A. Heisenberg approach:
H(t) . Ψ(r) = E . Ψ(r)
matrix . vector = real scalar . vector
B. Schrödinger approach:
H: Ψ(r,t) = E . Ψ(r,t)
operator: functie = real scalar . functie
solutions to the Schrödinger-equation are plane waves:
Ψ(r,t) = N . exp[ i . (k ● r – ω . t) ] → Ψ(r,t) = N . exp[ i . (p ● r – E . t) / h-bar ]
according to classical mechanics:
H = KE + PE = ½ . m . v2 + U(r) = p2/2m + U(r)
but QM requires dynamical variables (here E, p and r (in 3D)) to be operators:
H-hat = i . ħ . δψ/δt
p-hat = ħ / i . δψ/δr = – i . ħ . δ../δr
( derived from: Ψ(r,t) = N . exp[ i . (p ● r – E . t) / h-bar ] )
i . ħ . δΨ/δt = i2 . ħ2/2m . δ2Ψ/δr2 + U(r) . Ψ(r,t)
this equation gives the time-evolution of a classical wave
Ψ contains all the physical info about the system
by itself Ψ has no physical meaning in QM
C. Klein-Gordon approach:
according to relativity, there is no difference between space and time
therefore, the derivatives w.r.t. time and space coordinates must be of the same order
but the last equation (= the SE!) is first-order in time and second-order in space
Klein & Gordon made their equation relativistic by using two second derivatives
Dirac made his equation relativistic by using two first derivatives
according to relativity: E2 = ( p2. c2 + m2. c4 ) (to replace H)
again, QM requires dynamical variables (like E, p and x (in 1D)) to be operators:
E-hat = i . ħ . δ../δt
p-hat = ħ / i . δ../δx
i2 . ħ2 . δ2../δt2 = (ħ2 / i2) . δ2../δx2. c2 + m2. c4
– ħ2 . δ2../δt2 = – ħ2 . c2 . δ2../δx2 + m2. c4
(and yes, we have two second derivatives)
and with E-hat and p-hat acting on a state Ψ we get:
2
– ħ . δ2Ψ/δt2 = – ħ2 . c2 . δ2 Ψ/δx2 + m2. c4 . Ψ(x,t)
– ħ2 . δ2Ψ/δt2 + ħ2 . c2 . δ2 Ψ/δx2 – m2. c4 . Ψ(x,t) = 0