Standing Waves Reminder Confined waves can interfere with their reflections Easy to see in one and...
Transcript of Standing Waves Reminder Confined waves can interfere with their reflections Easy to see in one and...
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