Page 99 - 71 the abc of dft_opt

B.2. HARMONIC OSCILLATOR 197 198 APPENDIX B. RESULTS FOR SIMPLE ONE-ELECTRON SYSTEMS
B.2 Harmonic oscillator B.3 H atom

For the Hamiltonian
1 d 2 1
ˆ 2 2
H = − 2 + ω x (B.12)
2 dx 2
the energies are
< 1 =
& n = n + ω, n = 0, 1, 2, . . . , (B.13)
2
% 1
The lengthscale is given by x 0 = , and the wavefunctions are
ω
  1 ( ) 2 < =
1 2 1 x x
φ n (x) =  √  e − 2 x 0 H n , (B.14)
n
x 0 π2 n! x 0
where H n (y) are the Hermite polynomials. The ground state and first 2 excited states wave
functions are
  1 ( ) 2
1 2 1 x
φ 0 (x) =  √  e − 2 x 0 , (B.15)
x 0 π
  1 < = ( ) 2
2 2 x 1 x
φ 1 (x) =  √  e − 2 x 0 , (B.16)
x 0 π x 0
 1
 < = 2 ( ) 2
1 2 x 1 x
φ 2 (x) =  √  (2 − 1)e − 2 x 0 . (B.17)
2x 0 π x 0