Wavefunction free particle

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). 

Incidentally, the telegrapher s equation (17) with x = ihjlmc2 is satisfied by the Klein-Gordon (also Dirac) wavefunction for a free particle, if the factor exp [—(ijh)mc2f is split off from it. Thus, the time lag according to relativity corresponds to an imaginary relaxation time x. 

In order to understand the wavefunction of an electron emitted from an atom by a certain ionization process, the wavefunction of a free particle with wavenumber k travelling along the positive z-axis will first be considered. The space and time dependence of this wavefunction follows from the time-dependent Schrodinger equation with zero potential1- and is given by... 

We refer to the wavefunction (2.98) in the present context as a wavepacket It is a local function in position space that can obviously be expanded in the complete set of free particle waves (2.82). Below we see that this function is also localized in momentum space. 

As an example consider the free particle wavefunctions (r) = A expfz k r) and V 2(i ) = cos(k r). From Eq. (2.123) it is clear that the flux associated with 1/ 2 is zero. This is true for any wavefunction that is real or can be made real by multiplication by a position independent phase factor. On the other hand, using... 

For example, the free particle wavefunction (2.200), a solution of a differential equation of the second-order, is also characterized by two coefficients, and we may choose 5 = 0 to describe a particle going in the positive x direction or ff = 0 to describe a particle going in the opposite direction. The other coefficient can be chosen to express normalization as was done in Eq. (2.82). 

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 same equation holds for the free particle wavefunctions V l and -02 which are defined by V (r) = 0. Thus ... 

An analogous expression is obtained in three dimensions. We now need to consider periodic systems. As we have discussed, the wavefunction for a particle on a periodic lattice must satisfy Bloch s theorem. Equation (3.85). The wavevector k in Bloch s theorem plays the same role in the study of periodic systems as the vector k does for a free particle. One important difference is that whereas the wavevector is directly related to the momentum for a free particle (i.e. k = p/h) this is not the case for the Bloch particle due to the presence of the external potential (i.e. the nuclei). However, it is very convenient to consider Hk as analogous to the momentum and it is often referred to as the crystal momentum for this reason. The possible values that k can adopt are given by ... 

For a free particle in quantum mechanics, V(x) = 0 everywhere. The Schrodinger Eq. (8.93) still applies but now with no restrictive boundary conditions. Any value of k is allowed (- cc,k < oo) and thus > 0. There is no quantization of energy for a free particle. The wavefunction is conventionally written as... 

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. 

A FIGURE 2.4 Real parts cos(fcx) of the free particle wavefunctions at selected energies. 

The phase function was given as the solution to the free particle wavefunction in Eq. 2.22, but (with a a real number) was not mentioned, although it can also satisfy the Schrodinger equation (Eq. 2.21),... 

Find the general solution for the wavefunction of the free particle in three dimensions. It is not possible to normalize these wavefunctions please don t try. 

We can t begin with 6, because part of the 0-dependence is locked up in the second term (l/sin 0)O /O0 and can t be solved until we have the second derivative with respect to . Also, the part of our wavefunction that depends on cf)—let s call it 4>(d>)—is easier to solve because the operator df /dcf is similar to the d jdx - operator we saw in the free particle Schrodinger equation (Section 2.2). The free particle has a Hamiltonian (Eq. 2.19)... 

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. 

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