Single particle

The importance of the particle levitation methods is that they allow the study of how a single particle responds to changes in environment. The infrared molecular spectroscopy of single particles is possible [253], as are photophysical studies using adsorbed or dissolved dyes. 

An illustrative example is provided by investigating the possible momenta for a single particle travelling in the v-direction, p First, one writes the equation that defines the eigenvalue condition... 

It is admittedly inconsistent to begin a section on many-particle quantiun mechanics by discussing a problem that can be treated as a single particle. Flowever, the hydrogen atom and atomic ions in which only one... 

Since indistinguishability is a necessary property of exact wavefiinctions, it is reasonable to impose the same constraint on the approximate wavefiinctions ( ) fonned from products of single-particle solutions. Flowever, if two or more of the Xj the product are different, it is necessary to fonn linear combinations if the condition P. i = vj/ is to be met. An additional consequence of indistinguishability is that the h. operators corresponding to identical particles must also be identical and therefore have precisely the same eigenfiinctions. It should be noted that there is nothing mysterious about this perfectly reasonable restriction placed on the mathematical fonn of wavefiinctions. 

It should be mentioned that the single-particle Flamiltonians in general have an infinite number of solutions, so that an uncountable number of wavefiinctions [/ can be generated from them. Very often, interest is focused on the ground state of many-particle systems. Within the independent-particle approximation, this state can be represented by simply assigning each particle to the lowest-lying energy level. If a calculation is... 

This missing synuuetry provided a great puzzle to theorists in the early part days of quantum mechanics. Taken together, ionization potentials of the first four elements in the periodic table indicate that wavefiinctions which assign two electrons to the same single-particle fiinctions such as... 

Regardless of how many single-particle wavefiinctions i are available, this number is overwhelmed by the number of n-particle wavefiinctions ( ) (Slater detenninants) that can be constructed from them. For example, if a two-electron system is treated within the Flartree-Fock approximation using 100 basis fiinctions, both of the electrons can be assigned to any of the % obtained m the calculation, resulting in 10,000 two-electron wavefimctions. For water, which has ten electrons, the number of electronic wavefiinctions with equal numbers of a and p spin electrons that can be constructed from 100 single-particle wavefimctions is roughly... 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

For non-mteracting particles in a box, the result depends on the particle statistics Fenni, Bose of Boltzmamr. The state of a quanPim system can be specified by the wavefrmction for that state, Tv(Qi> Q2 . qyy). is the vth eigensolution to the Scln-ddinger equation for an A -particle system. If the particles are noninteracting, then the wavefrmction can be expressed in temis of the single-particle wavefrinctions given... 

Sj Uj, and if the yth single-particle state has energy then the energy of the system in the state v is =... 

Wlren the single-particle states j are densely packed within any energy interval of k T, the sum over j can be replaced by an integral over energy such that... 

This is the classical Boltzmaim distribution m which (Uj)/(A, tire probability of finding a particle in the single-particle state j, is proportional to the classical Boltzmaim factor... 

This result is identical to that obtained from a canonical ensemble approach in the thennodynamic limit, where the fluctuations in N vanish and (N) = N. The single-particle expression for the canonical partition fiinction = (-" can be evaluated using ih r rV i f<,2M) or a particle in a cubical box of volume V. 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

To make further progress, consider first the PF of a single particle in a potential field E(x) moving in one dimension. The Flamiltonian operator... 

Strictly unimolecular processes—sometimes also called monomoiecuiar—involve only a single particle ... 

This mechanism as a whole is called unimolecular since the essential isomerization step equation (A3.4.26) only involves a single particle, viz. CH NC. Therefore it is often simply written as follows ... 

The biggest change associated with going from one to tliree dimensional translational motion refers to asymptotic boundary conditions. In tiiree dimensions, the initial scattering wavefiinction for a single particle... 

An important point for all these studies is the possible variability of the single molecule or single particle studies. It is not possible, a priori, to exclude bad particles from the averaging procedure. It is clear, however, that high structural resolution can only be obtained from a very homogeneous ensemble. Various classification and analysis schemes are used to extract such homogeneous data, even from sets of mixed states [69]. In general, a typical resolution of the order of 1-3 mn is obtained today. 

Note that ( ), ) p( )d E is dimensionless for all cases and yields unity for a single particle when... 

Here is the original, many-body potential energy fiinction, while Vq is a sum of single-particle spring potentials proportional to As X —> 0 the system becomes a perfect Einstein crystal, whose free energy... 

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