Quantum mechanical model of the atom

Louis de Broglie Theoretical Idea: Electrons as a type of standing wave

Louis de Broglie In 1923 Lois de Broglie proposed that matter, like light or other radiant energy has both wave and particle characteristics. He derived an equation for a matter wave: λ=h/mv

Louis de Broglie The electron can be compared with a standing wave.

Standing Wave For any standing wave: Node = place of zero vibration or amplitude (dead spot)

Standing Wave Any standing wave has restrictions eg. Vibrating string must contain multiples of exactly ½ 

De Broglie’s Electron Standing Wave: The electron is like a standing wave of negative charge density, wrapped around the nucleus. If the wave pattern is not positioned just right it will destructively interfere with itself and be annihilated. De Broglie’s Electron Standing Wave:

The electron wave must fit seamlessly around the nucleus. Only certain wave patterns are possible Allowed electron wave patterns must be multiples of 1 deBroglie wavelength

Wave-particle duality of Electrons Light has been regarded as having wave- particle duality. Waves: a continuous traveling disturbance Particles: discrete bundles Distinctions appear to break down on the atomic level.

Evidence for the electron as a particle Milliken’s oil drop experiment determined the charge on one electron F (gravity, downwards) = mg = F (electrostatic upwards) = kqq2 / r2

Evidence for the electron as a particle A specific type of wave will always have the same velocity in a given medium But electrons can travel a variety of velocities all in the same medium simultaneously (this suggests individuality!)

Diffraction Patterns from Interference between Waves

Evidence of Electron’s Wave Properties Electron beam on Nickel Crystal Interference Pattern Electron Beam on Gold foil Interference Pattern

Wave-particle duality of Electrons Neither the wave model nor the particle model is capable of explaining ALL behaviours of electrons. Both are required.

Heisenberg’s Uncetainty Principle

Heisenberg’s Uncetainty Principle It is impossible to determine simultaneously the exact position and momentum of a single atomic particle x mv > h Planck’s constant (6.626 x 10-34 J s ) Uncertainty in position Uncertainty in momentum

Heisenberg’s Uncetainty Principle Our “probe” to see is light. Light shining on a large object leaves the object unaffected. However, Light would interfere with the position and momentum of an electron in an unpredictable way. (In the process of measuring, we affect the property we attempt to measure)

Heisenberg’s Uncertainty Principle Applied:

Heisenberg’s Uncertainty Principle There is a physical limit to the precision with which we can simultaneously measure pairs of properties like location and momentum. This limit is Planck’s constant , h (6.626 x 10-34 Js) We cannot know the exact position of the electron; only where the electron is most likely to be.

Schrodinger’s Wave Equation

Schrodinger’s Wave Equation An equation with 2 unknowns: E = allowed energy level of atom 𝞇 = wavefunction; a mathematical description of the electron H= the “hamiltonian”; not a variable, a set of mathematical instructions to be performed on 𝞇

Schrodinger’s Wave Equation Only certain Energy values will result in answers (wavefunctions), 𝞇 For any given E there may be 1 or more than one wavefunction possible.

Schrodinger’s Wave Equation Looks like Fun!!!

The Wavefunctions (Ψ) of the Hydrogen atom the prize!!!

Schrodinger’s Wave Equation 𝞇 gives us information on a particular electron waveform. (called an “orbital”) An equation for a line, y = mx + b may be plotted in two dimensions: 𝞇 contains the imaginary number, i (square root of -1) ∴We plot |𝞇2| instead

Plotting the wavefunction to “see” the electron orbital: |𝞇2| is proportional to the probability of the electron being at a given point in space. Making a 3-d plot of |𝞇2| gives a probability picture: At a given Energy, where the electron is most likely to be.

Orbitals: 𝞇2 for E n plotted in 3-d A fuzzy picture of where the electron is most likely to be at a given energy level

Orbitals: 𝞇2 for E 1 (n=1) plotted in 3-d: Probability Pictures of Electron Waveforms

n, l, ml , ms Quantum Numbers: Like any standing wave, there are some physical restrictions built in. These boundary conditions require that certain constants enter into the solution of Schrodinger’s Wave equation These constants are called quantum numbers. n, l, ml , ms

Tell us the characteristics of the Quantum Numbers: A set of these quantum numbers give information about each orbital and each electron n, l, ml , ms Tell us the characteristics of the electron waveforms

Quantum Numbers: Principal quantum number n = positive integers (n≠ 0) symbol name Value restrictions Characteristic described Special conventions n Principal quantum number n = positive integers (n≠ 0) shell Average distance from the nucleus n shell K L M etc…

l Quantum Numbers: Secondaryquantum number symbol name Value restrictions Characteristic described Special conventions l Secondaryquantum number l = all integers in the range: 0 to n-1 indicates shape of orbital Specifies subshell l subshell 0 s p d f

Quantum Numbers: Magnetic quantum number symbol name Value restrictions Characteristic described Special conventions ml Magnetic quantum number ml = all integers in the range: -l to l indicates orientation of orbital Specifies an orbital n/a

Quantum Numbers: Spin quantum number ms = + ½, -½ symbol name Value restrictions Characteristic described Special conventions ms or s Spin quantum number ms = + ½, -½ indicates inherent magnetic field generated by electron Specifies an electron n/a