lecture 6 9/12/2024 Quantum Mechanics Notes Topic: Quantum Mechanics: Wave properties of an e- result in interference, causing only certain allowed E states (quantized E's) and locations (probabilities) the e- can exist in. Overall prob: Where likely to find e-. Standing and traveling waves 1D e- in SPACE -> Ø restrict -> traveling wave no consequences of wave prop. Standing waves: imposed limits (boundary conditions) to a wave -> interference pattern unable to move 4th harmonic 2nd harmonic Wave bounces off limit & interferes with itself (constructive or deconstructive) Fundamental * nodes, no movement Constructive only at particular wavelengths. L = nλ/2 (n=1) L= length, n=1,2,3… to find allowed λs Going from one solution of allowed λs -> next… ↑ # of nodes 2D In between certain Vs, there is no wave/ring (destructive) radial nodes angular nodes on each side of node -> destructive int. can have both ↑↑ (ex) +: out -: in e- can cross node but 0 prob. of being found there; think of as a diffuse cloud 3D angular node radial node phase change across node Restricted to Sphere ("bowl") Orbital Plane and spherical node restricted to nucleus, keeping e- in place (not charge) ↳ dumbbell shape λ = h/p e- in 1D box * λ must fit as standing wave * lowest allowed E state. ↑ have to add a quantum of E so e- moves to a new allowed state * = greatest prob of finding e- Wave Function Section wave function: a set of equations derived from kinetic and potential energies taking into account wave properties of an electron Ψ²(x,y,z) = probability of finding an e- at (x,y,z) a point Ψ²(r,θ,φ) = probability of finding an e- at (r,θ,φ) a point Y(r,θ,φ) = R(r) * Y(θ,φ) r: distance from center point (direct) θ: angle w respect to z-axis; how far off (in line with z-axis -> θ = 0) Φ: projection of the vector (ie. its shadow from z-axis) and its angle from x-axis 3D coordinate diagram included, labeled x, y, and z with r, theta and phi shown x = rsinθcosφ y = rsinθsinφ z = rcosθ Questions and Answers what is the shape in which r can be anything, θ can only be 90°, and φ can be anything? ↳ x-y plane what is the shape in which r must be 1, but θ and φ can be anything? ↳ shell of a sphere (would be a sphere if r was 1 to 0) Schrödinger Equation Notes schrödinger equation: to know the energy of an e- in an atom, you must consider kinetic and potential E (total E) as well as wave properties/solutions/are equations size ⟵ radial portion: R(r) ; where the e- is likely to be found in terms of distance from the nucleus; solely based on r shape ⟵ angular portion: Y(θ,φ); gives the shape of the orbital; based on θ and φ

CHEM 1810 - Lecture 6

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