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at time t. r is not restricted to be a physical position, but contains all measures of
interest about the system. Ĥ is called the “Hamiltonian operator” and calculates
the total energy of the system if applied to ψ (r, t).
3.1.1 Time Dependent versus Time Independent Part
The Schrödinger equation is usually solved by separation of variables using:
ψ (r, t) = ψ r (r) ψ t (t) (3.2)
Plugging into Equation 3.1:
i ∂ t
(
ψ r (r) ψ t (t) ) = Ĥ ψr (r) ψ t (t) (3.3)
If Ĥ is independent of time, dividing both sides by ψr (r) ψ t (t) gives:
i ∂ t ψ t (t)
ψ t (t)
= Ĥ ψr (r)
ψ r (r)
(3.4)
For this equality to hold for all values of t and r, both sides must equal a constant:
i ∂ t ψ t (t) = E ψ t (t) (3.5)
Ĥ ψ r (r) = E ψ r (r) (3.6)
The solution to Equation 3.5 is simple and describes the time evolution of the
states of the system:
ψ t (t) = e −iEt/ (3.7)
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