Hamiltonian function partial differential equation

In order to construct the change of coordinates back to the original coordinates, we make use of the generating function W. Indeed we simply have to evaluate Poisson brackets but without solving any partial differential equations. Therefore the computational effort is much smaller than the one corresponding to the calculation of the normal-form Hamiltonian and the generating function. 

This reduces the Schrodinger equation to = 4/. To solve the Schrodinger equation it is necessary to find values of E and functions 4/ such that, when the wavefunction is operated upon by the Hamiltonian, it returns the wavefunction multiplied by the energy. The Schrodinger equation falls into the category of equations known as partial differential eigenvalue equations in which an operator acts on a function (the eigenfunction) and returns the... 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

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