Plan for Mon, 3 Nov 08• Exam 2, Quiz 5, and Quiz 4 regrades returned

Wednesday• Bring a USB drive to lab on Wednesday

• Lecture– Cheer up, it’s only quantum mechanics! – Wavefunctions, energies, and the Hamiltonian for the

H atom (not in book)– Quantum numbers (7.6)– Orbital shapes and energies (7.7)– Electron spin and the Pauli Principle (7.8)

Particles and Waves• EM radiation can behave as either a wave or a particle

depending on the situation• “Light has properties that have no analogy at the

macroscopic level, and thus, we have to combine two different ideas to describe its behavior.” (Cracolice & Peters, 3/e)

Famous Opinions of QM“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, 1920)

“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?‘” 

(Albert Einstein, 1954)

“Those who are not shocked when they first come across quantum physics cannot possibly have understood it.”

(Niels Bohr, 1958)  

Famous Opinions of QM

The one great dilemma that nails us…day and night is the wave-particle dilemma.

(Erwin Schrodinger, 1959)  

I think I can safely say that nobody understands quantum mechanics.

(Richard Feynman, 1965)  

Wavefunctions• A wavefunction is a probability amplitude.

The “square” of a wavefunction gives the probability density…the likelihood of finding the particle in region of space.

• The wavefunctions and kinetic energies available to a quantum particle are quantized if the particle is subject to a constraining potential.

• We can determine the wavefunctions and KEs available to our system by considering the field of force (the PE) our system is subject to.

The Hamiltonian• Erwin Schrodinger developed a mathematical formalism

that incorporates the wave nature of matter.• H, the “Hamiltonian,” is a special kind of function that

gives the energy of a quantum state, which is described by the wavefunction, .

• H contains a KE part and a PE part:

• By solving the Schrodinger equation (below) with a known Hamiltonian, we can determine the wavefunctions and energies for quantum states.

H E

2 2 2 2

2 2 2 2ˆ ( , , )

8

h d d dH V x y z

m dx dy dz

H-atom wavefunctions• In the H atom, we are interested in describing the regions in

space where it is likely we will find the electron, relative to the nucleus…we want the wavefunction for the electron.

• We can model the attraction of the H atom’s single electron to its single proton with a “Coulombic” potential curve:

e-

P+r

r0

V (r) e2

r

• The V(r) potential becomes part of the Hamiltonian for the electron.

H-atom wavefunctions (cont.)• The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates.

p+

e-

r = interparticle distance (0 ≤ r ≤ )

= angle from “xy plane” (/2 ≤ ≤ - /2)

= rotation in “xy plane” (0 ≤ ≤ 2)

H-atom wavefunctions (cont.)• Then the Schrodinger equation for the hydrogen atom becomes:

H E

2 22

2 2 2 2

2

1 1sin

8 sin sin

hr

r r r

ZeE

r

3-dimensional KE operator in spherical polar coordinates

Radial Coulombic PE operator

H-atom wavefunctions (cont.)

• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized:

En Z 2

n2

me4

802h2

2.178x10 18 J

Z 2

n2

n is the principle quantum number, and can have integer numbers ranging from 1 to infinity.

The higher n, the greater the distance between the nucleus (+) and the electron (-).

H-atom wavefunctions (cont.)

• In solving the Schrodinger Equation, two other quantum numbers become evident:

…the orbital angular momentum quantum number. Ranges in value from 0 to (n - 1 ).

ml … the “z component” of orbital angular momentum. Ranges in value from - to 0 to .

• We can characterize the hydrogen-atom orbitals using the quantum numbers: n, , ml

Orbitals and Quantum Numbers• Naming the electron orbitals is done as follows

– n is simply referred to by the quantum number– (0…n - 1) is given a letter value as follows:

• 0 = s• 1 = p• 2 = d• 3 = f

- ml (- …0… ) is usually “dropped”

For example: for n = 3, = 2 “3d orbital”

Quantum Mechanical Model

The Bohr model is deterministic…uses fixed “orbits” around a central nucleus to describe electron structure of atoms.

The QM model is probabilistic…uses probabilities to describe electron structure.

A probabilistic electron structure is much more difficult to visualize. HOWEVER, the electronic energy levels are still quantized.

Deterministic vs. Probabilistic• In the Bohr model, you can

always find the electron in an atom, just like you can always find the moon as it orbits the earth.

• You can always determine the relative location of the nucleus and electron in Bohr’s model.

• This is because the electron follows a particular orbit around the nucleus.

• In the QM model, the electron does not travel along a particular path around the nucleus.

• You can never determine the electron’s exact location…you can only find where it is likely to be.

• The Bohr orbit is replaced by orbital which describes a volume of space in which the electron is likely to be found.

Quantum Numbers and Orbitals (cont.)

• Table 7.2: Quantum Numbers and Orbitals

n Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1 1 3p -1, 0, 1 3

2 3d -2, -1, 0, 1, 2 5

Incr

easi

ng

Ene

rgy

Orbital Shapes (cont.)

• Example: Write down the orbitals associated with n = 4.

Ans: n = 4 l = 0 to (n - 1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f

4s (1 ml sublevel)4p (3 ml sublevels)4d (5 ml sublevels4f (7 ml sublevels)

Which of the following sets of quantum numbers (n, l, m) is not allowed?

A. (3, 2, 2).

B. (0, 0, 0).

C. (1, 0, 0).

D. (2, 1, 0).

Electron OrbitalsOrbitals represent a probability space where an electron is likely to be found.

All atoms have all orbitals, but many of them are not occupied.

The shapes of the orbitals are determined mathematically...they are not intuitive.

s orbitals

p orbitals

d orbitals

Electron Orbital Shapes• The “1s” wavefunction has no angular dependence

(i.e., it only depends on the distance from the nucleus).

1s 1

Z

ao

32

e

Z

a0

r

1

Z

ao

32

e

*Probability =

• Probability is spherical

Electron Orbital Shapes (cont)s (l = 0) orbitals

as n increases, orbitals demonstrate n - 1 nodes.

Node: an area of space where the electron CAN’T be, ever, no matter how much it wants to.

Aside: What’s a node??• Remember the guitar-string standing wave analogy?

• A standing wave is a motion in which translation of the wave does not occur.

• In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs.

Electron Orbital Shapes (cont.)

not spherical, but lobed.

labeled with respect to orientation along x, y, and z.

2p (l = 1) orbitals

2px 2py 2pz

2 pz

1

4 2Z

ao

32

e

2 cos

Electron Orbital Shapes (cont.)

• more nodes as compared to 2p (expected.).

• still can be represented by a “dumbbell” contour.

3 pz

2

81 Z

ao

32

6 2 e 3 cos

3p (l = 1) orbitals

Orbital Shapes (cont.)3d (l = 2) orbitals labeled as dxz, dyz, dxy, dx2-y2 and dz2.

Orbital Shapes (cont.)

We will not show the exceedingly complex probability distributions associated with f orbitals.

4f (l = 3) orbitals

Electron Orbital Energies in the H-atom

• energy increases as 1/n2

• orbitals of same n, but different l are considered to be of equal energy (“degenerate”).

• the “ground” or lowest energy orbital is the 1s.

Orbital Summary• Orbital E increases with n.

– At higher n, the electron is farther away from the nucleus...this is a higher energy configuration.

• Orbital size increases with n. – There is a larger area of space where you are likely to find an

electron at higher E’s. • Orbital shape is the same no matter the value of n.

– 3s looks like 1s, except it’s bigger and has more nodes. Same for p, d, f, etc.

• Number of nodes in an orbital goes as n - 1. – 1s has zero nodes, 2s has one node, 3s has two nodes...– 2px, 2py, 2pz each have one node, 3px, 3py, 3pz each have two nodes– the 3d orbitals each have two nodes, 4d have three, etc.– Note that number of nodes indicates relative energy!

• All atoms have all orbitals…but in an unexcited atom, only those closest to the nucleus will be occupied by electrons

Orbital Quiz• The shape of a given type of orbital changes as

n increases.

• The number of types of orbitals in a given energy level is the same as the value of n.

• The hydrogen atom has a 3s orbital.

• The number of lobes on a p-orbital increases as n increases. That is, a 3p orbital has more lobes than a 2p orbital.

• The electron path is indicated by the surface of the orbital.

T

F

T

F

F