Quantum Mechanics

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Quantum Mechanics

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Title: Quantum Mechanics

1
Quantum Mechanics

Chapter 5.
The Hydrogen Atom

The solution of the problem of atomic spectra
was a great triumph for the Schreodinger
equation.
Although this equation does not yield exact
solutions for atoms containing more than one
electron, it permits approximations which can be
applied, in principle, to any problem and to any
desired degree of accuracy.
The ultimate result is enormous accuracy in
theoretical calculations of hydrogen energy
levels, taking into account such factors as the
spin of the electron, a previously unknown
quantity.

Subsequently, P.A.M. Dirac showed that electron
spin emerges as a natural consequence of a
relativistic wave equation..
Experimenters have responded to these
theories with equally accurate experiments to
test these calculations.
This accuracy has not been sought simply to
demonstrate the prowess of physics and
physicists proof of the existence of each small
contribution to the energy in the amount
predicted by the theory is an indication of the
correctness of our fundamental ideas concerning
the nature of matter.

4
5.1 Wave Function for More Than One Particle

If the proton were of infinite mass, it would be
a fixed center of force for the electron in the
hydrogen atom, and we could solve the problem by
the methods of Chapter 2.
The wave function would be a function of the
coordinates of a single particle, the electron.
However, because the proton also moves, we must
incorporate this fact into our wave function.

According to Postulates 1 and 2 (Section 3.3),
each dynamical variable for each particle must be
represented by an operator whose eigenvalues are
the allowed values of the variable.
As a logical extension of these postulates, we
now assert that there mast be a wave function for
a system which is capable of generating all the
dynamical variables of the system.
Therefore, for a hydrogen atom the wave function
may be written as F(xp, yp, zp, xe, ye, ze, t),
where xp is the x coordinate of the proton, xe is
the x coordinate of the electron, etc.

It is now clear that the wave function is simply
a mathematical construction there is no physical
wave in the sense of a simple displacement that
exists at each point of space and time, for the
wave function is a function of six space
coordinates in this case, rather than three.
Our single-particle problems of the previous
chapters enabled us to make useful analogies with
conventional waves. but we must now go into a
more abstract realm of theory.
This does not mean that the problems are
necessarily more difficult to solve. It simply
means that we must beware of visualizations of
the solutions

the limited experience that we have gained
through our senses is not sufficient to permit
this.
As usual. the wave function is an
eigenfunction of the Schreodinger equation. The
Schreodinger equation, in turn, contains the
operator for the total energy of the system, as
before.
The total energy of the proton-electron system is
ET pp2/2mp p2/2me V(r)
(9.1)
or, in operator form
where r is the proton-electron distance, mp is
the proton's mass, me is the electron's mass.
V(r) -e2/4pe0r is the Coulomb potential, and
the symbol ?2 represents

or the equivalent expression in spherical
coordinates ?p2 operates on the proton's
coordinates, and ?e2 operates on the electron's
coordinates in the wave function.
The time-independent Schreodinger equation for
the hydrogen atom is therefore
but when the equation is written in this form,
the energy eigenvalue ET includes the kinetic
energy of translation of the center of mass of
the whole atom. a quantity in which we are not
interested at the moment.

The states that tell us about the hydrogen
spectrum are the internal energy statesstates of
the relative motion of proton and electron.
Fortunately, the potential energy is a
function only of the relative coordinates of
proton and electron, and we can rewrite Eq.(9.3)
in terms of the coordinates X, Y, and Z of the
center of mass and the coordinates x, y, and z of
the electron relative to the proton.
These coordinates are related to the
coordinates of the individual particles as
follows

It is not difficult to use Eqs.(9.4) to write the
kinetic energy in terms of the relative velocity
and the velocity of the center of mass the
result is
Total kinetic energl
(me mp)(X2 Y2 Z2)/2 mr(x2 y2
z2)/2 (9.5)
where x, y, and z are the x, y, and z
components of the relative velocity, and

mr memp/(me mp) is the reduced mass, which
we previously encountered in discussing the Bohr
model of hydrogen.
In terms of momentum variables defined as Px
(me mp)X, px mrx, etc., the kinetic energy
may be written as
so that. according to the rules for writing the
energy and momentum operators, the Schreodinger
equation becomes

where the operators ?c2 and ?2 operate on center
of mass and relative coordinates, respectively.
Equation (9.7) could also have been obtained
by direct transformation of the partial
derivatives in Eq.(9.3), making use of Eqs.(9.4)
to find the derivatives with respect to the new
variables.
Separation of Variables
The wave function ? is now a function of X, Y,
Z, x, y, z, and t. Let us assume that it is
possible to write the space dependence of ? as a
function of x, y, and z multiplied by another
function of X, Y, and Z. We write

?(x,y,z,X,Y,Z,t) u(x,y,z)U(X,Y,Z)e-i(EE')t/?
(9.8)
where E is the energy of the relative motion and
E' the energy of translation of the center of
mass.
From Eq.(9.7) we may now extract the two
time-independent equations
Equation (9.9) is simply the equation of
motion of the center of mass of the whole
hydrogen atom it tells us nothing about the
atom's internal energy levels.

Equation (9.10) is the Schreodinger equation for
the motion of the electron relative to the
proton. It is identical to the equation for a
single particle of mass mr moving under the
influence of a fixed potential energy V(r).
As in the analysts of the Bohr atom, the
fact that the proton and electron both move is
completely accounted for by using the reduced
mass mr instead of the actual mass of the moving
electron (or proton).
The eigenvalues E are the energy levels of the
hydrogen atom in the frame of reference in which
the center of mass of the atom is at rest.

15
5.2 Energy Level of The Hydrogen Atom

Coulomb Potential and Effective Potential
Because Eq.(9.10) is identical to the
one-particle equation treated in Chapter 2, we
already know that this equation can be separated
into an angular equation and a radial equation
and that the function u can be written in
spherical coordinates as u(r,?,?)
R(r)Yl,m(?,?), where Yl,m(?,f) is a spherical
harmonic, a solution of the angular equation.

To find the energy levels, we need to solve only
the radial equation,
or, in this case
where l(l1)h2/2mrr2 is the centrifugal
potential. whose introduction, as we saw (Chapter
4), results from eliminating the angular
dependence in the equation.

Equation (9.11) is identical to the equation
for one-dimensional motion of a panicle in the
potential field Veff V(r) l(l 1)h2/2mrr2.
This effective potential, the sum of the Coulomb
potential and the centrifugal potential Vcent, is
sketched in Figure 9.1.
In hydrogen, an electron of negative total
energy is trapped in the potential well formed
by the effective potential. Classically, the
particle would describe an elliptical orbit under
these conditions it would oscillate between the
two values of r at which its total energy would
be equal to the potential energy.

FIGURE 9.1 The effective potential Veff (solid
line), the sum of the centrifugal potential
Vcent, and the Coulomb potential V(r) -e2/4pe0r2
for the hydrogen atom.

In quantum theory, we may find a wave function
for this well just as we did for the
one-dimensional wells considered before.
The general method of solution of Eq.(9.11)
begins with the assumption that the solution is
the product of a polynomial and an exponential.
Details of this solution are given in Appendix
E, which treats the general case of a
one-electron ion with nuclear charge Ze.
The polynomial contains n l terms, where l is
the angular momentum quantum number and n is a
new quantum number, the radial quantum number..

The statement that R(r) contains n - l, terms may
be considered to be the definition of n.
Degeneracy of Solutions
It is a curious feature of the solutions that the
energy depends only on n, not on l.
For example. the energy is the same for the l
1 solution with a two-term radial solution and
for the l 2 solution with a one-term radial
solution in both cases n 3.
The energy levels which result are identical to
the levels predicted by Bohr for the hydrogen
atom or any one-electron ion

although n now has a completely different
interpretation from that of Bohr.
Figure 9.2a shows the effective potentials
for l 0, 1, and 2 and the energy levels for n
l, 2, 3, and 4. Because the energy depends only
on n and not on l, the same levels which are
allowed for any given l are also allowed for all
lower values of l.
The only effect of l on the levels appears
through the condition that n l 1, so that
lower energy levels are possible for smaller l,
because smaller n values are then possible.

FIGURE 9.2 (a) Effective potential and energy
levels of the hydrogen atom for l 0, l 1, and
l 2. The lowest four levels are shown the
number of levels is infinite. (b) Radial
probability amplitudes rRnl(r). Notice the points
of inflection where E Veff, at the classical
turning points of the motion.

For example. E3 is an energy eigenvalue for
all three effective wells shown in Figure 9.2a.
but E2 is an eigenvalue only for l 1 and l 0.
The fact that all of these different wells have
the same set of energy levels is a remarkable
property peculiar to the Coulomb potential.
Because Eq.(9.11) is identical to the
one-dimensional Schreodinger equation, we can use
a graphical analysis, to gain more understanding
of the form of the eigentunctions.
As we saw in Chapter 3, the product rR(r) is the
probability amplitude for the radial coordinate.

Figure 9.2b shows graphs of the probability
amplitudes rRnl(r) for the radial probability
amplitudes rR20, rR21, and rR31, whose energy
eigenvaluse are shown in Figure 9.2a as E2, E2,
and E3, respectively.
Notice that the probability amplitude curves
away from the axis in the region where E lt Veff
(the classically forbidden region) and it curves
toward the axis, tending to oscillate, where E gt
Veff. the classical turning point, where E
Veff, is a point of inflection for the
eigenfunction.
You may also notice that the function rR31
curves more rapidly than rR21 in the allowed
region, because the curvature is proportional to
the kinetic energy.

As usual, each eigenfunction contains one more
node than the eigenfunction immediately lower in
energy.
This point is reflected in the fact that the
polynomial factor in R(r) has n - l terms, and
thus there are n - l roots to the equation R(r)
0.
Because each of the functions rR(r) is a
probability amplitude, its absolute square
is the probability density P(r) for finding
the electron at a radius between r and r dr.
(See Section 4.1 for details.)
Thus Figure 2 shows where the electron is likely
to be, as far as the r coordinate is concerned.

It indicates how the average radius of the
orbit increases as n increases it is evident
from the figure that this average radius must be
close to the value given by the original Bohr
theory.
Table of Wave Functions
The complete normalized wave functions
unlm(r,?,f) for the lowest energy states of the
hydrogen atom are given in Table 9.1.
These wave functions also apply to any
one-electron ion. if one use the appropriate
values for the atomic number Z and the nuclear
mass M.
The probability densities associated with some
of these wave functions are shown in Figure 9.3.

TABLE 9.1 Normalized wave Functions for Hydrogen
Atoms and Hydrogen like lons

Comparison with the Bohr Model Correspondence
Principle and Orbits in Hydrogen
Figure 1.7 shows elliptical orbits in the Bohr
model of the hydrogen atom, for n 4. The major
axis of each ellipse has a length of 16a0.

29
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30

If you measure the orbits, you see that the
most eccentric of these has a minimum value of r
that is less than a0 (0.053 nm), and the maximum
value of r in that orbit is greater than 31a0, or
about 1.64 nm.
In that orbit the angular momentum is equal
to h, so this would correspond to n 4 and l 1
in the Schreodinger equation.
The graph of the probability density for this
pair of quantum numbers is shown in Figure 9.4a.
You can see that the classically allowed
region does indeed extend from less than 0.5nm to
greater than 1.6nm.

FIGURE 9.4 Radial probability densities For
three different states of the hydrogen atom. (a)
For n 4, l 1, the allowed region stretches
from r lt 0.5 nm to r gt l.6 nm. (b) For n 4, l
3. the center of the allowed region is unchanged,
but the region is narrower, extending from r ?
0.6 to r ? 1.1 (c) For n 100, l 99, the
allowed region is centered on r 530nm. the
radius of the Bohr orbit for n 1OO. It extends
from r ? 490 to r ? 570. This is the narrowest
possible allowed region for this value of n. and
it is much narrower than curve (b), relative to
the value of ltrgt.

In Figure 9.4b, with the same n but l 3, the
allowed region is much narrower. because the
larger value of l means a less eccentric orbit.
For another example, you see in Figure 9.4c
that the allowed region for n 100, l 99 is
centered on approximately 530nm, just the Bohr
radius for n 100.
You recall that this state has the maximum
angular momentum of all the states with n 100,
and as such the function has no nodes.
Notice how much narrower the classically allowed
region appears with n 100.

It is actually broader, but as a fraction of the
radius it has become smaller. (See Exercise 9.)
Spectroscopic Symbols
For historical reasons associated with
observation of the various series of lines in
atomic spectra, the l value of each state is
designated by a letter, as follows
Letter s P d f g h i
l 0 l 2 3 4 5 6
The letters go, in alphabetical order for l
gt 3. Each state is then identified by the nunber
for n followed by the letter for l for example.
3d for n 3. l 2

34
5.3 Solution of the Radial Equation for the
Hydrogen Atom

Simplification of the radial equation
The radial Schroedinger equation is given by
(E1)
To simplify the subsequent equations, we
introduce the symbols

With these substitutions Eq.(E1) becomes
(E2)
To solve this equation, we begin with the limit
In that case (E3).
One solution of this equation is
and it can be shown that any function of
the form
is also a solution of Eq.(E3)
Therefore we write
, where g(r) is a power series whose form we
now seek.
Substitution into Eq.(E2) yields
(E6)

Substitution of the power series
into Eq.(E6) yields the following four series
To satisfy Eq.(E6), the sum of these four series
must equal zero for all values of r. This can be
true only

if the sum of the coefficients of each power of r
is zero.
The lowest power,with exponent s-2, appears only
twice. The sum of the coefficients of is
thus
The reasonable solution for s is sl1.
Setting the coefficients of equal to
zero gives us a relation
(E8)
The value of is determined by the
normalization condition and we can then determine
any of the

value b from Eq.(E8)
However,it can be shown that an infinite series
with these coefficients goes to infinity as r
tends to infinity.
We therefore need to find a value of
for which this series has a finite
number of terms. This will happen if one
coefficient, is zero Eq.(E8) will then
ensure that all succeeding coefficients will be
zero as well.
When this occurs (and ), Eq.(E8)
shows that
and therefore
(E10)

From the definitions of and C we now find
the energy in agreement with Eq.(9.12)
(E11)
where n is the radial quantum number,equal to
l1q. The possible values of n are therefore
l1, l2,lt, where t is the number of terms in
the radial function R(r).
Equation (E8) can be used to write each radial
function explicitly. For nl1we have only one
term, and with the help of Eq.(E5) we have

where the value of is determined by the
normalization.
When nl2, R(r) has two terms.Using Eq.(E8),
first with q0 and again with q1,we have
and so, from (E14),
(E15)
and Eq.(E13) then becomes
(E16)
and therefore
(E17)

where again is determined by normalization.
The polynomials in brackets in Eq.(E17), as well
as analogous polynomials for nl3, nl4, etc.,
are called Laguerre polynomials.

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