A Free Particle

For a free particle, the potential energy is constant and we can choose it to be zero. The time-independent Schrodinger equation in one dimension then becomes simply 

Let us consider first the simplest case free particle motion. Free motion means the case with U = 0, i.e., the motion of a particle on which no irrelevant forces are exerted. Such a particle is moving uniformly and in a straight line. Let the jc-axis be directed along the particle motion. The Schrodinger equation in this case has the form 

As here depends on only one variable, the partial derivatives are replaced onto the full derivative. The energy in this case is kinetic energy, becanse potential energy for free particles is zero. Therefore, 

Because of the fact that we are dealing with the uniform Schrbdinger equation and the derivation from the wavefunction is taken on the coordinate x, this solution can be multiplied by any coefficient not depending on the coordinate. Let this coefficient be e . Then the solution accepts the form (x,t) = A exp(-iLt ) exp(icu0 = A exp(ico t-ikx). The real part of it is therefore the wave running in x direction 

The calculation of the probability density to find a particle on any point of the JC-axis gives 

Thereby, if a particle momentum is strictly defined (as in our case), its position in space is indefinite there is no priority to find it on any point of the x-axis, the particle is as if it is smashed along the x-axis. It is worth noting that this result completely corresponds to the uncertainty principle (refer to Section 7.2). 

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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,... 

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 ... 

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... 

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. 

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). 

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... 

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... 

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... 

Equations (3.75 and 3.76) describe the motion of a free particle in a rotating frame of reference, when the angular velocity is relatively small and the z-axis is directed along the plumb line. Certainly, a presence of terms with a> is related to the rotation of the earth. Besides, the gravitational field also contains a term with this frequency. It may be proper to emphasize again that in deriving these formulas we assumed... 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

The classical, non-relativistic energy E for a free particle, i.e., a particle in the absence of an external force, is expressed as the sum of the kinetic and potential energies and is given by... 

Since a free particle is represented by the wave packet I (jc, i), we may regard the uncertainty Ajc in the position of the wave packet as the uncertainty in the position of the particle. Likewise, the uncertainty Ak in the wave number is related to the uncertainty Aj3 in the momentum of the particle by Ak = hsp/h. The uncertainty relation (1.23) for the particle is, then... 

Another Heisenberg uncertainty relation exists for the energy E ofa particle and the time t at which the particle has that value for the energy. The uncertainty Am in the angular frequency of the wave packet is related to the uncertainty A in the energy of the particle by Am = h.E/h, so that the relation (1.25) when applied to a free particle becomes... 

We now wish to derive the energy-time uncertainty principle, which is discussed in Section 1.5 and expressed in equation (1.45). We show in Section 1.5 that for a wave packet associated with a free particle moving in the x-direction the product A A/ is equal to the product AxApx if AE and At are defined appropriately. However, this derivation does not apply to a particle in a potential field. 

Equation (6.11) is the Schrodinger equation for the translational motion of a free particle of mass M, while equation (6.12) is the Schrodinger equation for a hypothetical particle of mass fi. moving in a potential field F(r). Since the energy Er of the translational motion is a positive constant (Er > 0), the solutions of equation (6.11) are not relevant to the structure of the two-particle system and we do not consider this equation any further. 

We should also mention that basis sets which do not actually comply with the LCAO scheme are employed under certain circumstances in density functional calculations, i. e., plane waves. These are the solutions of the Schrodinger equation of a free particle and are simple exponential functions of the general form... 

In Eq. (67) the classical energy of a free particle, a = mu2, has been substituted, with u its velocity and mv its momentum. Equation (67) is of course the well-known relation of deBroglie. 

The function 4> k) is known as the wave function in momentum space. The Fourier integral represents the superposition of many waves of different wave vectors. This construct defines a wave packet, once considered as the theoretically most acceptable description of a wave-mechanical particle5. Schrodinger s dynamical equation (4) for a free particle... 

In wave mechanics the wave function 4>a(x) of a system in a state is specified by the label a. For example, in the case of a free particle a could be the momentum p, and ipa(x) would be the de Broglie wave function,... 

Closely related to the problem of a particle on a line is that of a particle confined to a hollow sphere. Such a particle is described by the same Hamiltonian as a free particle (V = 0), i.e. 

Fig. 7. The two types of contributions to pfPfp. t). (a) Free particle acceleration, (b) Correlated particle acceleration.

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