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