ABSTRACT
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that
( ) ( ) t
tx dt
txd ∂
Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes
ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i
dt dt t t t
⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4)
On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have
( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞
∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5)
( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦
⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫
∗∗ ˆˆˆˆˆ =
From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if
Ψ=∂ Ψ∂ H t
i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is
given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
