Schrödinger equation, Hamiltonian II

Let’s derive the Schrödinger equation from the postulates of an Isolated System.

Let t0t_0 be the initial time and tt be the time after some time has passed.

Let

∣ψ(t)⟩=U(t,t0)∣ψ(t0)⟩|\psi(t)\rangle = U(t, t_0) |\psi(t_0)\rangle

So

∂∂t∣ψ(t)⟩=∂U(t,t0)∂t∣ψ(t0)⟩\frac{\partial}{\partial t} |\psi(t)\rangle = \frac{\partial U(t, t_0)}{\partial t} |\psi(t_0)\rangle

Due to its inverse property where U(t0,t)U(t,t0)=IU(t_0,t)U(t,t_0)=I,

=∂U(t,t0)∂t U(t0,t)∣ψ(t)⟩= \frac{\partial U(t, t_0)}{\partial t} \, U(t_0, t) |\psi(t)\rangle =∂U(t,t0)∂t U†(t,t0)⏟Λ(t,t0)∣ψ(t)⟩= \underbrace{\frac{\partial U(t, t_0)}{\partial t} \, U^\dagger(t, t_0)}_{\Lambda(t, t_0)} |\psi(t)\rangle

Properties

  1. Λ(t,t0)\Lambda(t, t_0) is anti-Hermitian.

Note:

Λ†(t,t0)=U(t,t0) ∂U†(t,t0)∂t \Lambda^\dagger(t, t_0) = U(t, t_0) \, \frac{\partial U^\dagger(t, t_0)}{\partial t}

Differentiate U(t,t0)U†(t,t0)=IU(t, t_0) U^\dagger(t, t_0) = I:

∂∂t ⁣[U(t,t0)U†(t,t0)]=∂U(t,t0)∂tU†(t,t0)+U(t,t0)∂U†(t,t0)∂t=0 \frac{\partial}{\partial t}\!\left[U(t, t_0) U^\dagger(t, t_0)\right] = \frac{\partial U(t, t_0)}{\partial t} U^\dagger(t, t_0) + U(t, t_0) \frac{\partial U^\dagger(t, t_0)}{\partial t} = 0 ⇒Λ(t,t0)+Λ†(t,t0)=0 \Rightarrow \Lambda(t, t_0) + \Lambda^\dagger(t, t_0) = 0 ⇒Λ=−Λ†is anti-Hermitian. \Rightarrow \Lambda = -\Lambda^\dagger \text{is anti-Hermitian.}
  1. Λ(t,t0)\Lambda(t, t_0) is independent of t0t_0
Λ(t,t0)=∂U(t,t0)∂tU†(t,t0) \Lambda (t,t_0)=\frac{\partial U(t, t_0)}{\partial t} U^\dagger(t, t_0) =∂U(t,t0)∂tU(t0,t1)U†(t0,t1)U†(t,t0) =\frac{\partial U(t, t_0)}{\partial t} U(t_0,t_1)U^\dagger(t_0,t_1) U^\dagger(t, t_0) =∂∂t[U(t,t0)U(t0,t1)][U(t,t0)U(t0,t1)]† =\frac{\partial}{\partial t}[U(t,t_0)U(t_0,t_1)][U(t,t_0)U(t_0, t_1)]^\dagger =∂U(t,t1)∂tU†(t,t1)=Λ(t,t1) =\frac{\partial U(t,t_1)}{\partial t}U^\dagger(t,t_1)=\Lambda (t,t_1)

For any t0,t1t_0,t_1

Let

Λ(t)≜Λ(t,t0)\Lambda (t)\triangleq \Lambda (t,t_0) ⇒ddt∣ψ(t)⟩=∂∂tU(t,t0)U†(t,t0)∣ψ(t)⟩\Rightarrow\frac{d}{dt}\left\lvert \psi(t) \right\rangle = \frac{\partial}{\partial t}U(t, t_0)U^\dagger(t,t_0) |\psi(t)\rangle =Λ(t)∣ψ(t)⟩=\Lambda (t)\left\lvert \psi(t) \right\rangle

Then, let

H(t)≜iℏΛ(t)=iℏ∂U(t,t0)∂tU†(t,t0)\boxed{H(t)\triangleq i\hbar \Lambda (t) = i\hbar \frac{\partial U(t, t_0)}{\partial t} U^\dagger(t, t_0)}

This is known as the Hamiltonian, which has the units of energy. In natural units where ℏ=1\hbar=1, it is simply inverse time s−1s^{-1}.

We arrive at the Schrödinger equation:

iℏddt∣ψ(t)⟩=H(t)∣ψ(t)⟩\boxed{i\hbar \frac{d}{dt}\left\lvert \psi(t) \right\rangle=H(t)\left\lvert \psi(t) \right\rangle}

Remarks

  1. Derived Schrödinger equation from only the assumption of unitary time evolution. The Hamiltonian operator is derived from U(t,t0)U(t,t_0)

  2. Converse is true where we can construct U(t,t0)U(t,t_0) from knowledge of H(t)H(t)

  3. in many physical systems, it easier to find H(t)H(t) than finding U(t,t0)U(t,t_0)