Hamiltonian operator
The Hamiltonian operator gives us the energy of a wavefunction. Generally the Hamiltonian is
H^=2mp^2+Vwhere p^ is the momentum operator and V is the potential.
In position space the Hamiltonian is
H^=−2mℏ2∂x2∂2+V.The eigenfunctions un(x) of the Hamiltonian are the eigenstates of the system, the eigenvalues are their respective energies. A wave equation can be constructed a superposition of eigenstates and their time-dependent functions (see Schrödinger equation).
Ψ(x,t)=n=1∑∞Cnun(x)e−iEnt/ℏ.The coefficients Cn are given by the projection of the wavefunction onto the eigenstate Cn=⟨un∣ψ⟩.
Since the Hamiltonian is hermitian, it is a generator of
time evolution. This Hamiltonian generates the
time evolution that we observe physically. It’s not obvious why this
is the case, but we can think of the classical Hamiltonian (generator
of time translation classicaly) as a special case of this operator.
For some time translation operator U(t,t0), the Hamiltonian is
H(t)=iℏ(∂tU)U†. We can find U in a few special
cases.
For the simplest case, assume H(t)=H is time independent. Then it
can be shown that
U(t,t0)=U~e−iHt/ℏ=U(0,t0)e−iHt/ℏ.Or assume [H(t),H(t′)]=0. Then guess
U(t,t0)dtdG(t,t0)[G′(t,t0),G(t,t0)]=exp[−ℏi∫t0tdt′H(t′)]=:eG(t,t0)=−ℏiH(t)=ℏ21[H(t),∫t0tdt′H(t′)]=−ℏ21∫t0tdt′[H(t),H(t′)]=0.
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