Topic 9: The Electron

T9VA: Duality of Nature

Particle v.s Wave

• Light as a wave

• Matter as something that has mass and position

Reminder

λ represents wavelength

• distance between 2 consecutive troughs or peaks

Amplitude:

• height of the wave as defined from the centre to the top of a trough of peak

Characteristics of a Wave

1. Diffraction: when a wave moves through a small slit, it will spread out in a process called diffraction

   

   1. HOWEVER, a particle hitting a slit will either bounce back or go straight through, NOT diffract

2. Interference: When waves move through a double slit they both diffract and thus interfere with one another

   

   1. However, particles will go through the slit or bounce off of it without necessarily interacting

NOTE: wave interference can either be constructive or destructive

• we can understand interference by looking at what happens when we add two waves together

Constructive Interference

Combining two waves that have their peaks and troughs aligned will result in a wave with a greater amplitude than that of the combining waves

• These two similar waves are said to be in phase

These waves interact by the ADDITION of their amplitudes

Destructive Wave Interference

Combining two waves that DO NOT have their peaks and troughs aligned will result in a wave with a lower amplitude than the original waves

• these two waves are said to be out of phase

• sometimes, the resulting wave will have no amplitude

These waves interact by the SUBTRACTION of their amplitudes

Characteristics of particles

• recoil (bouncing off of stuff)

• mass

By the end of the 19th century, scientists knew that

• light is a wave because it shows interference and diffraction patterns

HOWEVER, photons act like particles

• The energy of the photon is proportional to the frequency of the light, and not to its amplitude

Based on this, scientists posed the question:

If we can model light, in photons, as a particle, shouldn’t it also have a measurable mass ?

Einstein proposed that E = mc², determining that energy does have mass.

The mass of a photon can be theoretically computed with this equation:

The Compton effect also describes the fact that light (photons) have mass. Remember, that when light hits something, it diffracts. When a particle hits something, it recoils. Compton shot light at a metal.

• If light were a wave, it would have diffracted; however

if ended up kicking an electron out of the metal and recoiling, just like a billiard ball would

Compton used this experiment and Einsteins equation to demonstrate that the mass of the displaced electron and the recoil of photons was consistent with the predicted mass of a photon

So light is both a wave and a particle at the same time, depending on how it is measured

If light can be both a wave and a particle, what about matter ?

Matter also has a wavelength

• h: Planck’s Constant = 6.626×10^-34 J-s

• m = mass (kg)

• v = speed (m/s)

  • this is similar to Einstein’s equation except, particles don’t move at the speed of light, so instead of the constant c, it is a variable speed (v)

This λ is called the deBroglie wavelength, and it describes the wavelength of matter

NOTE: for macroscopic matter, the wavelength would be too small, and we can’t make experiments to measure them. So on a macroscopic level, we cannot experimentally determine that matter is a wave. However, the wavelengths of electrons are on a large enough order of magnitude that their wavelengths can physically be measured using solid crystals.

In summary, both matter and light have both particle and wave behavious

LIGHT:

• particle:

  • Compton mass

  • recoiling

  • E_photon is quantized and related to its frequency and mass:

    

• wave

  • interference

  • diffraction (double slit experiment)

MATTER:

• particle

  • observable mass

• wave

  • electrons and neutrons show diffraction patterns (double slit experiment)

  • the wavelength of matter can be determined from its mass:

NOTE: v in this equation is NOT the speed of light

T9VB: Birth of Quantum Mechanics

After the establishing of matter and light being both particles and waves, the planetary model (Niels Bohr) was deemed unproductive bc

• electrons cant orbit a central nucleus without losing energy and therefore spiralling inwards

• atoms would have decayed if this were accurate

So scientists needed to come up with a new model for the structure of the atom

Remember that the fundamental equation of classical mechanics is the equation f = ma

• atoms cannot be explained this way

Quantum Mechanics

• from quantum mechanics we get the principle that not all energies are allowed (only those in specific states); and that

• we cannot actually predict with certaintly where small particles are found (they are not localised)

Fundamental equation of quantum mechanics, the Schrodinger Equation:

E: energy → gives us the energy levels of the atom

H: hamiltonian operator → contains the mathematic operations of partial derivatives

Ψ = the critical wave function → contains all the information we need about the electron

This equation relates the particle nature of an electron to its wave nature. A stable atom can be predicted using Schrodinger’s Equation

• there are always two types of solutions from this equation:

  • E: the energy levels; or

  • Ψ: the wave functions

• small particles are described by these wave functions (expressed with Ψ)

The principle of quantization is that matter and electrons are restricted to certain energy states, and that they cannot “land” between levels. Why does this happen ?

Think of standing waves, waves that are physically anchored at the end, like the strings of a guitar. If we pluck a guitar string, we hear fixed tones from fixed frequencies. This is because the ends of the strings are anchored, and are effectively quantizing the energy of the vibration, or restricting the energy to certain levels. See below:

The string’s anchors on the guitar are equivalent to boundary conditions for the wave function (Ψ). Like the guitar string, the anchoring of boundary conditions is what produces fixed and stable energy states for an atom.

Anchoring (boundary conditions) leads to quantization of energy levels.

• energy level and wave functions are labelled by quantum numbers

T9VC: Quantum Numbers

Reminder: we solve Schrodinger’s Equation for two things: energy and wave functions (Ψ)

An important outcome to the Schrodinger equation is a set of numbers used to label and identify these energies and wavefunctions: these are called quantum numbers

• n

• ℓ

• m_ℓ

• m_s (this will be discussed in T9VD)

n

• the principle quantum number

• represents the orbital energy level

• it is an integer value

  • NOTE: in the Rydberg equation, E is restricted by the value of n. If n could be any real number, all the E would be possible, so E would not be quantized.

• n = 1, 2, 3, …

ℓ

• angular momentum quantum number

• represents the orbital shape such as s, p, d, etc

  • ℓ = 0: s orbital

  • ℓ = 1: p orbital

  • ℓ = 2: d orbital

  • ℓ = 3: f orbital

  • ℓ can continue with larger integer values***

• possible ℓ values are restricted by the n value, we can determine the value of ℓ from n

  • ℓ = n-1, … 0

    • the maximum ℓ can be is n-1

    • the minimum ℓ can be is 0

    • ℓ also includes all the integers between n-1 and 0

***NOTE: elements in ground state can have electron configurations through the f-block. However, when electrons become excited, the energy levels can go higher, the value of n can be extended, so the value of ℓ can also be extended. ℓ values above 3 (those that represent orbitals above s, p, d, and f, are labelled with the alphabet, starting from f, g, h, and going on)

m_ℓ

• magnetic quantum number

• represents the number of orbitals, which represents the orbital orientation

  • the number of m_ℓ values dictates the number of orbitals

    • EXAMPLE: ℓ = 1: p orbital; m_ℓ = -1, 0, 1: 3p orbitals

• possible m_ℓ are restricted by the ℓ value

  • m_ℓ= +ℓ, …, 0, …, -ℓ

    • if ℓ = 4, m_ℓ = -4, -3, -2, -1, 0, 1, 2, 3, 4

  • always integers

T9VD: Electron Spin

There is also a quantum number m_s

When a stream of electrons goes through a magnetic field, the stream bends in one direction or another. This suggests that the electron itself has a specific spin (electronic spin) that can be redirected or manipulated by magnetic interference. An electron’s spin is represented with the quantum number m_s

m_s

• spin quantum number

• represents the spin on an electron

• the value of m_s = + ½, - ½ (spin up and spin down)

Putting all the quantum numbers together means that all electrons in an atom each have four quantum numbers: n, l, m_l, and m_s. No two electrons in an atom can have the same set of four quantum numbers

• this means that electrons that sit in the same orbital must have opposite spins

Example of labelling an electron with its quantum numbers. Let’s say we’re given this configuration:

n: the electron is in the third shell, so n = 3

l: the electron is in the d suborbital, so l = 2

m_l: based on l = 2, m_l can be -2, -1, 0, 1, 2; we don’t know what the orbital looks like based on this image, so we can pick any of these numbers

• let’s pick -2

m_s: this electron is either down-spin or up-spin, so m_s is either + ½, - ½. we can pick either one because we don’t have further information about this electron

• let’s pick + ½

So, we are left with the set of these four quantum numbers: 3, 2, -2, + ½. All of the other electrons in the 3d suborbital have to have a different set, but since they are all 3d electrons, the n and l remain the same, and only the m_l and m_s change.

T9VE: Wave Functions

Let’s explore the wave functions (Ψ) for the hydrogen atom. Why hydrogen ? The hydrogen atom wave functions serve as models for all the other atoms.

NOTE: wave functions (Ψ) themselves are not observable. However, Ψ² gives us the electronic density, which is the PROBABILITY of finding an electron in a given area of the atom

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