LETTERS ha E., = 0.125 -
To the Editm: The problem of determining the energy of an electron in the lowest energy level of a spherical potential box and showing that i t is lower than the energy of the lowest energy level of a cubical potential box of equal volume requires advanced quantum mechanics for exact solution. However, two approximate solutions can be obtained in the following ways. I. The energy, E, of an electron in a potential box with infinite potential barriers is given by the wellknown equation
where m = 1 in the problem considered, 2a is the distance between the boundaries of the box, h is planckjs constant, and pis the mass of the electron. This box has no "lid" on it or thickness, but if one assumes that -i -t has a -~~~ lid a distance 2a from the bottom and a thickness Za, then it is a cubical box. If the box were trulv such a cubical box the eauation savs that the energy is inversely proportional to the cross-sectional area of the box. By making the assumption that for a spherical box the energy is also inversely proportional to the cross-sectional area, the problem can be carried through. k c t h e area of the cross section of the sphere be and that of the box of equal volume be 4be = 4 ( ~ / 6 ) ' 9, / ~ where s is the radius of the sphere and b is the semiedge of the box. Replacing ( 1 / 2 ~in ) ~equation (1) by the reciprocal of the cross-sectional areas of the sphere and the new cube, the lowest energy level of the sphere is ~~
~~
~~
~~
~
~
S'P
and
However, their ratio E J E , = 1.21 while the ratio of the correct value is 1.155. 11. The energy, E, of an electron in the lowest energy level of a cubical box is given by the exact equation
where a is the length of an edge. Following the general procedure above by substitnting the reciprocal of the cross-sectional area of the sphere, ma,and of the cross-sectional area of a cube of ~ , ( l / ~in )equation ~ ( Z ) , the equal volume ( 4 ~ / 3 ) % for approximate energy of the lowest level of the 'phere is
~
and the lowest energy level of the cube is
The results show qualitatively that the lowest level for the sphere has a lower energy than the lowest level for the cubical box. These figures are in enor by a factor of about 3, the correct values being
zs g3 . h q = 0.119 ;hai;
E . = 1
T;I
which is only per cent low, and the exact energy of the lowest level of the cubical box is .- . . .. .
.
-
In this case, again
These approximate solutions given may prove useful as problems in courses in elementary quantum mechanics. They are based, in part, upon material from the following sources: ROJANSKY, V. B., "Introductory Quantum Mechanics," Prentice-Hall, Inc., New York, 1938, p., 452; DUSHMAN, S., "Elements of Quantum Mechanics," John Wiley and Sons, Inc., New York, 1938, p. 55; and PAULING, L., AND E. B. WILSON,JR., "Introduction to Quantum Mechanics," McGraw-Hill Book Company, Inc., New York, 1935, p. 95. DANA. H. OLSON GENERAL PETROLEUM CORPORATION Los ANGELES, CALIFORNIA
Pearls don't lost forever. Whole strings of pearls have crumbled into dust when removed from ancient tombs. Nowadays, rarely worn pearls are given frequent airings topreserve their lustre.
