Chapter 6: Quantum Theory of the Hydrogen Atom
• The first problem that Schrödinger tackled with his new wave equation was that of the
hydrogen atom.
• The discovery of how naturally quantization occurs in wave mechanics:
◦ “It has its basis in the requirement that a certain spatial function be finite and single-
valued.”
6.1 SCHRÖDINGER’S EQUATION FOR THE HYDROGEN ATOM
• Symmetry suggests spherical polar coordinates.
• A hydrogen atom consists of a proton, a particle of electric
charge +e, and an electron, a particle of charge -e which is 1836
times lighter than the proton.
• For the sake of convenience, we shall consider the proton to be
stationary, with the electron moving about in its vicinity but
prevented from escaping by the proton’s electric field.
Schrödinger’s equation for the electron in three dimensions, which is
what we must use for the hydrogen atom, is
• Since U is a function of r rather than of x, y, z, we cannot
substitute Eq. (6.2) directly into Eq. (6.1).
• There are two alternatives.
1. One is to express U in terms of the cartesian coordinates x, y, z
by replacing r by
2. The other is to express Schrödinger’s equation in terms of the
spherical polar coordinates r, θ, defined in Fig. 6.1.
Figure 6.1 (a) Spherical
polar coordinates.
(b) A line of constant
zenith angle θ on a sphere
is a circle whose plane is
perpendicular to the z
axis. (c) A line of
constant azimuth angle
is a circle whose plane
includes the z axis.
•
• Equation (6.4) is the partial differential equation for the wave function ψ of the electron in a
hydrogen atom.
• Together with the various conditions ψ must obey,
◦ ψ be normalizable
◦ ψ and its derivatives be continuous and single-valued at each point r, θ, this equation
completely specifies the behavior of the electron.
• In order to see exactly what this behavior is, we must solve Eq. (6.4) for ψ.
• A particle in a three-dimensional box needs three quantum numbers for its description, since
there are now three sets of boundary conditions that the particle’s wave function ψ must
obey:
◦ ψ must be 0 at the walls of the box in the x, y, and z directions independently.
◦ In a hydrogen atom the electron’s motion is restricted by the inverse-square electric field
of the nucleus instead of by the walls of a box.
6.2 SEPARATION OF VARIABLES
• A differential equation for each variable.
• Here the wave function ψ (r, θ, ) has the form of a product of three different functions:
1. R(r) which depends on r alone;
2. Θ(θ) which depends on θ alone;
3. Φ() which depends on alone.
• The function R(r) describes how the wave function ψ of the electron varies along a radius
vector from the nucleus, with θ and constant.
• The function Θ(θ) describes how ψ varies with zenith angle θ along a meridian on a sphere
centered at the nucleus, with r and constant (Fig. 6.1c).
• The function Φ() describes how ψ varies with azimuth angle along a parallel on a sphere
centered at the nucleus, with r and θ constant (Fig. 6.1b).
• The third term of Eq. (6.6) is a function of azimuth angle only, whereas the other terms are
functions of r and θ only.
6.3 QUANTUM NUMBERS
• Three dimensions, three quantum numbers.
• The first of these equations, Eq. (6.12), is readily solved. The result is
• From Fig. 6.2, it is clear that and +2 both identify the same
meridian plane. Hence it must be true that Φ()=Φ(+2), or
which can happen only when m
l
is 0 or a positive or negative integer (±1, ±2,
±3, . . .).
Figure 6.2 The angles and +2π both indentify the same meridian plane.
• The constant m
l
is known as the magnetic quantum number of the hydrogen atom.
• The differential equation for Θ(θ), Eq. (6.13), has a solution provided that the constant l is
an integer equal to or greater than m
l
, the absolute value of m
l
.
• The constant l is known as the orbital quantum number.
• The solution of the final equation, Eq. (6.14), for the radial part R(r) of the hydrogen atom
wave function also requires that a certain condition be fulfilled.
• Recognize that this is precisely the same formula for the energy levels of the hydrogen atom
that Bohr obtained.
• Another condition that must be obeyed in order to solve Eq. (6.14) is that n, known as the
principal quantum number, must be equal to or greater than l+1.
• Hence, we may tabulate the three quantum numbers n, l, and m together with their
permissible values as follows:
• The electron wave functions of the hydrogen atom
Example 6.1
Find the ground-state electron energy E
1
by substituting the radial wave function R that corresponds
to n=1, l=0 into Eq. (6.14).
6.4 PRINCIPAL QUANTUM NUMBER
• Quantization of energy.
• Two quantities are conserved -that is, maintain a constant value at all times- in planetary
motion:
• the scalar total energy
• the vector angular momentum of each planet.
• Classically the total energy can have any value whatever, but it must, of course, be negative
if the planet is to be trapped permanently in the solar system.
• In the quantum theory of the hydrogen atom the electron energy is also a constant, but while
it may have any positive value (corresponding to an ionized atom), the only negative values
the electron can have are specified by the formula E
n
=E
1
/n
2
.
• The quantization of electron energy in the hydrogen atom is therefore described by the
principal quantum number n.
6.5 ORBITAL QUANTUM NUMBER
• Quantization of angular-momentum magnitude.
• The kinetic energy KE of the electron has two parts, KE
radial
due to its motion toward or
away from the nucleus, and KE
orbital
due to its motion around the nucleus.
• The potential energy U of the electron is the electric energy
• Hence the total energy of the electron is
• This peculiar code originated in the empirical classification of spectra into series called
sharp, principal, diffuse, and fundamental which occurred before the theory of the atom
was developed.
• Thus, an s state is one with no angular momentum, a p state has the angular moment,
and so forth.
• The combination of the total quantum number with the letter that represents orbital angular
momentum provides a convenient and widely used notation for atomic electron states.
• In this notation, a state in which n=2, l=0 is a 2s state, for example, and one in which n=4,
l=2 is a 4d state. Table 6.2 gives the designations of electron states in an atom through n=6,
l=5.
6.6 MAGNETIC QUANTUM NUMBER
• Quantization of angular-momentum direction.The orbital quantum number l determines the
magnitude L of the electron’s angular momentum L.
• However, angular momentum, like linear momentum, is a vector
quantity, and to describe it completely means that its direction be
specified as well as its magnitude. (see Fig. 6.3)
• What possible significance can a direction in space have for a
hydrogen atom? The answer becomes clear when we reflect that an
electron revolving about a nucleus is a minute current loop and has a
magnetic field like that of a magnetic dipole.
• Hence an atomic electron that possesses angular momentum
interacts with an external magnetic field B.
Figure 6.3 The right-hand rule for angular momentum.
• The magnetic quantum number m
l
specifies the direction of L by determining the
component of L in the field direction. This phenomenon is often referred to as space
quantization.
• If we let the magnetic-field direction be parallel to the z axis, the component of L in this
direct
ion is
• The space quantization of the orbital angular momentum of the
hydrogen atom is show in Fig. 6.4.
• An atom with a certain value of m
l
will assume the corresponding
orientation of its angular momentum L relative to an external magnetic
field if it finds itself in such a field.
• In the absence of an external magnetic field, the direction of the z axis
is arbitrary.
• What must be true is that the component of L in any direction we
choose is m
l
h.
• What an external magnetic field does is to provide an experimentally
meaningful reference direction.
Figure 6.4 Space quantization of orbital angular momentum. Here the orbital quantum number
is l=2 and there are accordingly 2l+1=5 possible values of the magnetic quantum number m
l
,
with each value corresponding to a different orientation relative to the z-axis.
6.7 ELECTRON PROBABILITY DENSITY
• No definite orbits.
• In Bohr’s model of the hydrogen atom the electron is visualized as revolving around the
nucleus in a circular path. This model is pictured in a spherical polar coordinate system in
Fig. 6.7.
• It implies that if a suitable experiment were performed, the electron would always be found
a distance of r=n
2
a
0
from the nucleus and in the equatorial plane θ=90
o
, while its azimuth
angle changes with time.
• The quantum theory of the hydrogen atom modifies the Bohr model in two ways:
• From Eq. (6.15) we see that the azimuthal wave function is given by
• The likelihood of finding the electron at a particular azimuth angle
is a constant that does not depend upon at all.
• The electron’s probability density is symmetrical about the z axis
regardless of the quantum state it is in, and the electron has the
same chance of being found at one angle as at another.
• The radial part R of the wave function, in contrast to Φ, not only
varies with r but does so in a different way for each combination
of quantum numbers n and l.
• Figure 6.8 contains graphs of R versus r for 1s, 2s, 2p, 3s, 3p, and
3d states of the hydrogen atom.
◦ Evidently R is a maximum at r 0 -that is, at the nucleus itself-
for all s states, which correspond to L=0 since l=0 for such
states.
◦ The value of R is zero at r=0 for states that possess angular
momentum.
Figure 6.8 The variation with distance from the nucleus of the radial part of the electron
wave function in hydrogen for various quantum states. The quantity a
0
=0.053 nm is the radius of the first Bohr orbit.
Probability of Finding the Electron
• As Θ and Φ are normalized functions, the actual probability P(r)dr of finding the electron in
a hydrogen atom somewhere in the spherical shell between r and r+dr from the nucleus is
• Equation (6.25) is plotted in Fig. 6.11 for the
same states whose radial functions R were
shown in Fig. 6.8. The curves are quite
different as a rule.
• The most probable value of r for a 1s electron
turns out to be exactly a
0
, the orbital radius of
a ground-state electron in the Bohr model.
Figure 6.11 The probability of finding the electron in a hydrogen
atom at a distance between r and r+dr from the nucleus for the
quantum states of Fig. 6.8.
Example 6.2
Verify that the average value of 1/r for a 1s electron in the hydrogen atom is 1/a
0
.
Example 6.3
How much more likely is a 1s electron in a hydrogen atom to be at the distance a
0
from the nucleus
than at the distance a
0
/2?
6.9 SELECTION RULES
• Some transitions are more likely to occur than others.
• The general condition necessary for an atom in an excited state to radiate is that the integral
not be zero, since the intensity of the radiation is proportional to it.
• Transitions for which this integral is finite are called allowed transitions, while those for
which it is zero are called forbidden transitions.
• It is found that the only transitions between states of different n that can occur are
◦ those in which the orbital quantum number l changes by +1 or -1
◦ and the magnetic quantum number ml does not change or changes by +1 or -1.
• That is, the condition for an allowed transition is that
Figure 6.13 Energy-level diagram for hydrogen showing
transitions allowed by the selection rule Δl=±1. In this
diagram, the vertical axis represents excitation energy above
the ground state.
6.10 ZEEMAN EFFECT
• How atoms interact with a magnetic field.
• In an external magnetic field B, a magnetic dipole has an
amount of potential energy U
m
that depends upon both the
magnitude of its magnetic moment and the orientation of this
moment with respect to the field (Fig. 6.15).
• The torque on a magnetic dipole in a magnetic field of flux
density B is τ=μBsinθ.
Figure 6.15 A magnetic dipole of moment at the angle relative to a magnetic field B.
• Set U
m
=0 when θ=π/2=90
o
, that is, when μ is perpendicular to B.
• The potential energy at any other orientation of μ is equal to the external work that must be
done to rotate the dipole from θ
0
=π/2 to the angle θ that corresponds to that orientation.
Hence
When μ points in the same direction as B, then U
m
=-μB, its minimum value.
• The magnetic moment of a current loop has the magnitude
Figure 6.16 (a) Magnetic moment of a current
loop enclosing area A. (b) Magnetic moment of
an orbiting electron of angular momentum L.
• In a magnetic field, then, the energy of a particular atomic state depends on the value of m
l
as well as on that of n.
• A state of total quantum number n breaks up into several substates when the atom is in a
magnetic field, and their energies are slightly more or slightly less than the energy of the
state in the absence of the field.
• This phenomenon leads to a “splitting” of individual spectral lines into separate lines when
atoms radiate in a magnetic field. The spacing of the lines depends on the magnitude of the
field.
• The splitting of spectral lines by a magnetic field is called the Zeeman effect (first observed
in 1896). The Zeeman effect is a vivid confirmation of space quantization.
Figure 6.17 In the normal Zeeman effect
a spectral line of frequency υ
0
is split
into three components when the
radiating atoms are in a magnetic field
of magnitude B. One component is υ
0
and the others are less than and greater
than υ
0
by U
m.
There are only three
components because of the selection rule
Δm
l
=0, ±1.
Example 6.4
A sample of a certain element is placed in a 0.300 T magnetic field and suitably excited. How far
apart are the Zeeman components of the 450-nm spectral line of this element?
