Eigenfunction free particle

Note that these states are eigenfunctions corresponding to a free particle, and strictly speaking not physically acceptable, since they are not square integrable. Again, they... 

Equation (3.51) expresses the general solution for A(x, t) as a sum over independent normal modes. qi(,t), obtained from Eq. (3.52), determines the time evolution of a mode, while Z>/(x), the solution to Eq. (3.53), detennines its spatial structure in much the same way as the time-independent Schrodinger equation determine the intrinsic eigenfunctions of a given system. In fact, Eq. (3.53) has the same structure as the time-independent Schrodinger equation for a free particle, Eq. (2.80). It admits similar solutions that depend on the imposed boundary conditions. If we use periodic boundary conditions with period L we find, in analogy to (2.82),... 

In the absence of the periodic potential our problem is reduced again to that of a free particle. Eigenfunctions of H = T that satisfy the periodic boundary conditions are of the form... 

Even though in similarity to free particle wavefunctions the Bloch wavefunctions are characterized by the wavevector k, and even though Eq. (4.80) is reminiscent of free particle behavior, the functions V nkfr) are not eigenfunctions of the momentum operator. Indeed for the Bloch function (Eqs (4.78) and (4.79)) we have... 

The eigenfunctions of the free particle Hamiltonian can be written as free waves, lAkCr) = fi /2exp(zk r). Bloch states have the fonn (r) =... 

Because the Hamiltonian of any central potential quantum system, H p/ commutes with the operators and H, they also have common eigenfunctions, including the situation of confinement by elliptical cones. Although Ref. [8] focused on the hydrogen atom. Ref. [1] included the examples of the free particle confined by elliptical cones with spherical caps, and the harmonic oscillator confined by elliptical cones. They all share the angular momentum eigenfunctions of Eqs. (98-101), which were evaluated in Ref. [8] and could be borrowed immediately. Their radial functions and their... 

Readers are invited to do their own reading of Ref. [1], including the effects of the confinement by elliptical cones on the energy spectra and eigenfunctions of the familiar free particle and harmonic oscillator. 

This section is the counterpart of Section 4.1 aimed to illustrate the generation of the complete radial and spheroconal angular momentum eigenfunction for the free particle in three dimensions using an alternative representation of the same operator. 

Comparing the free-particle wave function (2.30) with the eigenfunctions (3.36) of Px, we note the following physical interpretation The first term in (2.30) corresponds to positive momentum and represents motion in the +jc direction, while the second term in (2.30) corresponds to negative momentum and represents motion in the -X direction. 

The approach is rather different from that adopted in most books on quantum chemistry in that the Schrbdinger wave equation is introduced at a fairly late stage, after students have become familiar with the application of de Broglie-type wavefunctions to free particles and particles in a box. Likewise, the Hamiltonian operator and the concept of eigenfunctions and eigenvalues are not introduced until the last two chapters of the book, where approximate solutions to the wave equation for many-electron atoms and molecules are discussed. In this way, students receive a gradual introduction to the basic concepts of quantum mechanics. 

Although the box potential has so far been considered, the potential-free (F = 0) case may be close to the readers image of translational motion. Such potential-free particles are called free particles. In the one-dimensional case, the eigenfunctions of free particles are given by... 

As before, we need a functional description of the wavefunction at f = 0,4 (x, 0). We will then expand that in terms of eigenfunctions of the free-particle hamiltonian. As we have seen in Section 2-5, the free-particle eigenfunctions may be written exp( i-Jim Ex/h), where E is any noimegative number. These are also eigenfunctions for the momentum operator, with eigenvalues -JlmEh. 

We discuss here some representative examples of how the TISE is solved to determine the eigenvalues and eigenfunctions of some potentials that appear frequently in relation to the physics of solids. These include free particles and particles in a harmonic oscillator potential or a Coulomb potential. 

Thus, the free-particle eigenfunctions are an example where the energy eigenfunctions are also eigenfunctions of some other operator in such cases, the hamiltonian and this other operator commute, that is,... 

Just as Eq. (15.3-38) represents a linear combination of waves moving in opposite directions, a wave eigenfunction for a three-dimensional free particle can consist of a superposition (linear combination) of waves moving in various directions with various energies ... 

---