![]() ![]() In Hamiltonian mechanics, the way a system changes with time can be interpreted geometrically as a flow in phase space (which I’ll explain later) and this geometric notion has a lot of really deep consequences to the nature of physics in general. The real importance and beauty of Hamiltonian mechanics comes from its geometric structure, which simply does not exist in the Lagrangian formulation. However, the importance of Hamiltonian mechanics is not so obvious as it can essentially be used to solve mechanics problems just as well as Lagrangian mechanics. ![]() ![]() In many cases, the benefits of the Lagrangian formulation are quite clear (which you can read more about in this article). I also have this article that compares Lagrangian and Hamiltonian mechanics and covers the main differences, advantages and applications of both formulations.īefore we get started on the actual details of the Hamiltonian formulation, I think it’s important to make explicitly clear why exactly you would want to learn and even consider Hamiltonian mechanics. Hamiltonian mechanics does have a little bit to do with Lagrangian mechanics, so I’d recommend reading this introductory article first if you don’t know anything about the Lagrangian formulation. The context of this article is more about what Hamiltonian mechanics means in classical mechanics, although I will also give some insights about Hamiltonian mechanics and its significance to other areas in physics. Hamiltonian mechanics also has its own advantages and applications in more advanced physics, such as in quantum mechanics and quantum field theory. ![]() Hamiltonian mechanics is based on the Lagrangian formulation and is also equivalent to Newtonian mechanics.Įven though Newtonian, Lagrangian and Hamiltonian mechanics are all equivalent in principle, what really makes Hamiltonian mechanics unique is its geometric interpretation and the concept of phase space. One of these formulations is called Hamiltonian mechanics.Īs a general introduction, Hamiltonian mechanics is a formulation of classical mechanics in which the motion of a system is described through total energy by Hamilton’s equations of motion. By an appropriate choice of the operators we obtain Heisenberg’s Uncertainty Principle: the product of the error in the measurement of the position multiplied and the error in the measurement of the momentum, must be greater than or equal to a strictly non-zero value.In classical mechanics, there are quite many different formulations, which all have their unique purposes and advantages. Using the positivity of a quadratic function we conclude that the discriminant must be negative, which gives a result about the expectation of two operators and their commutator. We then compute the commutator of the position and momentum operators and see that it is non-zero.įinally, the key proposition which is used to derive Heisenberg’s Uncertainty Principle is stated and proved. The definitions of the expectation and dispersion (standard deviation) of an operator are given in terms of a complex inner product, as these will be required later. This is then expressed in terms of the momentum and position operators which together make up the Hamiltonian. The equation for the remaining spatial component is referred to as the Stationary State Schrödinger Equation. By looking for a separable solution we are able to solve for the time dependent component as a complex exponential. The video begins with Schrödinger’s Equation for the quantum wave function. ![]()
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