm

Light out of Molecules

n =1

n =2

n =3n =4n =5

n =∞

E transition

hydrogen

Rhodamine

532 nm 570 nm

‘Fluorescence’

Degeneracy

Orbital energy levels for the hydrogen atom.

Beyond hydrogenHydrogen is the simplest element of the periodic table.

Exact solutions to the wave equations for other elements do not exist!

Polyelectric AtomsWhat do the orbitals of non-hydrogen atoms look like?

Multiple electrons: electron correlation

Due to electron correlation, the orbitals in non-hydrogen atoms have slightly different energies

Polyelectric AtomsScreening: due to electron repulsion, electrons in different orbits ‘feel’ a different attractive force from the nucleus

11+

e-

e-e-

e-

e-

e-

e-

e-

Sees a different effective charge!

Screening changes the energy of the electron orbital; the electron is less tightly bound.

Polyelectric AtomsPenetration: within a subshell (n), the orbital with the lower quantum number l will have higher probability closer to the nucleus

n =2 orbital n=3 orbital

Polyelectric Atoms

Hydrogen Polyelectric atom

Orbitals with the same quantum number n are degenerate

Degeneracy is gone:

Ens < Enp < End < Enf

Spectra of Polyelectric AtomsDue to lifting of degeneracy, many more lines are possible in the spectra of polyelectric atoms