Standing Waves Reminder Confined waves can interfere with their reflections Easy to see in one and two dimensions –Spring and slinky –Water surface –Membrane For 1D waves, nodes are points For 2D waves, nodes are lines or curves

Rectangular Potential Solutions  (x,y) = A sin(n x  x/a) sin(n y  y/b) Variables separate  = X(x) · Y(y) 0 0 b a U = 0U = ∞ Energies 2m2m 2h22h2 2 nxnx a nyny b 2 +

Square Potential Solutions  (x,y) = A sin(n x  x/a) sin(n y  y/a) 0 0 a a U = 0U = ∞ Energies 2ma 2 2h22h2 n x 2 + n y 2

Combining Solutions Wave functions giving the same E (degenerate) can combine in any linear combination to satisfy the equation A 1  1 + A 2  2 + ··· Schrodinger Equation U  – (h 2 /2M)  = E 

Square Potential Solutions interchanging n x and n y are degenerate Examples: n x = 1, n y = 2 vs. n x = 2, n y = 1 + – +–

Linear Combinations  1 = sin(  x/a) sin(2  y/a)  2 = sin(2  x/a) sin(  y/a) + – +– 1 + 21 + 2 + – 1 – 21 – 2 + – 2 – 12 – 1 + – –1 – 2–1 – 2 – +

Verify Diagonal Nodes Node at y = a – x 1 + 21 + 2 + –  1 = sin(  x/a) sin(2  y/a)  2 = sin(2  x/a) sin(  y/a) 1 – 21 – 2 + – Node at y = x

Circular membrane standing waves Circular membrane Nodes are lines Higher frequency  more nodes Source: Dan Russel’s pageDan Russel’s edge node onlydiameter nodecircular node

Types of node radial angular

3D Standing Waves Classical waves –Sound waves –Microwave ovens Nodes are surfaces

Hydrogen Atom Potential is spherically symmetrical Variables separate in spherical polar coordinates x y z   r

Quantization Conditions Must match after complete rotation in any direction –angles  and  Must go to zero as r  ∞ Requires three quantum numbers

We Expect Oscillatory in classically allowed region (near nucleus) Decays in classically forbidden region Radial and angular nodes

Electron Orbitals Higher energy  more nodes Exact shapes given by three quantum numbers n, l, m l Form  nlm (r, ,  ) = R nl (r)Y lm ( ,  )

Radial Part R  nlm (r, ,  ) = R nl (r)Y lm ( ,  ) Three factors: 1.Normalizing constant (Z/a B ) 3/2 2.Polynomial in r of degree n–1 (p. 279) 3.Decaying exponential e –r/a B n

Angular Part Y  nlm (r, ,  ) = R nl (r)Y lm ( ,  ) Three factors: 1.Normalizing constant 2.Degree l sines and cosines of  (associated Legendre functions, p.269) 3.Oscillating exponential e im 

Hydrogen Orbitals Source: Chem Connections “What’s in a Star?”

Energies E = –E R /n 2 Same as Bohr model

Quantum Number n n: 1 + Number of nodes in orbital Sets energy level Values: 1, 2, 3, … Higher n → more nodes → higher energy

Quantum Number l l: angular momentum quantum number l0123l0123 orbital type s p d f Number of angular nodes Values: 0, 1, …, n–1 Sub-shell or orbital type

Quantum number m l z-component of angular momentum L z = m l h l0123l0123 orbital type s p d f degeneracy Values: –l,…, 0, …, +l Tells which specific orbital (2l + 1 of them) in the sub-shell

Angular momentum Total angular momentum is quantized L = [l(l+1)] 1/2 h L z = m l h But the minimum magnitude is 0, not h z-component of L is quantized in increments of h

Radial Probability Density P(r) = probability density of finding electron at distance r |  | 2 dV is probability in volume dV For spherical shell, dV = 4  r 2 dr P(r) = 4  r 2 |R(r)| 2

Radial Probability Density Radius of maximum probability For 1s, r = a B For 2p, r = 4a B For 3d, r = 9a B (Consistent with Bohr orbital distances)

Quantum Number m s Spin direction of the electron Only two values: ± 1/2