The Particle in a Three-Dimensional Box

It is customary in quantum mechanics to denote integration over the full range of all the coordinates of a system by J dr. A shorthand way of writing (3.55) or (3.57) is 

Although (3.58) may look like an indefinite integral, it is understood to be a definite integral. The integration variables and their ranges are understood from the context. 

For the present, we confine ourselves to one-particle problems. In this section we consider the three-dimensional case of the problem solved in Section 2.2, the particle in a box. 

There are many possible shapes for a three-dimensional box. The box we consider is a rectangular parallelepiped with edges of length a, b, and c. We choose our coordinate system so that one corner of the box lies at the origin and the box lies in the first octant of space (Fig. 3.2). Within the box, the potential energy is zero. Outside the box, it is infinite  

Since the probability for the particle to have infinite energy is zero, the wave function must be zero outside the box. Within the box, the potential-energy operator is zero and the Schrodinger equation (3.47) is 

To solve (3.61), we assume that the solution can be written as the product of a function of X alone times a function of y alone times a function of z alone  

It might be thought that this assumption throws away solutions that are not of the form (3.62). However, it can be shown that, if we can find solutions of the form (3.62) that satisfy the boundary conditions, then there are no other solutions of the Schrodinger equation that will satisfy the boundary conditions. (For a proof, see G. F. D. Duff and D. Naylor, Differential Equations of Applied Mathematics, Wley, 1966, pp. 257-258.) The method we are using to solve (3.62) is called separation of variables. 

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Now we will apply the particle in a three-dimensional box model to a chemical problem. When sodium vapor is passed over a crystal of NaCl, the crystal exhibits a greenish-yellow color, which is the result of the process... 

A color center or F-center is formed from diffusion of a small quantity of M+ ion into an ionic crystal MX. Since the crystal must keep its charge neutrality, additional electrons readily move to fill the vacancies normally occupied by anions. Thus the composition of the crystal becomes (M+)i+i(X e ). The origin of the color is due to electronic motion, and a simple picture of an electron in a vacancy is illustrated by the particle in a three-dimensional box problem, which is discussed in Section 1.5.2. 

The majority of interesting problems involve more than one coordinate and momentum. Immediately the Schrodinger equation becomes a partial differential equation and the solutions become more complicated. One of the simplest cases that illustrates a general method of solving the partial differential equation is the example of the particle in a three-dimensional box. 

We did the same thing for the wave function of the particle in a three-dimensional box.) Therefore,... 

As we put more energy into one degree of freedom, the number of nodes along that coordinate increases. There maybe nodes at particular distances r from the nucleus, as well as at certain angles

three-dimensional box, some wavefunctions will have nodes along all three coordinates. 

As promised, the wavefunctions are still separable to a degree, in that each wavefunction may be factored into three distinct terms an r-dependent term, a 0-dependent term, and a -dependent term. Unlike the particle in a three-dimensional box wavefunctions, though, those three terms are not completely independent. The radial wavefunction R j(r) depends on the quantum number /,... 

Based on the Hamiltonians we ve seen for the particle in a three-dimensional box and the one-electron atom, write the Hamiltonian using cylindrical coordinates r,z, (f> for an electron travelling near a charged wire where the potential energy is U = —Cjr, where C is a constant. In cylindrical coordinates, z is the same as the Cartesian z coordinate, r is the distance from the z axis, and (f> has the same definition as for spherical polar coordinates. 

Section 2-7 The Particle In a Three-Dimensional Box Separation of Variables... 

The particle in a three-dimensional box can be used to model the translational energy of a gas phase atom or molecule. Consider a gas phase argon atom in a 1.00 m square box at 300. K. The average thermal energy of the argon atom is found by multiplying the temperature by Boltzmann s constant, k. 

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