Kinetic energy in quantum mechanics

Cohen, L. (1979) Local kinetic energy in quantum mechanics. J. Chem. Phys., 70, 788-789. 

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The other vital consideration affecting the possibility of solidification and influencing also almost all of the other properties of liquid helium is its very high zero-point energy. In quantum mechanics, the lowest permitted kinetic energy for a particle is not necessarily zero. If a particle of mass m is confined to a volume F, then its minimum kinetic energy, or zero-point energy, is... 

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

Schrodinger postulated that the form of the hamiltonian in quantum mechanics is obtained by replacing the kinetic energy in Equation (1.20), giving... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

In quantum mechanics the vector p is replaced by the operator Thtts, the Operator for the kinetic energy becomes... 

As shown above, the potential energy can be expressed as a sum of harmonic potentials. We now consider the expression for the kinetic energy in classical as well as quantum mechanical form. In classical mechanics,... 

In quantum mechanics the kinetic energy is represented by the operator... 

The chief content of the isomorphism displayed in Table 1 is embodied in the phrase "electride ion . By introducing at the outset in the electronic interpretation of chemistry the wave-like character of electrons and the Exclusion Principle through the concept of van der Waals-like electron-domains or electride ions , whose sizes indicate the magnitudes of the electrons kinetic energies, whose impenetrability8) simulates, at least approximately, the operation of the Exclusion Principle, and whose charges yield within the framework of the model easily foreseeable effects, one transforms the complex treatment of the covalent bond in quantum mechanics into a simpler, if less precise, exercise in classical electrostatics. 

To understand how STM works, it is vital to know what is tunneling current, and how it is related to all the experimental observations. Tunneling current is originated from the wavelike properties of particles (electrons, in this case) in quantum mechanics. When an electron is incident upon a vacuum barrier with potential energy larger than the kinetic energy of the electron, there is still a non-zero probability that it may traverse the forbidden region and reappear on the other side of the barrier. It is shown by the leak out electron wave function in Fig. 2. 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

A model based on early ideas of Lewis was proposed by Kimball, enclosing each pair of electrons in a sphere. The atom or molecule consist of spheres touching each other, and containing the nucleus in the smallest two-electron sphere of a given atom. Like in quantum mechanics, the kinetic energy of each electron confined in a sphere with radius R is proportional to R-2 and the proportionality constant is chosen by... 

The transformation to a quantum mechanical Hamilton operator (Sect. 2.3) has been discussed by Watson58. The operator resulting from the kinetic energy in Eq. (3.50), omitting the translational energy, is given by... 

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