slides for video a-d found at http://proteinsandwavefunctions.blogspot.dk/2012/08/very-quick-introduction-to-molecular.html

1. p =
h
λ
= k
De
Broglie:
∂2
Ψ(x,t)
∂t2
= v2 ∂2
Ψ(x,t)
∂x2
Classical
wave
equa3on:
E = hν = ω
Planck/Einstein:
∂2
∂x2
Ψ = −k2
Ψ ⇒ −
2
2m
∂2
∂x2
Ψ =
p2
2m
⎛
⎝⎜
⎞
⎠⎟ Ψ
∂
∂t
Ψ = −iωΨ ⇒ i
∂
∂t
Ψ = EΨ
Schrödinger
Ψ(x,t) = Aei(kx−ωt)
∂
∂x
eax
= aeax

2. i
∂
∂t
Ψ = EΨ =
p2
2m
+ V
⎛
⎝⎜
⎞
⎠⎟ Ψ
Time
dependent
Schrödinger
equa3on
Time
independent
Schrödinger
equa3on
(standing
wave
solu0on)
Ψ(x,t) = Ψ(x)e−iEt /
i
∂
∂t
Ψ = −
2
2m
∂2
∂x2
+ V
⎛
⎝⎜
⎞
⎠⎟ Ψ
i
∂
∂t
Ψ = ˆHΨ
i
∂
∂t
Ψ = EΨ
ˆHΨ(x) = EΨ(x)

3. −
2
2m
∇2
+ V
⎛
⎝⎜
⎞
⎠⎟ Ψn = EnΨn
V = −
1
r
H
atom
V =
0 0 < x < L
C x < 0 or x > L
Par3cle
in
a
box
V = 1
2 kx2
Harmonic
oscillator
(vibra0onal
spectroscopy)
∇2
=
1
r2
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂φ2
⎛
⎝⎜
⎞
⎠⎟
V = 0
Rigid
Rotor
(rota0onal
spectroscopy)

4. −
2
2m
∇2
−
1
r
⎛
⎝⎜
⎞
⎠⎟ Ψn = EnΨn
En =
−me4
22
4πε0( )2
n2
= −
13.6 eV
n2
n = 1,2,3,...
Ψ1 =
1
π
e−r
1s
Ψ2,0 =
1
8π
1−
r
2
⎛
⎝⎜
⎞
⎠⎟ e−r/2
2s
Ψ2,1 =
1
4 2π
xe−r/2
2p
P(x) = Ψ(x)
2
dx
Probability
Ψ(x)
2
Probability
density
(amplitude)

5. −
2
2m
∇2
+ V
⎛
⎝⎜
⎞
⎠⎟ Ψn = EnΨn
V = −
1
r
H
atom
V =
0 0 < x < L
C x < 0 or x > L
Par3cle
in
a
box
V = 1
2 kx2
Harmonic
oscillator
(vibra0onal
spectroscopy)
Ψ(x)
2
Probability
density
(amplitude)

6. −
2
2m
∇2
−
1
r
⎛
⎝⎜
⎞
⎠⎟ Ψn = EnΨn
En =
−me4
22
4πε0( )2
n2
= −
13.6 eV
n2
n = 1,2,3,...
Ψ1 =
1
π
e−r
1s
Ψ2,0 =
1
8π
1−
r
2
⎛
⎝⎜
⎞
⎠⎟ e−r/2
2s
Ψ2,1 =
1
4 2π
xe−r/2
2p
P(x) = Ψ(x)
2
dx
Probability
Ψ(x)
2
Probability
density