Quantum states, number

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

Let us now ask what fraction of the total number of molecules is in the quantum state having energy Ej. We define this fraction by... 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

Quanten-theorie, /. quantum theory, -zahl, /. quantum number, -zustand, m. quantum state. 

This represents a violation of the number of quantum states, since a two-fold increase in terms of the atomic core seems to occur on the introduction of an additional electron. Bohr s response was to maintain adherence to the... 

Isomers—Nuclides having the same number of neutrons and protons but capable of existing, for a measurable time, in different quantum states with different energies and radioactive properties. Commonly the isomer of higher energy decays to one with lower energy by the process of isomeric transition. 

The properties of the Slater determinant demonstrate immediately the Pauli exclusion principle, as usually taught. It reads No two electrons can have all four quantum numbers equal, that is to say that they cannot occupy the same quantum state. It is the direct result of the more general argument that the wavefunction must be antisymmetric under the permutation of any pair of (identical and indistinguishable) electrons. 

These restrictions, imposed above on electrons, apply equally to all pariiqles that are represented by antisymmetric wavefunctions, the so-called Fermions. The condition that no more than one particle can occupy a given quantum state leads immediately to the expression for the number of possible combinations. If C nhgi) is the number of combinations that can be made with g, particles taken tii at a time,... 

Under the conditions determined by the symmetric permutation of identical particles, any number of particles can be placed in each quantum state, lb distribute , of a given energy in gi quantum states, consider the fo Ibarriers separating the boxes are necessary. There are then nt particles and gi - 1 barriers. There fore, there are nf +g,- - 1 Bosons that can be permuted and (/i + gi -1) possible permutations. However, as the particles are indistinguishable, the elements of each ensemble of , can be permuted, leading to / possibilities. Clearly, the indistinguishability of the barriers corresponds to (gi - 1) permutations, The number of combinations is then given by... 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

In the lowest energy state (called the ground state) of Al, the electrons occupy various quantum states written Is2 2s2 2p6 3s2 3p Here the precursor numbers (called the principal quantum numbers) give the total energy of the state the letters describe the nature of the state and the superscripts give the number of electrons in each state (the sum of the superscripts = 13). The energy levels of the electrons in Al are ... 

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