Schrodinger equation, time-independent

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

Unfortunately, both Schrodinger equations (time-independent and time-dependent) are not invariant with respect to the Lorentz transformation and therefore are illegal. 

Unfortunately, both Schrodinger equations (time-independent and time-dependent) are not invariant with respect to the Lorentz transformation, and therefore, they are illegal. As a result, one cannot expect the Schrodinger equation to describe accurately objects that move with velocities comparable to the speed of light. 

Quantum mechanics describes molecules in terms of interactions between nuclei and electrons and molecular geometry in terms of minimum energy arrangements of nuclei. All quantum-mechanical methods ultimately trace back to Schrodinger s (time-independent) equation, which may be solved exactly for the hydrogen atom. For a multinuclear and multielectron system, the Schrodinger equation may be defined as ... 

This form of the Schrodinger equation is independent of time and so is applicable to steady state situations. The symbol v2 denotes the operator... 

Schrodinger s time-independent equation in one dimension is given by ... 

Equation (1-49) is Schrodinger s time-independent wave equation for a single particle of mass m moving in the three-dimensional potential field V. 

The mass variable is a strictly empirical assumption that only acquires meaning in non-Euclidean space-time on distortion of the Euclidean wave field defined by Eq. (2). The space-like Eq. (5), known as Schrodinger s time-independent equation, is not Lorentz invariant. It is satisfied by a non-local wave function which, in curved space, generates time-like matter-wave packets, characterized in terms of quantized energy and three-dimensional orbital angular momentum. The four-dimensional aspect of rotation, known as spin, is lost in the process and added on by assumption. For macroscopic systems, the wave-mechanical quantum condition ho) = E — V is replaced by Newtonian particle mechanics, in which E = mv +V. This condition, in turn, breaks down as v c. 

The electronic structure and properties of any molecule, in any of its available stationary states, may be determined in principle by solution of Schrodinger s (time-independent) equation. For a system of N electrons, moving in the potential field due to the nuclei, this takes the form... 

If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation... 

As the basis set becomes infinitely flexible, full Cl approaches the exact solution of the time-independent, non-relativistic Schrodinger equation. 

One of the great difficulties in molecular quantum mechanics is that of actually finding solutions to the Schrodinger time-independent equation. So whilst we might want to solve... 

Suppose that is the lowest energy solution to the Schrodinger time-independent equation for the problem in hand. That is to say,... 

As I mentioned above, it is conventional in many engineering applications to seek to rewrite basic equations in dimensionless form. This also applies in quantum-mechanical applications. For example, consider the time-independent electronic Schrodinger equation for a hydrogen atom... 

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

We normally take the constant of integration i7o to be zero. Solution of the time-independent Schrodinger equation can be done exactly. We don t need to concern ourselves with the details, I will just give you the results. 

Thus we wish to solve the time-independent Schrodinger equation... 

The Bom-Oppenheimer approximation shows us the way ahead for a polyelec-tronic molecule comprising n electrons and N nuclei for most chemical applications we want to solve the electronic time-independent Schrodinger equation... 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is... 

The total wavefunction will depend on the spatial coordinates ri and ra of the two electrons 1 and 2, and also the spatial coordinates Ra and Rb of the two nuclei A and B. I will therefore write the total wavefunction as totfRA. Rb fu fi)-The time-independent Schrodinger equation is... 

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are very light particles, and they cannot be described even qualitatively correctly by classical mechanics. We will in this and subsequent chapters concentrate on solving the time-independent Schrodinger equation, which in short-hand operator fonn is given as... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The description of electronic distribution and molecular structure requires quantum mechanics, for which there is no substitute. Solution of the time-independent Schrodinger equation, Hip = Eip, is a prerequisite for the description of the electronic distribution within a molecule or ion. In modern computational chemistry, there are numerous approaches that lend themselves to a reasonable description of ionic liquids. An outline of these approaches is given in Scheme 4.2-1 [1] ... 

The Schrodinger equation with its time-independent hamiltonian does not in fact constitute a dynamical theorem it is simply a description of the time-dependence of the probability field corresponding to steady states or equilibrium conditions. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

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