Quantum Numbers

1. Quantum Numbers Mr. Tsigaridis

2. Quantum Mechanics • We have discussed thus far in the course the wave-particle duality in some length • The Heisenberg uncertainty principle, as discussed last class briefly, is a fundamental principle of quantum theory • Heisenberg’s theory says that you can know a particles position, but you can not know that same particles momentum, or vice versa • Since that theory science has been trying to identify how to determine both of these quantities at the same time

3. Quantum Mechanics • Luckily, we have developed mathematics called statistics that can help us in predicting the likelihood of finding an electron at a particular point around the nucleus • In particular, an equation called Schrödinger’s wave equation

4. Schrödinger’s Wave Equation • The equation can be used to determine the behaviour of electrons in atoms, specifically, the probability that electrons are at a given point in space and time • With modern day computers, we can actually develop ways in which we can map the probability of electrons being found in a specific part of an atom. • Theses are called electron probability graphs

5. Electron Probability Graphs

6. Quantum Numbers • Quantum numbers can be used to describe the position and orientation, spin and energy level of each electron in an atom • To do this, values are placed on each electron that describe its location much like a complete address for a house

7. Principal Quantum Number • The principle quantum number describes the ground state of an element • It represents the energy level that the electron exists at • This number is denoted by the letter n • These numbers are whole numbers starting at n=1 and continue until n=∞

8. Principal Quantum Number • The principle quantum number represents the energy level of a certain electron and it also, and more importantly represents the size of the orbital • This has an affect on the physical and chemical properties of an atom which we will be discussing further • The important thing we have to remember about the first quantum number is that it is represents the energy contribution with the momentum contribution left out

9. Principal Quantum Number • This numbers only depends on the distance that an electron is from the nucleus (radial distance) • It is the probability of finding an electron at a specific distance from the nucleus

10. Azimuthal Quantum Number • The second of the quantum numbers, the Azithmal Quantum Number, is also known as the angular momentum quantum number because it includes the momentum of the electron in orbit • This quantum number is denoted by the letter l • It can have a value of anything up to n-1 • If n=1 then l = 0 • Similarly if n=3 then l = 0, l = 1 , or l = 2

11. Azimuthal Quantum Number • The second quantum number represents the angular momentum • Angular momentum is a vector quantity that represents the electrons position and its momentum • Each of the two are controlled by their own operators with x,y,z co-ordinates • Heisenberg’s uncertainty principle tells us that not all six of these quantities can be known simultaneously with arbitrary precision

12. Azimuthal Quantum Number • In chemistry this number is vital in its importance since it describes the shape of the orbital that an electron can delve in • It strongly influences chemical bonds and bond angles

13. Azimuthal Quantum Number • In some contexts and for the purposes of this course, the l = 0 represents the “s” orbital, l = 1 represents the “p” orbitals, or l = 2 represents the “d” orbitals

14. Magnetic Quantum Number • The third quantum number, The Magnetic Quantum Number, is denoted by ml • It is also known as the orbital orientation quantum number as it gives rise to the orientation or direction of each orbital in 3D space along the three axis • It is representative of the orbital angular momentum of an electron along a specified axis

15. Magnetic Quantum Number • Its values can range anywhere between –l to +l • If l =0 then ml =0 • Similarly if l =1, ml =-1, ml =0, ml =1 • When l =0 it corresponds to the s orbital, there is no specific directional values for ml • When l =1 it corresponds to the p orbitals, and gives rise to three possibilities for as stated before • The additional quantum numbers are required to specify the particular direction along the axis that these orbitals are alligned

16. Magnetic Quantum Number • Notice that as the “p” orbitals are placed together on the same Cartesian plane they form almost a sphere, much like the sphere of the “s” orbitals in which they will fit

17. Magnetic Quantum Number • Shown here are the shapes of the “d” orbitals

18. Magnetic Spin Quantum Number • The last of the quantum numbers, the magnetic spin quantum number, is denoted by ms • The number can have values that are either +½ or -½ • They represent the spin orientation of each electron • When electrons occupy orbitals they occupy them in pairs and a pair of electrons, in order to occupy the same orbital and part of orbital must have opposite spins

19. Magnetic Spin Quantum Number • Certain fundamental particles have associated with them a magnetic moment that can align itself in either of two directions with respect to an external magnetic field • The idea that no two identical electrons can occupy the same point in space is known as the Pauli Exclusion principle • By having different spins, the electrons are able to co-exist at the same energy level even though they are different

20. Quantum Numbers • Two electrons in a single orbital are often referred to as an electron pair • The electrons and their configurations are described by the four quantum numbers • Each number represents an ever increasing accuracy of identifying where in an atom an electron exists • As previously stated the electrons quantum numbers are like an address system for each electron, each one being unique

21. Quantum Numbers • The quantum numbers have the following analogy which helps us understand what the quantum numbers represent in terms of the location and orientation of electrons within atoms • Principal Quantum Number – What country • Azimuthal Quantum Number – What city • Magnetic Quantum Number – What street • Magnetic Spin Quantum Number – What house