Particles, free

The potential energy for a free particle is a constant (taken arbitrarily as zero) V = 0 and, therefore, energy E represents the kinetic energy only. The Schrodinger equation takes the form 

The special solutions to this equation are expfiKJi ) and exp(—ikjc). Their linear combination with arbitrary complex coefficients A and B represents the general solution  

For eigenvalue hK 0 the eigenfunction exp(iKJc) describes a particle moving towards +00. Similarly, exp(—ikjc) corresponds to a particle of the same energy, but moving in the opposite direction. The function F = 4 exp(iKJc) + B exp(—ikjc) is a superposition of these two states. A measurement of the momentum can give only two values kH with probability proportional to jA l or —kH with probability proportional to B.  

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Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

This is the density of microstates for one free particle in volume V= L. ... 

In an ideal molecular gas, each molecule typically has translational, rotational and vibrational degrees of freedom. The example of one free particle in a box is appropriate for the translational motion. The next example of oscillators can be used for the vibrational motion of molecules. 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

The solution to this is a Gaussian function, which spreads out in time. Hence the solution to the Bloch equation for a free particle is also a Gaussian ... 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

To understand how to describe a particle moving in a constant potential, consider the case of a free particle for which V(x) = 0. In this case the time-dependent Scln-ddinger equation is... 

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics... 

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

Check. The number of free particles with all momenta p in equilibrium with a gas bath of volume v at temperature T is the translational partition fiinction Z>. Since the fraction of particles with energy E is exp (-... 

Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

I he wavelength of this motion is h/p and the pcirameter k is equal to 2irplh. Thus k has units of 1 /length (i.e. reciprocal length). The energy for a free particle varies in a quadratic fashion u ith A and in principle any value of the energy is possible. 

We can recover the free-particle result (i.e. zero potential) from Equation (3.98) by setting all i)f Ihe Fourier coefficients Uq to zero, in which case the equation reduces to ... 

The solution of this equation requires that = i k + G /2m with the wavefunctions heiiig of the form oc exp[i(k + G) r]. Although cast in a slightly different form, this i.-, et[uivalent to our earlier expression for the wavefunction of a free particle. Equation (3.92). 

Fig. 2. (a) Energy, E, versus wave vector, k, for free particle-like conduction band and valence band electrons (b) the corresponding density of available electron states, DOS, where Ep is Fermi energy (c) the Fermi-Dirac distribution, ie, the probabiUty P(E) that a state is occupied, where Kis the Boltzmann constant and Tis absolute temperature ia Kelvin. The tails of this distribution are exponential. The product of P(E) and DOS yields the energy distribution... 

Where b is Planck s constant and m and are the effective masses of the electron and hole which may be larger or smaller than the rest mass of the electron. The effective mass reflects the strength of the interaction between the electron or hole and the periodic lattice and potentials within the crystal stmcture. In an ideal covalent semiconductor, electrons in the conduction band and holes in the valence band may be considered as quasi-free particles. The carriers have high drift mobilities in the range of 10 to 10 cm /(V-s) at room temperature. As shown in Table 4, this is the case for both metallic oxides and covalent semiconductors at room temperature. 

In conclusion of this section, we write out the expressions for the density matrix of a free particle and a harmonic oscillator. In the former case p(x, x P) is a Gaussian with the half-width equal to the thermal de Broglie wavelength... 

Combining the above expressions, we determine the number of times the particle velocity in a centrifuge is greater than that in free particle settling ... 

Imagine a free particle with charge Q and mass m. The Hamiltonian is... 

The reason for this can be seen as follows. In a perfect crystal with the ions held fixed, a positive hole would move about like a free particle with a mass m depending on the nature of the crystal. In an applied electric field, the hole would be uniformly accelerated, and a mobility could not be defined. The existence of a mobility in a real crystal derives from the fact that the uniform acceleration is continually disturbed by deviations from a perfect lattice structure. Among such deviations, the thermal motions of the ions, and in particular, the longitudinal polarisation vibrations, are most important in obstructing the uniform acceleration of the hole. Since the amplitude of the lattice vibrations increases with temperature, we see how the mobility of a... 

It is left to reader to verify that, under Lee .s discrete mechanics, both free particles and particles subjected to a constant force, behave in essentially the sa e way as they do under continuous equations of motion. Moreover, the time intervals At = t-i i — ti are all equal. While the spatial behavior for non-constant forces (ex particles in a harmonic oscillator V potential) also remains essentially... 

Example, Free Particle. In this case the hamiltonian is H = Pa/2m and the state vector t> satisfies the equation... 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

To conclude this section we make a few remarks concerning the physical interpretation of the covariant amplitude tfr(x). For a free particle one would surmise that adoption of the manifold of positive... 

In concluding this section we briefly establish the connection between the Dirac theory for a single isolated free particle described in the previous section and the present formalism. If T> is the state vector describing a one-particle state, iV T> = 1 T> consider the amplitude... 

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