Chapter 6

Quantum Theory of the

Hydrogen Atom

6 Quantum Theory of the Hydrogen Atom

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6 Quantum Theory of the Hydrogen Atom

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• 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

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• 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.

• 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

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.

The potential energy U here is the electric potential energy

of a charge -e when it is the distance r from another charge +e.

(6.2 Electric potential energy)

(6.1)

6.1 Schrödinger’s Equation for the Hydrogen Atom

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• Since U is a function of r rather than of x, y, z, we cannot

substitute Eq. (6.2) directly into Eq. (6.1). 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.

The spherical polar coordinates r, θ,  of the point P shown in Fig. 6.1

have the following interpretations:

(Spherical polar coordinates)

6.1 Schrödinger’s Equation for the Hydrogen Atom

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• In spherical polar coordinates Schrödinger’s equation is written

• Substituting Eq. (6.2) for the potential energy U and multiplying the

entire equation by r

sin

θ, we obtain

(6.3)

(6.4)

• 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.

6.2 Separation of Variables

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• 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.

(6.5 Hydrogen atom wave function)

• 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.

6.2 Separation of Variables: Ψ=R Θ Φ

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• 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).

When we substitute R Θ Φ for in Schrödinger’s

equation for the hydrogen atom and divide the

entire equation by R Θ Φ , we find that

(6.6)

6.2 Separation of Variables: Ψ=R Θ Φ

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• 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.

• Rearrange Eq. (6.6) to read

• This equation can be correct only if both sides of it are equal to the same constant,

since they are functions of different variables.

• It is convenient to call this constant m

. The differential equation for the function

 is

Next we substitute m

for the right-hand side of Eq. (6.7), divide the

entire equation by sin

θ, and rearrange the various terms, which yields

(6.7)

(6.8)

(6.9)

• Again we have an equation in which different variables appear on each side,

requiring that both sides be equal to the same constant.

6.2 Separation of Variables: Ψ=R Θ Φ

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• This costant is called l(l+1). The equations for the functions Θ and R

are therefore

(6.10)

(6.11)

(6.12 Equations for Φ )

Equations (6.8), (6.10), and (6.11) are usually written

• Each of these is an ordinary differential equation for a single function

of a single variable.

• Only the equation for R depends on the potential energy U(r).

(6.13 Equations for Θ )

(6.14 Equations for R )

6.3 Quantum Numbers

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• Three dimensions, three quantum numbers.

• The first of these equations, Eq. (6.12), is readily solved. The result is

(6.15)

From Fig. 6.2, it is clear that  and +2p

both identify the same meridian plane.

Hence it must be true that Φ()=Φ(+2p), or

Figure 6.2 The angles

 and  +2π both

indentify the same

meridian plane.

which can happen only when m

is 0 or a positive or

negative integer (1, 2, 3, . . .).

• The constant m

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

, the absolute

value of m

. m

=0,1, 2, 3, . . ., l

• The constant l is known as the orbital quantum number.

6.3 Quantum Numbers

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• 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

(6.16)

• 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. [l=0,1,2,…,(n-1)]

• Hence, we may tabulate the three quantum numbers n, l, and m

together with their permissible values as follows:

(6.17)

• The electron wave functions of the hydrogen atom

6.3 Quantum Numbers

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6.3 Quantum Numbers

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Example 6.1

Find the ground-state electron energy E

by substituting the radial wave

function R that corresponds to n=1, l=0 into Eq. (6.14).

6.3 Quantum Numbers

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6.4 Principal Quantum Number

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• Quantization of energy.

• Two quantities are conserved (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

• The quantization of electron energy in the hydrogen atom is therefore

described by the principal quantum number n.

6.5 Orbital Quantum Number

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• 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

Inserting this expression for E in Eq. (6.14) we obtain, after a slight rearrangement,

(6.19)

If the last two terms in the square brackets of this equation cancel each other out: a

differential equation for R(r) that involves functions of the radius vector r exclusively.

(6.21 Electron angular momentum)

(6.20)

6.5 Orbital Quantum Number: Designation of Angular-Momentum States

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• 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 and one in which

n=4, l=2 is a 4d state.

6.6 Magnetic Quantum Number

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• Quantization of angular-momentum direction.

• The orbital quantum number l determines the magnitude

L of the electron’s angular momentum L.

• However, angular 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)

Figure 6.3 The right-hand

rule for angular momentum..

• 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.

• The magnetic quantum number m

specifies the direction of L by

determining the component of L in the field direction.

• This phenomenon is often referred to as space quantization.

6.6 Magnetic Quantum Number

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• If we let the magnetic-field direction be parallel to the z-axis, the

component of L in this direction is

• The number of possible orientations of the angular-momentum vector

L in a magnetic field is 2l+1. (When l=2, L

may be 2ħ, ħ, 0,- ħ,-2ħ).

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 m

• 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

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

ħ.

• What an external magnetic field does is to provide an

experimentally meaningful reference direction.

(6.22 Space quantization)

6.7 Electron Probability Density

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Figure 6.7 The Bohr model

of the hydrogen atom in a

pherical polar coordinate

system.

• 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

from the nucleus and in the equatorial plane θ=90

while its azimuth angle  changes with time.

The quantum theory of the hydrogen atom modifies the Bohr model in two ways:

1.No definite values for r, θ,  or can be given, but only the relative

probabilities for finding the electron at various locations. This

imprecision is a consequence of the wave nature of the electron.

2.We cannot even think of the electron as moving around the nucleus in

any conventional sense since the probability density |ψ|

independent of time and varies from place to place.

6.7 Electron Probability Density

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Figure 6.8 The Bohr model of the

hydrogen atom in a spherical polar

coordinate system.

• The probability density |ψ|

that corresponds to the electron

wave function ψ= RΘΦ in the hydrogen atom is

(6.23)

• 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.

6.7 Electron Probability Density: Probability of Finding the Electron

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Figure 6.9 Volume element dV

in spherical polar coordinates

• The probability density of the electron at the point r, θ, ,

is proportional to |ψ|

, but the actual probability of

finding it in the infinitesimal volume element dV there is

|ψ|

dV. In spherical polar coordinates (Fig. 6.9)

(6.24 Volume element3)

• 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

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.

• Equation (6.25) is plotted in Fig. 6.11 for the

same states whose radial functions R were

shown in Fig. 6.8.

• The most probable value of r for a 1s electron

turns out to be exactly a

, the orbital radius of a

ground-state electron in the Bohr model.

(6.25)

6.7 Electron Probability Density

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Example 6.2

Verify that the average value of 1/r for a 1s electron in the hydrogen

atom is 1/a

6.7 Electron Probability Density

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Example 6.3

How much more likely is a 1s electron in a hydrogen atom to be at the

distance a

from the nucleus than at the distance a

/2?

6.9 Selection Rules

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• 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.

(6.34)

• In the case of the hydrogen atom, three quantum numbers are needed

to specify the initial and final states involved in a radiative transition.

• If the principal, orbital, and magnetic quantum numbers of the initial

state are n’, l’, m

’, respectively, and those of the final state are n, l,

, and u represents either the x, y, or z coordinate, the condition for an

allowed transition is

(6.25 Allowed transitions)

6.9 Selection Rules

• 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 m

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.

(6.36)

(6.37)

The change in total quantum number n is not restricted.

Equations (6.36) and (6.37) are known as the selection

rules for allowed transitions (Fig. 6.13).

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

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.

(6.38)

• Set U

=0 when θ=π/2=90

, 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

=π/2 to the

angle θ that corresponds to that orientation. Hence

When μ points in the same direction as B, then U

=-μB, its minimum value.

6.10 Zeeman Effect

• The magnetic moment of a current loop has the magnitude μ=IA

where I is the current and A the area it encloses.

• An electron that makes f rev/s in a circular orbit of radius r is

equivalent to a current of -ef, and its magnetic moment is therefore

Figure 6.16 (a) Magnetic moment of a current loop

enclosing area A. (b) Magnetic moment of an orbiting

electron of angular momentum L.

Because the linear speed of the electron is 2pfr its angular momentum is

Comparing the formulas for magnetic moment and angular momentum L shows

that

(6.39 Electron magnetic moment8)

(6.42 Bohr magneton)

(6.41 Magnetic ennergy)

6.10 Zeeman Effect

• In a magnetic field, the energy of a particular atomic state depends on

the value of m

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.

• Because m

can have the 2l+1 values of +l through 0 to -l, a state of

given orbital quantum number l is split into 2l +1 substates that differ

in energy by μ

B when the atom is in a magnetic field.

6.10 Zeeman Effect

• Because changes in m

are restricted to m

=0, 1, we expect a

spectral line from a transition between two states of different l to be

split into only three components, as shown in Fig. 6.17.

• The normal Zeeman effect consists of the splitting of a spectral line

of frequency 

into three components whose frequencies are

Figure 6.17 17 In the normal Zeeman effect a spectral line of frequency υ

is split into three components when the radiating atoms are in a

magnetic field of magnitude B. One component is υ

and the others are less than and greater than υ

by U

There are only three components

because of the selection rule Δm

=0, ±1.

(6.43)

6.10 Zeeman Effect

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?

6 Solved Problems

1. Find the normalization constant of the ground state wave function for

a particle trapped in the one-dimensional Coulomb potential energy.

6 Solved Problems

2. In the ground state of an electron bound in a one-dimensional

Coulomb potential energy, what is the probability to find the electron

located between x = 0 and x = a

6 Solved Problems

3. An electron in a hydrogen atom is in a 3d state. What is the most

probable radius at which to find it? (Hint: use P(r) not P(r)dr and

then to find the maximum set the derivative dP(r)/dr to zero.))

6 Solved Problems

4. Prove that the most likely distance from the origin of an electron in

the n = 2, l = 1 state is 4a

6 Solved Problems

5. For the n = 2 states (l = 0 and l = 1), compare the probabilities of the

electron being found inside the Bohr radius.

6 Solved Problems

6. What is the expectation value of the radius of an electron in

hydrogen atom in a 3d state? Just write the expression without

evaluating the integral.

6 Solved Problems

7. A stone with mass 1.00 kg is whirled in a horizontal circle of radius

1.00 m with a period of revolution equal to 1.00 s. What value of

orbital quantum number l describes this motion?

6 Solved Problems

8. Consider an atomic electron in the l=3 state. Calculate the magnitude

|L| of the total angular momentum and the allowed values of L

and

.

6 Solved Problems

9. Compute the change in wavelength of the 2p → 1s photon when a

hydrogen atom is placed in a magnetic field of 2.00 T.

5 Quantum Mechanics

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