p = h

λ= k

De  Broglie:  

∂2Ψ(x,t)∂t 2

= v2 ∂2Ψ(x,t)∂x2

Classical  wave  equa3on:  

E = hν = ω

Planck/Einstein:  

∂2

∂x2Ψ = −k2Ψ ⇒ −

2

2m∂2

∂x2Ψ =

p2

2m⎛⎝⎜

⎞⎠⎟Ψ

∂∂t

Ψ = −iωΨ ⇒ i ∂∂t

Ψ = EΨ

Schrödinger  

Ψ(x,t) = Aei(kx−ω t )

∂∂xeax = aeax

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 = HΨ

i ∂∂t

Ψ = EΨ

HΨ(x) = EΨ(x)

−2

2m∇2 +V

⎛⎝⎜

⎞⎠⎟Ψn = EnΨn

V = −1r

H  atom  

V =0 0 < x < LC x < 0 or x > L

Par3cle  in  a  box  

V = 12 kx

2

Harmonic  oscillator  (vibra0onal  spectroscopy)  

∇2 =1r2

1sinθ

∂∂θsinθ ∂

∂θ+

1sin2θ

∂2

∂φ 2⎛⎝⎜

⎞⎠⎟

V = 0

Rigid  Rotor  (rota0onal  spectroscopy)  

−2

2m∇2 −

1r

⎛⎝⎜

⎞⎠⎟Ψn = EnΨn

En =−me4

22 4πε0( )2 n2= −

13.6 eVn2 n = 1,2,3,...

Ψ1 =1πe−r 1s

Ψ2,0 =18π

1− r2

⎛⎝⎜

⎞⎠⎟e−r /2 2s

Ψ2,1 =1

4 2πxe−r /2 2p

P(x) = Ψ(x) 2 dxProbability    

Ψ(x) 2

Probability  density  (amplitude)    

−2

2m∇2 +V

⎛⎝⎜

⎞⎠⎟Ψn = EnΨn

V = −1r

H  atom  

V =0 0 < x < LC x < 0 or x > L

Par3cle  in  a  box  

V = 12 kx

2

Harmonic  oscillator  (vibra0onal  spectroscopy)  

Ψ(x) 2

Probability  density  (amplitude)    

−2

2m∇2 −

1r

⎛⎝⎜

⎞⎠⎟Ψn = EnΨn

En =−me4

22 4πε0( )2 n2= −

13.6 eVn2 n = 1,2,3,...

Ψ1 =1πe−r 1s

Ψ2,0 =18π

1− r2

⎛⎝⎜

⎞⎠⎟e−r /2 2s

Ψ2,1 =1

4 2πxe−r /2 2p

P(x) = Ψ(x) 2 dxProbability    

Ψ(x) 2Probability  density