A particle of mass m and fixed energy E is confined to a special three-dimension

Question

A particle of mass m and fixed energy E is confined to a special three-dimensional box where the z side-length on the order of atomic dimensions. The other two box dimensions are much larger than atomic dimensions. As in the text density-of-statees derivation the x ,y, and z side-lengths are a, b, c respectively (see Fig. P4.3); U(x, y, z) = constant everywhere inside the box.

a) Does the small size of the z-dimenstion hace any effect on the overall wavefunction solution embodied in Eqs. (4.6) throught (4.9)?

Explanation / Answer

The answer is yes. Basically when a, and b are large, the value of pi/a and pi/b are very small, so that the spectrum of k_x=pi/a, 2pi/a, 3*pi/a, etc and k_y are almost continuos.
In other words k_x and k_y could take almost any REAL CONTINUOUS values.

Contrary to this, since c is small, the value of pi/c will be high enough so that k_z = pi/c, 2*pi/c, 3*pi/c   will be very separated one from the other.
in other words k_z can take only DISCRETE values.

Back into equations (4.6) the wavefucntion will take continuos values (like a sinusoide) on the x and y directions, but will have only a few allowed values on the z direction. Hence on the z direction the position of electrons will become quantified (on discrete levels of energies).

Corespondingly (to the distribution on energies) the momentum on x and y directions (k_x and k_y) will be continous (like for a massive body) whereas the momentum on the z direction (k_z) will become quantified (eq. 4.7)

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