ELEMENTS of the QUANTUM THEORY* VZ. THE HYDROGEN ATOH
SAUL DUSHMAN Research Laboratory, General Electric Co., Schenectady, New York BOHR THEORY OF THE
Hence
HYDROGEN-LIKE ATOM
"W)
T
HE ESSENTIAL differences between the wave mechanics treatment and that based on ordinary where d r / d t = velocity along the line joining electron mechanics are brought out most effectively by to nucleus, and : considering the problem of the hydrogen atom from dr dr both points of view. . .. ., . , , t ~ d =r~ l : $ ( ~ ) ~ , d t The hydrogen-like atom consists of a massive nu. , cleus of positive charge, +Ze, and an electron of charge -e. The Coulomb force of attraction, which varies inversely as the square of the distance, r, between dr p dr electron and nucleus, is balanced by the repulsive = Z(Z): force due to the motion of the electron in its orbit. The = T, - TO (iw) problem of the nature of the electronic orbits thus resembles from the point of view of ordinary mechanics where TI is the value of the kinetic energy at the end the astronomical problem of planetary orbits (the of the motion, and To is the value at the initial point. From (ii) and (iu) it follows that problem of motion under the action of central forces, that is, of forces varying inversely as some power of r ) . V, - VO= TO- T, . , . In such problems, where the force acting between or . . V, + TI= Vo + TO= E the two particles is a function of the coordinates only; it is very convenient to introduce the potential energy where E is the total energy, which remains constant function, V , which is defined in terms of the force, F, during the motion. For such a system (which is desigacting between the particles thus: nated conseruatiue) only such changes can occur as will make the decrease in potential energy equal to the increase in kinetic energy. Let us now consider the changes in this function and If now we regard the potential energy of the system as the corresponding change in kinetic energy, as the elec- zero for r = w , it follows that V must become more and tron approaches the nucleus from an initial distance, more negative, as r is decreased. Consequently, for YO,to a final distance, rl (ro > rl). any value of r, From ( i ) it follows that the decrease in potential v = - -Zc' (4 energy is given by + and the force of attraction, (ii) VO - V, = Fdr
,
..
1;
where Vo and Vl are the value of V at 7 = ro and r =r,, respectively. According to Newton's second law of motion, force is d&ned as rate of change of momentum. That is,
where pv = momentum, and p is regarded as constant (non-relativistic case).
I t will be observed that as 5 result of the convention = 0 for r = w , the force of attraction is denoted by a function of r with a negative sign. Consequently, a force of repulsion must be represented with a positive sign. In order to simplify the problem, we shall assume that the orbit of the electron is circular and of fixed radius a. Then the orbital velocity is given by
V
-
* This is the seventh of a series of articles presenting a more detailed and extended treatment of the subject matter covered in Dr. Dushman's contribution to the symposium on Modernizing the Course in General Chemistry conducted by the Division of Chemical Education at the eighty-eighth meeting of the American Chemical Society. Cleveland, Ohio, September 12, 1934. The author reserves the right to publication in book form.
where o = frequency of r e ~ ~ l ~ t i ~ n . Hence, the centrifugal force, which is given Fa = !a2= ~ e ( 2 r w ) ~
is
Since the system is in equilibrium a t r = a, F A F,, = 0
+
and therefore (mi)
Since
E
=
v
T+
and (mi;)
it follows from (a) that
While a more exact calculatiou would consider the possibility of elliptic as well as circular orbits and would also consider the precession of the orbit about an axis of symmetry, the logic involved in the classical quantum theory was essentially as follows: Given the law of force acting between the particles constituting the system, Newtonian methods were applied to calculate the types of orbits and the relation between the total energy and orbital constants. Then the quantum condition was applied in order to determine the actual nature of the discrete series of orbits and energy states. AYDROGEN ATOM AS A POTENTIAL BARRIER PROBLEM
That is, the tufa1 energy of the system is negative and equal to half of the potential energy. It may be shown that this conclusion is valid for any type of orbit under the action of the inverse square law of force. Let
then it follows from (vii) that
.
Before discussing the solution of the appropriate S. equation for this problem, i t is of interest to point out a method by which an approximate solution may be derived for the values of the discrete series of energy states. As N. F. Mott has pointed out,* "a hydrogen atom is simply an electron bound by an electrostatic force, which pulls i t back if it tries to get away from the nucleus. The wave function therefore will vibrate in normal modes, and we shall only have wave functions describing the behavior of electrons of certain discrete energies.'' An a ~ ~ r o x i m a tcalculation e o f the maenitnde of these energy states may be made as follows: 111 that in the case of an It was shown in the discrete series of eleckon in a potential energy states is given by
-
..
-7
Now Newtonian mechanics places no limitations on the possible values of Wand 7. In order to account for the observed energy states of an atomic system it is necessary to assume some form of quantum condition, and BOGtherefore postulated that the hydrogen atom (80) can exist onlv in those states for which the total anm" lar momentum of the electron integrated over a com- where m is an integral value-and 2a is the extent of the plete orbit is equal to an integral multiple of Plauck's region between the barriers.' Also i t was shown that constant, h. In the case of a circular orhit this quan- for the lowest of these states (m =I), the corresponding tizing principle leads to the relation characteristic function is 2rpm,,nn = nh
( 4
where a, = radius of orbit of quantum number n, and v. = velocity in this orbit: From ( x ) and (xi) it follows th$
Hence, (zii)
and eliminating w between this equation and ( x ) it follows that
4 =2
4:.
A cos(2szlh)
(85)
where
In the case of a hydrogen atom, the field of force is defined by the potential energy, -Ze2/r. Since the total energy cannot exceed this value for any given value of r, i t follows, if 2a denotes the diameter of the electron orbit for the lowest state, that
and The energy states are therefore defined by the relation E
where R =
"-
RchZ4
,. .,a
zr2e4,, , is known as the hac
and c = velocity of light.
(206b)
h
,,=4a=-
(ii)
1 -
From (i) and (ii) it follows, by eliminating a , that
Rydberg constant, -
* N.
F. MOTT,"An outline of wave mechanics," p. 63
which differs from the expression in (206a) by the ratio 32/(2r2). If X had been assumed equal to r a instead of 4a, the two results would be identical. THE SCHROEDINGER EQUATION
In the Schroedinger method of calculating energy states the starting point is the same as in the classical method, that is, we consider a system consisting of nucleus of charge +Ze, and electron for which the potential energy is given by V = -Ze2/r, as before, and the kinetic energy, T,by per2. In terms of spherical polar coordinates, the total energy is given by
As shown in equation (154) Chapter V, the S. equation deduced from this relation is of the form
where 1 = 0, 1, 2, etc.
and
m = 4 , *(l- 1) . . . * 1,O
As pointed out in the case of the rigid rotator this signifies that the particular energy state corresponding to any given value, EL,is degenerate, inasmuch as i t may he represented by any one of 21 1 independent eigenfunctions. However, it will be observed that i t is not possible from the solution of equation (209) to determine the value of EL. For this purpose it is necessary to solve the radial equation (210), which, by substituting the value for C, takes the form,
+
where A = 8ra&lhP I
and where a2 = 8r2#/h2, and the Laplacian operator is given by the relation
THE RADIAL SCHROEDINGER EQUATION
In solving the last equation, which is known as the radial equation, we must be guided by the conclusions deduced in ordinary mechanics about the motion of a body in a central field of force. The best illustration of this is furnished by the investigations on the possible The solution of this partial diierential equation orbits of the planets in the gravitational field of the must yield = +(r,@,q),and as in the case of the .~@d sun. We know that in this case two types of orbits rotator, we assume that it is possible to express this are possible, (1) hyperbolic orbits for which E>O, and (2) elliptical (including circular) orbits, for which solution in the form EO. For very large denotes a function of the angle variables. values of r, all the terms in (212) involving l / r or l/r2 We thus separate equation (208) into two differential may he neglected, and the equation becomes equations, one in r, and the other in O,q, which are as follows :*
+
The solution of this equation is, evidently and
s = K@roper fuktions that lie between the limiting points for the coordinate i n question. This
It is, therefore, an exponential function of r, and possesses spherical symmetry. For n = 2, there are four eigenfunctions, one corresponding to 1 = 0, and three to 1 = 1, as indicated in Table 2. The normalized functions are as follows:
brings to mind the analogy of the vibrating string in which the ordinal number of an overtone is likewise measured by the number of nodes that lie between the fixed ends of the string." THE COMPLETE SOLUTION OP THE S. EQUATION FOR T I E HYDROGEN ATOM
As stated a t the beginning of this chapter, the complete solution of the S. equation for the hydrogen-like atom must be an expression of the form d.rn(r.0.d = S d d K&%n) (238) where
From equation (225) i t will bepbserved that all the eigenfunctions for any given value -of n have the same eigenvalue,
Km(O.d = XI~(@.Z&)
is a tesseral harmonic, which was defined in equation (187c), with the normalizing factor given by equation (190). Furthermore, according t o equations (165). (186), and (189), the individual normalized functions of 8 and 7 are given by the relations:
* This figure is taken from the treatise by H. E. WHITE.''Introduction to atomic spectra," McGraw-Hill Campany. New York. 1934. Plots of these functions were first oublished bv L. A;~PAULING. P ~ o cROY. . SOC.,114k 181 (19271. See also-the summary by the sameauthor in Chem. RN., 5,173 (1928), and G Chapter V in the recently published work by L. P A ~ L I NAND R. R . WLSON. TR.. "Introduction t o auantum mechanics." M c -. Graw-Hill Bwk ~ b .~, e YO& w 1935: t SOMMBRIIELD, '"Wave mechanics." English trans., p. 72. ~
.
~~
~
~~
~
The state is, therefore, of the degenerate type. As was shown previously, there are 21 1functions corresponding to any given value ofl. But for a given value of n, 1 can have the values n - 1, n-2,. . .O. Hence the total number of eigenfunctions corresponding to a given value of n is
+
This is an arithmetical progression of n terms with constant difference 2, the first term being 1, and the nth, 2n- 1. Hence
t A table of narmalizedfunctions is given by A. SONMERFELD, "Atoms, "Wave mechanics," p. 71; also in rum^ m UREY'S, molecules and quanta," p. 564, and in PAWLING w n WILSON'S work, Chapter V.
Thus the degree of degeneracy for quantum number n is na. This conclusion may be illustrated by the following table giving the diierent possible values of the quantum numbers n, 1, and m, and the corresponding spectroscopic designations for the lower energy states of a hydrogen-like atom.
1
0
o
0
o 0
1
-1 0 0 -1
0 1
1
z
2
2
is regarded as denoting the probability of occurrence of the electron in the element of volume dv, at the point whose coordinates are r, 8, and q. As pointed out in Chapter V,
?"
1
currence. In the case of the hydrogen atom, &,(r,8,s) is a function of three coordinates which has been defined in equations (238) and (239). Omitting the subscripts and considering only the normalized functions, the expression
du = ra sin E drdedn
while the element of area on the surface of the sphere at r is
o
-1 *Z
d A = r2 sin E d@dq
S
Electrons
P Electrons rn-0
I- 1
d Electrons - I
1-3
FlouRE
35.-ANGULAR
DISTRIBUTION FUNCTIONS
FOR
The integer n is known as the total guantum number. The number I, which is associated with the azimuthal angle, 8, is designated the azimuthal quantum number, while m, which is associated with the angle s, is designated the magnetic quantum number.
DIFFERENT ELECTRONIC STATES 011 H ~ R O D E ATOMS N
Since ~ ( s ) Z ( q )= 1/(2a), it is evident that the probability of occurrence of the electron is independent of the angle s. Hence
QUANTUM MECHANICS INTERPRETATION OF THE CHARACTERISTIC FUNCTIONS In quantum mechanics the eigenfunction @, obtained by solving a S. equation, has no direct physical interpretation. But $+ or +2 (in case the function is real) is interpreted as representing a probability of oc-
is the probability of occurrence of the electron per unit solid angle at any given value of 9. It is evident that this interpretation is equivalent to the statement in Chapter V that P is a measure of the probability of occurrence per unit area on the surface of a sphere de-
scribed about the origin of coordinates. That is, the magnitude of P at any given value of 0 is a measure of the electron density on the corresponding zone (since P is independent of v). In Figure 31 and Figure 32 plots were shown of the function P in the cases 1 =3, m =0 and 1=3, m =3, respectively. Figure 35, taken from the treatise by H. E. White,* shows similar plots of the probability
FXXIRE 36.-RADIAL
out that the electron is not confined to the shaded areas in each fimre. The mamitude of a straieht line ioinine the center and any point on a &en curve is a m & x e of ihe el&ttron's probabilxy of being found i n the direction of rhat line. "These i l y r c s indicate rhat for all m = 0 states, with the exceotion of s-electrons the charw drnsitv is mratcqt in ihr dirrcti& of the poles, i. e:, in the dTrectian = 6 and r. The exoonent of &-being zero implies that there is no motion in t h i 7 codrdinate and that the motion of the electron, i. e., the plane of the orbit. is in some one meridian olane throuch the n-axis. all
D ~ s r n r e a r oFUNCTION ~ (D)FOR D I P F E ~ ~ NELECTRONIC T STATES 80% ORBITS FOR COMPARISON
OF
HYDROGEN ATOM,AND
density distribution as a function of 8 for different elecFor m = *1, P has its maximum in the direction of tronic states of the hydrogen atom. In the case n = 1, the equatorial plane, while for 0 < 1 m < 1, P has maxima 2=0, the function P is spherically symmetrical, that is, oriented in definite directions. P is independent of 8. But in the case n = 2 , I= 1, These plots correspond very well with the deductions three states are possible, corresponding to m=O, * 1. based on the classical Bohr theory for the directions of orientation of the electronic orbits. In the following "For these three states," as White describes the plots, "P gives section it be shown that on the basis of the S, the charge distributions shown a t the right and top in Figure 35. momentum Of the 'Iectron in Each curve is shown plotted symmetrically an etch side of the theOry*the vertical axis in order to represent a cross section of the three- its orbit is given by M = m h / ( 2 r ) and that .dimensional plot. Three-dimensional curves are obtained by the component of momentum about the lraxis rotating each figure about its vertical axis. I t should he pointed (the axis through 8 =0 and r ) is Ms = mhl(2rr). This means that for absolute values of m greater than zero * "Introduction to. atomic spectra," p. 63. I n this hook the .symbol ed is used for the function P ,(COS ~ 8). and the latter is but less than 2, the orbits are oriented with respect to used to designate the derivative d"'Pz(x)/dxm. the z-axis.
I
Figure 35 shows the classical oriented orbits for each state, below the corresponding plot of P. The orbits are drawn slightly tilted out of the normal plane in order to show an orbit rather than a straight line. It should also be added that in the plots "for states m = 0 the scale is approximately I/(/ 1) times that of the other states having the same value." Let us now consider the radial function. In this case, the square of the normalized function, {&(r) ] ', should evidently be interpreted as the probability of occurrence of the electron per unit length a t the point whose distance from the nucleus is given by the value of r. For a spherically symmetrical form of @, it follows that { S , L ( ~ also ) ) ~ represents the probability of occurrence per unit volume. However a much better physical interpretation of the behavior of the electron in a hydrogen atom is obtained by use of the so-called distribution function, designated by D, which is derived from S(7) thus. The volume of the spherical shell between the radii r and r dr is evidently given by
+
Figure 37t shows the function +loo a t the top, p= in the center, and D a t the bottom, each plotted as a function of T. The distribution function exhibits a maximum a t exactly that value of r, which was deduced by Bohr, as the radius of the circular electronic orbit in the normal state. On the basis of the Bohr theory, i t was also shown that the orbits for the 2p and 3d states should be circular and of radii 4ao and 9ao, respectively. A simple calculation shows that these are exactly the values a t which the corresponding Dfunctions exhibit single maxima, as indicated by the plots in Figure 36. Figure 38 gives a more graphical interpretation of this function for the case n = 2, 1 = 0. "If we could imagine the electron in a hydrogen atom replaced by an infinitesimal source of light, the net effectof the fluctnaCZ~~IOO
+
Hence, 4d(S,,(r))Vr = Ddr
(243)
is a measure of the probability of occurrence of the electron i n a spherical shell of thickness dr and of radius r. Figure 36, taken from the treatise by H. E. White* gives values of D as functions of r/ao for diierent electronic states of the hydrogen atom. It will be observed that in all cases D = 0 for r = 0. This arises from the fact that the volume of the spherical shell of thickness dr becomes infinitesimally small as r decreases to zero. Stated in other terms, D = 0 a t r = 0 because of the factor r2. On the other hand, $4 plotted as a function of 7, or S2(r)may be quite large a t r = 0. This is illustrated by considering the case n = 1, I =0 (the normal state of the hydrogen atom). From equation (24Q) i t follows that
tion in the instantaneous location of the electron over a period of time would result in an image which would be brightest a t those points where the proba' bility of occurrence is greatest."$ Figure 39** gives a photograph of a three-dimensional representation of the distribution function for n = 2 , I = 1, m =0. Most of the charge lies in the region' indicated by the boundaries, but actually the density That is, the probability density is a maximum a t the decreases exponentially with 7 in the space outside the origin and decreases exponentially with increase in 7. boundaries. The difference between the wave mechanics point of Thus for r = a&, the value of $@is l / e times (36.8 view and that of the Bohr theory is brought out by comper cent.) its value a t r = 0. paring this conception of a probability density distriThe function D for this state is given by bution for the electron with that of an electron revolving about the nucleus in a definite elliptic orbit. According to the older theory, the magnitude of the major axis of ellipse is determined by the value of n, while the It has a maximum a t a value r = 7 , which may be ratio of minor axis to major axis is given by the value calculated as follows: of k / n where k is designated the azimuthal quantum number. In the new theory, the latter has to be reFor r = r,, dD/dr 0
-
-
Hence, 21. 2Zri/ao = 0 For Z = 1, r, = ao. * W m , LOG. tit., p. 68. Similar plots are given in the book by PAWLING AND WILSON already cited.
With this modification, the placed by 1/1
