- #1
jelathome
- 6
- 0
I was wondering how to derive hamilton's equation (in the form of poisson brackets) from Heisenberg's equation of motion
jelathome said:I was wondering how to derive hamilton's equation (in the form of poisson brackets) from Heisenberg's equation of motion
Hamilton's equation is a mathematical equation derived from the Heisenberg equation of motion, which describes the evolution of a quantum mechanical system over time. It relates the time derivative of an operator to the commutator of the operator with the system's Hamiltonian.
Hamilton's equation is derived by replacing the time derivative in the Heisenberg equation of motion with the commutator of the operator with the Hamiltonian. This results in a set of equations that describe the evolution of the system's observables over time.
Hamilton's equation is a fundamental equation in quantum mechanics, as it allows us to calculate the time evolution of a quantum system's observables. It is also closely related to the Schrödinger equation, which is another important equation in quantum mechanics.
No, Hamilton's equation cannot be used to solve for the wave function of a quantum system. It only describes the time evolution of the system's observables, not the system's wave function. The Schrödinger equation is used to solve for the wave function of a quantum system.
Hamilton's equation has a similar form to the equations of motion in classical mechanics, which describe the evolution of a system's position and momentum over time. However, in quantum mechanics, these quantities are represented by operators, and the commutator in Hamilton's equation takes the place of the Poisson bracket in classical mechanics.