Schrodinger equation time-evolution operator

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

A formal possibility to obtain the reduced time-evolution operator involved in Eq. (114) is to solve the following Schrodinger equation within the... 

This time-evolution operator is the solution of the Schrodinger equation ... 

The key to performing a wavepacket calculation is the propagation of the wavepacket forward in time so as to solve the time-dependent Schrodinger equation. In 1983, Kosloff proposed the Chebyshev expansion technique [5, 6, 7, 8] for evaluating the action of the time evolution operator on a wavepacket. This led to a huge advance in time-dependent wavepacket dynamics [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Several studies have compared different propagation methods [30, 31, 32] and these show that the Chebyshev expansion method is the most accurate. 

Remembering that the time evolution operator satisfies the time-dependent Schrodinger equation (55) allows (66) to be written as... 

Schrodinger equation (p. 70) bound state (p. 73) wave function matching (p. 73) mathematical solution (p. 76) physical solutions (p. 76) wave function evolution (p. 76) time-evolution operator (p. 77) algebraic approximation (p. 80) two-state model (p. 81) first-order perturbation theory (p. 82) time-independent perturbation (p. 83) Fermi golden rule (p. 84) periodic perturbation (p. 84)... 

The time evolution operator in the Schrodinger equation (time-independent Hamiltonian H) is equal to ... 

With the unitary time-evolution operator of the unperturbed system, Ud t) = we obtain the Schrodinger equation in the interaction... 

The matrix representation in equation (28) of the time dependent Schrodinger equation has a general solution in equation (31) in terms of the matrix representation U of the time evolution operator U ... 

Schrodinger equation. The time evolution of a state vector is (in the non-relativistic case) governed by the Schrodinger equation which gives rise to a rotation of the state vector within the Hibert space H. The time evolution of a state vector can be described by a linear, unitary time evolution operator U t). That is, a normalized state l (O)) at a time t = 0 evolves into a normalized state l (T)) = U(T)l (0)) after a time T. Quantum mechanics is linear, therefore a superposition evolves according to... 

The particular form of the Schrodinger equation and the time evolution operator depends on the given physical context. 

The time evolution in the Schrodinger representation may be described in terms of a development operator T(t, t0) by the equation... 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

Another consequence of the Schrodinger equation is that the time evolution of the expectation value of a physical observable, represented by an operator A (with no explicit time dependence), is given by... 

Time reversal in quantum systems also leads to different results compared to classical systems. The basic relationship between the Hamiltonian (or energy) operator and time evolution in quantum mechanics is defined by Schrodinger s equation as... 

From the above-presented theory, we can conclude that an ensemble from a pure state always proceeds to a mixed state a consequence of irreversibility. Thus, it is impossible to describe the evolution of the pure state of a damped oscillator in the Schrodinger picture. Consequently, it is impossible to construct a linear Schrodinger equation in which the position and the momentum operator are time independent. 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators ... 

To end this section we note that the entire time evolution referred to in the above discussion arises from the Schrodinger equation. In general the operator may have an explicit dependence on time, in which case the transformation to the Heisenberg representation may again be carried out, however, the resulting Heisenberg equation... 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

The eigenfunctions are often chosen to depend on time, while the operators are chosen time independent. This choice is referred to as the Schrodinger representation. The change of time of a quantity is called evolution. The evolution of the wave function towards equilibrium is determined by the Schrodinger equation. 

If the Hamilton operator depends on time in a harmonic fashion, the time dependence can be eliminated by transformation into a rotating reference frame in analogy to the transformation of the Bloch equations. A representation in the rotating frame is also called interaction representation in quantum mechanics. If the time dependence is more general, the Schrodinger equation is solved for small enough time increments, during which H is approximately constant. For each of the n time increments At a solution of the form (2.2.46) applies. The complete evolution operator is the time ordered product of the incremental evolution operators. This operator is written in short hand as... 

In the previous section, we showed that the RRGM can be used to calculate individual state-to-state transition probabilities, Ptft). Another way of studying the IVR processes involves the explicit calculation of the time evolution of the initial state i) (75,76). We will begin by recalling that the solution to the Schrodinger time-dependent equation can be written in terms of the evolution operator (propagator)... 

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