standing waves reminder confined waves can interfere with their reflections easy to see in one and...
TRANSCRIPT
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(nxx/a) sin(nyy/b)
• Variables separate = X(x) · Y(y)
00
b
a
U = 0 U = ∞
• Energies 2m2h2
2nx
any
b
2
+
Square Potential
• Solutions (x,y) = A sin(nxx/a) sin(nyy/a)
00
a
a
U = 0 U = ∞
• Energies 2ma22h2
nx2 + ny
2
Combining Solutions
• Wave functions giving the same E (degenerate) can combine in any linear combination to satisfy the equation
A11 + A22 + ···
• Schrodinger Equation
U – (h2/2M) = E
Square Potential
• Solutions interchanging nx and ny are
degenerate
• Examples: nx = 1, ny = 2 vs. nx = 2, ny = 1
+
–+ –
Linear Combinations
• 1 = sin(x/a) sin(2y/a)
• 2 = sin(2x/a) sin(y/a)
+–
+ –
1 + 2
+–
1 – 2
+–
2 – 1
+–
–1 – 2
–+
Verify Diagonal Nodes
Node at y = a – x 1 + 2 +–
1 = sin(x/a) sin(2y/a)
2 = sin(2x/a) sin(y/a)
1 – 2 +– Node at y = x
Circular membrane standing waves
Circular membrane• Nodes are lines
• Higher frequency more nodesSource: Dan Russel’s page
edge node only diameter node circular 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, ml
• Form nlm(r, , ) = Rnl(r)Ylm(, )
Radial Part R
nlm(r, , ) = Rnl(r)Ylm(, )
Three factors:
1. Normalizing constant (Z/aB)3/2
2. Polynomial in r of degree n–1 (p. 279)
3. Decaying exponential e–r/aBn
Angular Part Y
nlm(r, , ) = Rnl(r)Ylm(, )
Three factors:
1. Normalizing constant
2. Degree l sines and cosines of (associated Legendre functions, p.269)
3. Oscillating exponential eim
Hydrogen Orbitals
Source: Chem Connections “What’s in a Star?” http://chemistry.beloit.edu/Stars/pages/orbitals.html
Energies
• E = –ER/n2
• 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
l
0123
orbital type
spdf
• Number of angular nodes• Values: 0, 1, …, n–1• Sub-shell or orbital type
Quantum number ml
• z-component of angular momentum Lz = mlh
l
0123
orbital type
spdf
degeneracy
1357
• 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
• Lz = mlh
• 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
• ||2dV is probability in volume dV
• For spherical shell, dV = 4r2dr
• P(r) = 4r2|R(r)|2
Radial Probability Density
Radius of maximum probability
•For 1s, r = aB
•For 2p, r = 4aB
•For 3d, r = 9aB
(Consistent with Bohr orbital distances)
Quantum Number ms
• Spin direction of the electron
• Only two values: ± 1/2