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What are Hamiltonian equations?
Hamiltonian equations are a representation of Hamiltonian mechanics. Please see the link.
Is momentum hamiltonian operator is hermitian operator?
The hamiltonian operator is the observable corresponding to the total energy of the system. As with all observables it is given by a hermitian or self adjoint operator. This is true whether the hamiltonian is limited to momentum or contains potential.
Why is energy expressed as the second-order partial differential of a wave function in quantum mechanics?
You are referring to the Schrodinger Equation. This is because it comes from the classical view that the total energy is equal to the hamiltonian of a system:Kinetic Energy + Potential Energy = Total energy.Classically the kinetic energy is (1/2)mv2 = p2/(2m) ; where m is mass, v is velocity, p is momentum (p=mv).Now the momentum operator in QM is p=iħ∇ ;where ∇ is the gradient operator.This therefore yields the QM hamiltonian [-ħ2∇2/(2m) + V(x,y,z)]Ψ = EΨNow a more fun question to ask would be "Why is the Hamiltonian a function of the second-order partial differential with respect to position but the time dependent is only a function of a first-order differential with respect to time?"meaningHΨ = -iħ(dΨ/dt) or[-ħ2∇2/(2m) + V(x,y,z)]Ψ = -iħ(dΨ/dt)hint: Think Maxwell's Equations!
Why Hamilton's equations are called canonical equations?
The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
How do you solve hamilton jacobi equations of motion?
This method was governed by a variational principle applied to a certain function. The resulting variational relation was then treated by introducing some unknown multipliers in connection with constraint relations. After the elimination of these multipliers the generalized momenta were found to be certain functions of the partial derivatives of the Hamilton Jacobi function with respect to the generalized coordinates and the time. Then the partial differential equation of the classical Hamilton-Jacobi method was modified by inserting these functions for the generalized momenta in the Hamiltonian of the system.
