quantum 1

E = c p <--- E = h f , p = h / $\lambda$, f $\lambda$ = c
space of states
 classic mechanics ... 'set' space
 quantum mechinics ... 'vector' space (over C:complex number Hilbert space)
  abstract 'vector' space <-> 'pointer' (position, velocity over R:real number)
  'vector' collection of things
| a > ket 'vector'
$\alpha$ |a> = |b>   ; $\alpha$ arbitrary complex number
|a> + |b> = |c>
$\alpha$|a> + $\beta$|b> = |c'>

$\psi(x) = \psi_R(x) + i \psi_I(x)$ set of complex function values forms complex 'vector' space over C

column vectors
${a_1;a_2;a_3;a_4} + {b_1;b_2;b_3;b_4}$ a_# b_# arbitrary Complex number

"Complex number itself is complex number space"

Dual space

Lecture 1 | Modern Physics: Quantum Mechanics (Stanford)