Eigenvalue equation level systems

This equation is an eigenvalue equation for the energy or Hamiltonian operator its eigenvalues provide the energy levels of the system... 

In general the relaxation to equilibrium of E(t) is nonexponential, since the rate matrix in the master equation has an infinite number of (in principle) nondegenerate eigenvalues if there are an infinite number of states n). There are, however, two instances where the relaxation is approximately exponential. In the first instance one assumes that the initial nonequilibrium state has appreciable population only in the first two oscillator eigenstates, and further that k,. 0 k,, m and k0. t k0 m for m > 2. If one neglects terms involving these small rate constants, the master equation reduces to a pair of coupled rate equations for a two-level system ... 

The eigenvalues a represent the possible measured values of the variable A. The Schrodinger equation (2.38) is the best-known instance of an eigenvalue equation, with its eigenvalues corresponding to the allowed energy levels of the quantum system. 

It can be seen that the residuals on position level are growing with increasing time leading to physically meaningless solutions because the invariant is not fulfilled. This instability is called drift-off effect. For linear differential equations with constant coefficients this effect can be explained by Theorem 2.7.3, which states that index reduction introduces multiple zero eigenvalues into the system. Multiple zero eigenvalues cause a polynomial instability, i.e. errors due to perturbations of the initial conditions grow like a polynomial. Errors due to discretization and due truncation of the corrector iteration can be viewed as perturbations of the system. This is reflected by the fact, that the choice of smaller step sizes diminishes also the violation of the constraint. 

When this is attempted, solutions of the differential equation can only be obtained for certain values of the energy E. Such energy values are known as the eigenvalues of the equation. They are the permitted quantized energy levels for the system, and we see that the quantum restriction has appeared naturally, and not arbitrarily as in... 

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