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
b
U=∞
U=0
0
0
a
• Solutions y(x,y) = A sin(nxpx/a) sin(nypy/b)
• Variables separate y = X(x) · Y(y)
• Energies
p2h2
2m
nx
a
2
+
ny
b
2
Square Potential
a
U=∞
U=0
0
0
a
• Solutions y(x,y) = A sin(nxpx/a) sin(nypy/a)
2h2
p
2 + n 2
• Energies
n
y
2ma2 x
Combining Solutions
• Schrodinger Equation
Uy – (h2/2M)y = Ey
• Wave functions giving the same E
(degenerate) can combine in any linear
combination to satisfy the equation
A1y1 + A2y2 + ···
Square Potential
• Solutions interchanging nx and ny are
degenerate
• Examples: nx = 1, ny = 2 vs. nx = 2, ny = 1
–
+
+
–
Linear Combinations
• y1 = sin(px/a) sin(2py/a)
–
+
• y2 = sin(2px/a) sin(py/a)
+ –
y1 + y2
–
+
y1 – y2
–
y2 – y1
+
+
–
–y1 – y2
–
+
Verify Diagonal Nodes
y1 = sin(px/a) sin(2py/a)
y2 = sin(2px/a) sin(py/a)
y1 – y2
y1 + y2
–
+
–
+
Node at y = x
Node at y = a – x
Circular membrane
• Nodes are lines
Circular membrane standing waves
edge node only
diameter node
circular node
Source: Dan Russel’s page
• Higher frequency  more nodes
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
z
q
r
y
f
x
Quantization Conditions
• Must match after complete rotation in any
direction
– angles q and f
• 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 ynlm(r, q, f) = Rnl(r)Ylm(q, f)
Radial Part R
ynlm(r, q, f) = Rnl(r)Ylm(q, f)
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
ynlm(r, q, f) = Rnl(r)Ylm(q, f)
Three factors:
1. Normalizing constant
2. Degree l sines and cosines of q
(associated Legendre functions, p.269)
3. Oscillating exponential eimf
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
Number of angular nodes
Values: 0, 1, …, n–1
Sub-shell or orbital type
l
orbital type
0
1
2
3
s
p
d
f
Quantum number ml
• z-component of angular momentum Lz = mlh
• Values: –l,…, 0, …, +l
• Tells which specific orbital (2l + 1 of them)
in the sub-shell
l
0
1
2
3
orbital type degeneracy
s
p
d
f
1
3
5
7
Angular momentum
•
•
•
•
L = [l(l+1)]1/2 h
Total angular momentum is quantized
Lz = mlh
z-component of L is quantized in
increments of h
• But the minimum magnitude is 0, not h
Radial Probability Density
• P(r) = probability density of finding electron
at distance r
• |y|2dV is probability in volume dV
• For spherical shell, dV = 4pr2dr
• P(r) = 4pr2|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