The Free Particle in One Dimension

We shall see that the correlation of wavefunction symmetries in such a manner as this is a powerful technique in understanding and predicting chemical behavior. 

EXAMPLE 2-5 Fig. 2-15 shows energy levels for states when the barrier has finite height. When the barrier is made infinitely high, the levels at Ei and E2 merge into one level. Where does the energy of that one level lie— below Ei, between E and E2, or above E2—and why  

SOLUTION It lies above 2- When the barrier is finite, there is always some penetration, so X is always at least a little larger than is the case for the infinite barrier. If X is larger, E is lower. 4 

Suppose a particle of mass m moves in one dimension in a potential that is everywhere zero. The Schrodinger equation becomes 

Chapter 2 Quantum Mechanics of Some Simple Systems 

By a free particle, we mean a particle subject to no forces whatever. For a fi ee particle, integration of (1.12) shows the potential energy remains constant no matter what the value of X is. Since the choice of the zero level of energy is arbitrary, we may set V x) = O.The Schrodinger equation (1.19) becomes 

What boundary condition might we impose It seems reasonable to postulate (since dx represents a probability) that il will remain finite as x goes to oo. If the energy E is less than zero, then this boundary condition will be violated, since for 0 we have 

The free-particle problem is an unreal situation because we could not actually have a particle that had no interaction with any other particle in the universe. 

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The Hamiltonian operator for the free particle in one dimension is just the kinetic energy operator, because there are no forces present to generate potential energy ... 

This is analogous to expressing the kinetic energy of a free classical particle as a sum of three terms involving the momentum components (pj + Py+Pz)/2wi-Returning to the free particle in one dimension, we note that if either were zero or B were zero in Equation 8.27, the wavefunction would be an eigenfunction of p. ... 

We now argue that, for finite t, rejecting moves is the choice with the smaller time-step error when the importance-sampled Green function is employed. Sufficiently close to a node, the component of the velocity perpendicular to the node dominates all other terms in the Green function and it is illuminating to consider a free particle in one dimension subject to the boundary condition that have a node at x = 0. The exact Green function for this problem is... 

A proo f of the second part of the uncertainty principle is lifficult indeed, the statement itself is vague (the exact meaning of A/, etc., not being given). We shall content ourselves with the discussion of a simple case which lends itself to exact treatment, namely, the translational motion in one dimension of a free particle. 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

In general, a dispersed particle is free to move in all three dimensions. For the present, however, we restrict our consideration to the motion of a particle undergoing random displacements in one dimension only. The model used to describe this motion is called a onedimensional random walk. Its generalization to three dimensions is straightforward. 

A free particle is a particle subject to no forces, so that V = 0 everywhere. For a free particle moving in one dimension, the Schrodinger equation is (1.122) and its solution is [Eq. (1.98)]... 

To illustrate how the Schrodinger equation might be applied to a familial" situation, consider the case of a free particle, that is, a particle moving along at a constant velocity with no force acting on the particle (V = 0) (Fig. E.l). For simplicity, let us consider motion in one dimension, the x direction. For the time-independent Schrodinger equation, we have... 

The time-independent Schrodinger equation in one dimension for the free particle reads... 

Parabolic dispersion relation for free particle E(k) =t)2k2/2m in one dimension, folded over and constrained into the first and second Brillouin zones. 

Consider now the time evolution of a free particle moving in one dimension that starts at Z = 0 in the normalized state... 

Another theory which is used to describe scattering problems and which blends together classical and quantum mechanics is the semiclassical wavepacket approach [53]. The basic procedure comes from the fact that wavepackets which are initially Gaussian remain Gaussian as a function of time for potentials that are constant, linear or quadratic functions of the coordinates. In addition, the centres of such wavepackets evolve in time in accord with classical mechanics. We have already seen one example of this with the free particle wavepacket of equation (A3.11.7). Consider the general quadratic Hamiltonian (still in one dimension but the generalization to many dimensions is straightforward)... 

Polymeric nanocomposites are an important class of new emerging nanomaterials that exhibits remarkable improvanents of material properties compared with conventional micro- and macrocomposite materials. The small dimension of the filler particles and, accordingly, large surface of the phase separation give the final product characteristics, which considerably exceed traditional ones at minimal filler concentration (Mikitaev et al. 2008). The formation of the polymeric nanocomposite may be represented as the process of filling of the free space in disperse phase with polymer in the form of melt or solution or with monomer followed by its in situ polymerization by chemical or radiation influence on the formed composite structure. The scheme of the polymeric nanocomposite synthesis under radiation is shown in Figure 18.5. 

This equation actually provides an estimate for the mean free path, because equation 19.37 considers a cylindrical volume swept out by travel in one dimension, whereas the average volume V/N is more of a three-dimensional average volume. This also assumes that the particle is moving while all other particles are fixed, which is not the case—as we argued in the previous section, gas particles are moving with respect to each other. This means that the average velocity is actually /2v, which reduces the mean free path by a factor of Vl. Thus, we have... 

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... 

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