1. Structure & Properties of Matter
Lesson # 2: Quantum Mechanics

2. Definitions
Classical mechanics – branch of physics that studies the motion of macroscopic objects.
Quantum mechanics – study of motion at the atomic level, where the laws of classical mechanics do not apply because particles behave like waves.

3. Schrödinger’s Contribution
Schrödinger decided to study the wave-like properties of electrons. He thought that electrons followed the path of a standing wave, similar to the vibration that occurs when plucking a guitar string, and follow whole-number wavelengths as it moves.
He thought that the first orbital was one wavelength, the second two, the third three, etc.

4. Schrödinger’s Cat
The problem was that if the electron travels in waves, how can we pinpoint its exact location at any time in its energy level? We know that it’s contained to its orbital based on its energy, but it is so tiny that it could really be anywhere in that space!

5. Heisenberg’s Uncertainty Principle
Heisenberg came up with a statistical approach to locating electrons. He said that in large objects, you must know both the speed and location to track them, but with subatomic particles, it is impossible to simultaneously know the exact position and speed of an electron.
A wave function is used to describe the orbital in an atom, where an electron of a certain energy is likely to be found. It’s not like the Bohr model – the electron is not moving around the nucleus in a circle. We actually have no exact idea in what path electrons move.

6. Wave Function

7. Quantum Mechanical Model
Using the wave function to predict a rough path, scientists have created three-dimensional electron probability densities which show the greatest probability of finding an electron. They look like electron “clouds”.
In hydrogen, the probability is spherical, and the densest area is actually roughly the value he estimated as the radius of the first orbital.
Overall, the quantum mechanical model means that electrons can be in different orbitals by absorbing or emitting quanta of energy. It is much different from other models of the atom as it is based completely on uncertainty.

8. Quantum Numbers
Schrödinger’s wave equation is very complicated and its solutions give us many wave functions that describe various types of orbitals.
Each of these orbitals has a set of four numbers called quantum numbers, which describe various properties of the orbital.
The numbers are like addresses that help us locate the position of an electron around an atom.

9. 1. The Principal Quantum Number (n)
This describes the size and energy of an atomic orbital. We know it as the “shell” number. n ranges from 1-7, which is another way the periodic table is organized – 7 periods.
Note that the spaces between orbitals are not equal – this is because as n increases, the energy required for an electron to occupy that orbital increases.
Each orbital is also larger as the distance from the nucleus increases, meaning that electrons in the far orbitals spend more time there, and that they are less tightly bound to the nucleus.
Example – if n = 3, then that electron is in the third shell.

10. 2. The Secondary Quantum Number (l)
When Bohr was looking at the line spectrum of hydrogen, he noticed that the lines were not actually just one line for each energy level, but a few smaller lines close together. We now know this is because there are subshells within each orbital.
The secondary quantum number describes the shape of an atomic orbital. It has whole-number values, and ranges from 0 to n-1 for each orbital.

11. Example When n = 1, l = When n = 2, l = When n = 3, l =
We use letters to represent the each value of l.
l = 0 – s orbital (“sharp”)
l = 1 – p orbital (“principal”)
l = 2 – d orbital (“diffuse”)
l = 3 – f orbital (“fundamental”) – also have g, h, and I (just use alphabet, no special names).
So if n = 3, l = then the atom has electrons in:

12. 3. The Magnetic Quantum Number (ml)
This describes the orientation of the orbital relative to other orbitals in the atom. It has whole number values ranging from +l to –l, including zero
Example - If l = 2, ml =
The number of different values that ml can have equals the number of orbitals that are possible in that energy level range, so when l = 2, there are 5 possible orbitals. There are 5 different d orbitals.

13. Quantum Number Summaryl
Sublevel designation ml
Number of orbitals
Number of electrons in each energy level (2n2) 1 1s 2 2s 8 2p
-1,0,1 3 3s 18 3p 3d
-2.-1,0,1,2 5 4 4s 32 4p 4d
-2,-1,0,1,2 4f
-3,-2,-1,0-1,2,3 7

14. Shapes & Orientations of Orbitals
S orbitals
Spherical shape.
They actually have areas of low and high probability alternating from the center. The areas of zero probability are called nodes (like in waves).

15. Shapes & Orientations of Orbitals
P orbitals
Two lobes separated by a node at the nucleus
They can exist in any orientation of the 3 dimensions (x, y, and z – hence the 3 ml options). The x orientation has its lobes centered across the x axis.

16. Shapes & Orientations of Orbitals
D orbitals
Four lobes separated by a node at the nucleus.
They can exist in 5 different orientations (5 ml options). They are xz, yz, xy, x2-y2 and z2.
xz, yz, and xy lie along the two planes. X2-y2 is between the x and y planes, and z2 is unique – two lobes along the z-axis and a belt centered in the xy plane.

17. 4. The Spin Quantum Number (ms)
This describes the magnetic property of atoms, called a magnetic moment, when the atom is placed in a magnetic field.
The moment has two orientations, called “spin”.
We assign values of +½ and -½, each spinning in opposite directions.

18. The Pauli Exclusion Principle
In a given atom, no two electrons can have the same set of four quantum numbers.
If electrons are in the same orbital then they have the same n, l and ml values, so they must have different spin values.
This is why no specific orbital can hold more than two electrons. This is also whey we tend to pair electrons up when we draw Bohr-Rutherford diagrams.