Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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6.1 The hydrogen atom 163

Figure 6.4 The electron

angular momentum, L.

= m

Figure 6.5 The electron

orbital magnetic moment,

, and angular

momentum, L.

an electron possesses a negative electric charge. Such a circular current

with the surface area S = πr

corresponds to an orbital magnetic moment

with a magnitude

= I

S =−

evr

. (6.27)

The orbital magnetic moment, µ

, is a vector, whose direction is perpendicular

to the plane of the electron orbit. The direction of the electron angular momentum,

L = r × p (see Fig. 6.4) is deﬁned by the right-hand rule and the magnitude of

the electron angular momentum is equal to

L = m

vr. (6.28)

The angular momentum, L, is also perpendicular to the plane of the electron

orbit. Since the charge of the electron is negative, the direction of the vector of

orbital magnetic moment, µ

, is opposite to the direction of the vector L (see

Fig. 6.5).

Let us introduce the so-called gyromagnetic ratio, γ , which deﬁnes the rela-

tion between the electron angular momentum and orbital magnetic moment

164 Quantum states in atoms and molecules

in an atom:

γ =



. (6.29)

Quantum-mechanical calculations of the gyromagnetic ratio, which will be con-

sidered in Example 6.2, lead to expression (6.29).

Thus, in any quantum-mechanical state the electron in the hydrogen atom has

not only angular momentum, L, with magnitude

L = h



l(l + 1), (6.30)

but also orbital magnetic moment µ

with magnitude

= γ L = µ



l(l + 1). (6.31)

Here, we introduced the constant µ

, known as the Bohr magneton:

= 9.27 × 10

−24

−1

, (6.32)

which serves as a unit for measuring the orbital magnetic moment of an atom.

The physical meaning of the orbital magnetic quantum number, m, can be

understood by taking into account that the wavefunction ψ

nlm

(r,ϕ,θ)ofan

electron in the hydrogen atom is simultaneously the eigenfunction of the operator

of the projection of angular momentum,

nlm

= mh

nlm

. (6.33)

It follows from Eq. (6.33) that the projection of angular momentum, L

, onto the

chosen direction (for example, the z-axis) can have only the following certain

values:

= mh

. (6.34)

The axis direction is chosen taking into account the direction of an external

magnetic or electric ﬁeld. The maximum value of L

= lh

is always smaller

than L, which is deﬁned by Eq. (6.30) (see Fig. 6.6). The possible values of the

projection of the atom’s orbital magnetic moment, µ

, along the z-axis, are also

quantized and are determined by the orbital magnetic quantum number, m:

=−γ L

=−mµ

. (6.35)

Example 6.2. Find the orbital magnetic moment of the hydrogen atom µ

, which

occurs because of the orbital motion of an electron and ﬁnd the gyromagnetic

ratio, γ .

Reasoning. Let us use Eq. (2.190) for the probability current, j. Since the wave-

function given in spherical coordinates depends on variables r , θ, and ϕ,for

the projections of the vector j onto the radial, polar, and azimuthal directions,

6.1 The hydrogen atom 165

Figure 6.6 Projections of

angular momentum, L,on

the z-axis. The maximum

value of L

for l = 2

zmax

= 2h

) is smaller

than the modulus of L

(|L|=L = 2h

√

1.5).

respectively, we get



nlm

∗

nlm

− ψ

∗

nlm



= 0,



nlm

dθ

∗

nlm

− ψ

∗

nlm

dθ

nlm



= 0,

r sin θ



nlm

dϕ

∗

nlm

− ψ

∗

nlm

dϕ

nlm



r sin θ

nlm

(6.36)

The projections j

and j

are equal to zero because the parts of the wavefunction

which depend on r and θ are real. In order to calculate the projection j

used the explicit form of the angular part of the wavefunction (Eq. (4.37)). The

probability current density, j, corresponds to the electric current density, j

,as

=−ej, j

=−ej

. (6.37)

Let us present the orbital magnetic moment of an atom as a sum of moments

from the elementary circular currents with radius r sin θ and cross-section ds.

The magnitude of such an elementary current is

= j

ds =−ej

ds. (6.38)

The area of the circular orbit is S = πr

sin

θ. The orbital magnetic moment

created by the elementary current is equal to

dµ

= S dI

=−πr sin θ

meh

nlm

ds. (6.39)

The total orbital magnetic moment, µ

, may be written as

=−

meh



nlm

dV =−

m, (6.40)

where dV = 2πr sin θ ds is the volume of a tube, and integration is carried out

over the entire space. Taking into account that L = h

m, we ﬁnd the gyromagnetic

166 Quantum states in atoms and molecules

n=1

n=2

n=3

n=4

free electron

Balmer

series

Lyman

series

Figure 6.7 The electron

energy spectrum in the

hydrogen atom; E

, E

,andE

are the

discrete energy levels of

theelectroninthe

potential well of the

hydrogen atom.

ratio, γ :

γ =



. (6.41)

6.2 The emission spectrum of the hydrogen atom

Figure 6.7 shows the electron energy spectrum of the hydrogen atom, which was

calculated using Eq. (6.7). If the electron has positive energy then it is free and

its energy spectrum is continuous as shown in Fig. 6.7. In the region of negative

values of the total energy the energy spectrum of the electron of the hydrogen

atom is discrete. In the stationary state the electron can be in one of the energy

states with E

< 0. The undisturbed atom is in its ground state with energy E

.If

the atom is given additional energy, i.e., if the atom is excited, then it may make

a transition to a state with higher energy E

> E

(see Fig. 6.7, where transition

1 for the electron from level E

to level E

is shown). If the energy given to the

atom is sufﬁciently high, then the electron may overcome the Coulomb attraction

of the nucleus and become free. This process is called ionization of the atom.The

minimal energy necessary for ionization, E

, of a hydrogen atom (see transition

2inFig.6.7)is|E

|=13.6eV.

An atom may stay in the excited state only for a short time τ . This time

is called the lifetime of the atom in the given quantum state and for different

quantum states it can be of the order of τ = 10

−6

–10

−10

s. During time τ the

electron spontaneously makes transitions to levels with lower energy, including

the ground state. During such a transition the atom emits an energy quantum in

the form of electromagnetic radiation, i.e., the atom emits a photon. Experiments

show that the emission from hydrogen atoms corresponds to certain regions of

the electromagnetic spectrum and that it has the form of narrow spectral lines.

6.2 The emission spectrum of the hydrogen atom 167

Knowing the structure of the energy levels, it is easy to ﬁnd the structure

of the optical emission spectrum of a hydrogen atom. During the transition of

the electron from the mth energy level to the nth level (m > n, E

> E

), the

electron loses energy E

− E

by emitting a photon with angular frequency

− E

. (6.42)

By substituting the expressions for energies E

and E

deﬁned by Eq. (6.7)into

Eq. (6.42) we obtain the equation for the emission frequency of the hydrogen

atom for the different transitions from higher energy levels to lower:

= 2π cR

∞



−



, m > n. (6.43)

Here, we introduced the so-called Rydberg constant R

∞

4π h



4π



= 1.0974 × 10

−1

, (6.44)

whose value was ﬁrst found experimentally; later it was obtained theoretically

by Niels Bohr.

According to Eq. (6.43), there must be a series of spectral lines in the emission

spectrum of a hydrogen atom corresponding to different parts of the optical

spectrum. Such series were experimentally found a long time before they were

theoretically predicted. The most well known of these series are the following.

The Lyman series

The Lyman series corresponds to transitions to the ground level n = 1from

higher energy levels with m = 2, 3, 4,...Its spectral lines are in the ultraviolet

region. The minimal emission frequency in this series is ω

= 1.55 × 10

−1

which corresponds to the maximal wavelength

2πc

= 121.4nm. (6.45)

The maximal frequency, which corresponds to the short-wavelength boundary,

i.e., when m →∞,isω

1∞

= 2.07 × 10

−1

, which corresponds to λ

1∞

91 nm.

The Balmer series

The Balmer series has the following n and m numbers: n = 2 and m = 3, 4, 5,...

The spectral lines of this series are in the visible range. The structure of these

spectral lines is shown in Fig. 6.8. The symbols Hα,Hβ,Hγ, and Hδ label

characteristic lines of this series. The spectral line Hr is the short-wavelength

boundary of the series, which corresponds to m →∞and maximal frequency

2∞

= 5.17 × 10

−1

. The spectral line Hα corresponds to the minimal fre-

quency ω

= 2.87 × 10

−1

. The color of Hα is red and Hδ is in the ultraviolet

region.

168 Quantum states in atoms and molecules

, nm

656.3

486.1 434.0 410.2 364.6

Hαβ

HHH H

E,eV

1.88 2.55 2.86

3.02

3.40

Figure 6.8 The Balmer

series. The symbols Hα,

Hβ,Hγ,andHδ label

characteristic spectral

lines of this series. Hα is a

red line, whereas Hδ is in

the ultraviolet region.

The Paschen series

The Paschen series has the following n and m numbers: n = 3 and m =

4, 5, 6,... All spectral lines of this series are in the near-infrared region of

the electromagnetic spectrum. The Brackett (n = 4, m ≥ 5) and Pfund (n = 5,

m ≥ 6) series are in the same spectral region.

The absorption spectra have the same spectral rules. Absorption occurs when

an electron in a lower energy state absorbs a photon and is promoted to a

higher energy state. Thus, emission and absorption of energy quanta in the

form of photons happen as a result of electron transitions from one energy level

to another. However, not all the transitions in an atom are allowed. The only

transitions allowed are those between quantum states whose orbital quantum

numbers, l, differ by unity:

l =±1, (6.46)

i.e., transitions between s- and p-states, between p- and d-states, and so on are

allowed. At the same time the orbital magnetic quantum number, m, must either

stay the same or change by unity, i.e.,

m = 0, ±1. (6.47)

Equations (6.46) and (6.47) are called the selection rules. They are derived from

the calculations of transition probabilities between different stationary states with

emission or absorption of a photon.

The existence of these rules for the quantum transitions of an electron is related

to the conservation of the square of angular momentum, L

, and its projection

. Indeed, as a result of the corresponding transition of an electron from one

energy level to another a photon is emitted (or absorbed). The photon takes away

(or transmits to the atom) not only an energy quantum, E = h

ω, but also a

quantum of angular momentum, L = h

. This fact leads to the condition deﬁned

by Eq. (6.46) for the orbital quantum number, l.

The selection rule for the orbital magnetic quantum number, m =±1, shows

that the emitted or absorbed photon may impart to the atom angular momentum

with a certain projection on a chosen axis. This projection may be equal to

±h

, which corresponds to the left-handed or right-handed circular polarization

6.3 The spin of an electron 169

of the photon. The selection rule m = 0 shows that a photon having angular

momentum h

does not have a certain projection on the chosen axis and does not

take away (or transmit) any projections of angular momentum. This happens in

the case when the photon has linear polarization, which can be presented as a

sum of two opposite circular polarizations – left-handed and right-handed.

The study of spectral lines using high-resolution spectrometers revealed that

the spectral lines that correspond to the transitions between the energy levels with

l ≥ 1aredoublets. The energy difference between the lines constituting doublets

is very small. Therefore, it is said that spectral lines have ﬁne structure. Such a

splitting of spectral lines may be connected only with the splitting of energy levels

themselves. Indeed, the experimentally established value of the splitting of the

energy level of the 2p-state of the hydrogen atom is equal to E = 4.5 ×10

−5

and the corresponding splitting frequency of the main spectral line of the Lyman

series is equal to ν = ω/(2π) = 11 GHz. However, such a splitting cannot

be obtained from the solutions of the Schr¨odinger equation which we wrote for

the electron in a spherically-symmetric potential. Understanding the atomic ﬁne

structure became possible only after the discovery that the electron has intrinsic

angular momentum and an intrinsic magnetic moment not related to the electron

orbital motion. In previous chapters we have mentioned this property called spin,

and now we will consider it in more detail.

6.3 The spin of an electron

The following experimental facts that could not be explained by the quantum

theory developed by Bohr and Schr¨odinger forced scientists to revise their con-

cepts.

1. Analysis of the spectral lines of alkali metals shows that p-, d-, etc. terms are doublets,

i.e., consist of two closely spaced spectral lines, whereas the s-term stays a singlet.

2. The Stern–Gerlach experiment showed that a beam of silver atoms splits into two

beams in the presence of an inhomogeneous external magnetic ﬁeld, a result that was

not expected. Since the outermost electron in a silver atom is in its ground state its

orbital quantum number, l, must be equal to zero and thus its orbital magnetic quantum

number, m, must also be equal to zero, and the beam must not split.

3. Many atoms in the presence of an external magnetic ﬁeld have even multiplets in the

spectra, whereas it was expected that multiplets must be only odd:2l + 1 for all values

of l is odd.

To explain these experimental facts George Uhlenbeck and Samuel Goudsmit

suggested in 1925 that an electron, in addition to its orbital motion, also has

intrinsic angular momentum, S, which they called spin. Later it was established

that spin is not connected with the rotation of an electron around its own axis

as was ﬁrst suggested, but is a quantum-mechanical and at the same time a

relativistic internal property of the electron. Together with the spin angular

170 Quantum states in atoms and molecules

momentum, S, an electron has an intrinsic spin magnetic moment, µ

. Similarly

to angular momentum, the values of electron spin angular momentum, S, and

spin magnetic moment, µ

, are deﬁned by the same expressions:

S = h



s(s + 1), (6.48)

= 2µ



s(s + 1), (6.49)

where s is the spin quantum number, which is also called simply spin.

It has been established experimentally that for electrons s = 1/2, and therefore

S =

√

, (6.50)

√

3µ

. (6.51)

The spin gyromagnetic ratio, which relates the electron spin magnetic moment,

, and spin angular momentum, S,asµ

= γ

S, is equal to

2µ

. (6.52)

Its magnitude is twice the orbital gyromagnetic ratio deﬁned by Eq. (6.29).

As in the case of orbital motion, the magnitude of the spin angular momentum,

S, may have the following values:

= h

s(s + 1). (6.53)

Its projections onto an arbitrarily chosen axis (for example the z-axis) are equal

= h

, (6.54)

where the magnetic quantum number, m

, is called the spin magnetic quantum

number and may take 2s + 1 values:

= s, s − 1,...,−s + 1, −s. (6.55)

Since for an electron s = 1/2, two spin states are possible. These states corre-

spond to two different projections with m

=±1/2 of spin angular momentum

onto the chosen axis (see Fig. 6.9).

The existence of the electron spin magnetic moment, µ

, and orbital magnetic

moment, µ

, leads to an additional interaction called spin–orbit coupling.Its

energy is described by

(r) =

4π

· µ

, (6.56)

where r is the radius of the corresponding orbit of the electron in the atom. The

spin–orbit coupling is not taken into account in the Schr¨odinger equation (6.2).

The existence of such an interaction leads to the splitting of energy levels that

correspond to the states with orbital quantum number l ≥ 1.

6.3 The spin of an electron 171

= +1/2

)

1/2)

Figure 6.9 Projections of

spin angular momentum,

S, onto the chosen z-axis

with two different values

of m

1/2

3/2

5/2

3/2

1/2

= 11 GHz

Figure 6.10 The splitting

of energy levels due to

spin–orbit coupling in the

hydrogen atom. The

quantum transitions that

belong to the Lyman and

Balmer series are shown.

Figure 6.10 shows the scheme of the splitting for the lower levels of the

hydrogen atom. It also shows the quantum transitions which correspond to two

doublet lines in the Lyman and Balmer series.

The quantum state of the electron in the hydrogen atom can be deﬁned by the

set of four quantum numbers whose values are given in Table 6.3.

Let us note again that many elementary particles have their own intrinsic

angular momentum, i.e., spin. Particles whose spin is a half-integer are called

Fermi particles or fermions. Electrons, protons, and neutrons are examples of

fermions. For these particles s = 1/2. For other types of particles spin is an

172 Quantum states in atoms and molecules

Table 6.3. The four quantum numbers, n, l, m, and m

, which deﬁne

the state of an electron in a hydrogen atom

Quantum number Symbol Possible values

Principal n 1, 2, 3, . . .

Orbital l 0, 1, 2,...,n − 1

Magnetic (orbital) m −l, −l + 1,...,−1, 0, 1,...,l − 1, l

Magnetic (spin) m

−1/2, +1/2

integer and these particles are called Bose particles or bosons. Some of the

mesons (π and K), photons, and the α-particle (the nucleus of a

helium

atom) are bosons. Ensembles of particles consisting of Fermi and Bose particles

behave differently, and we will talk about this subject later.

Example 6.3. Estimate the splitting of energy level E

in the hydrogen atom

which is due to spin–orbit coupling. Compare it with the splitting caused by

the interaction of the spin magnetic moment of the electron, µ

, with the spin

magnetic moment of the nucleus (proton), µ

Reasoning. The magnitude of splitting of the E

energy level has to be of the

order of the energy of dipole–dipole interaction of the corresponding magnetic

moments, which is deﬁned by Eq. (6.56):

(r) =

4π

· µ

, (6.57)

(r) =

4π

· µ

. (6.58)

Here, r is the effective distance between the magnetic moments. This distance is

of the order of the radius of the second Bohr orbit, r

= 2.1 × 10

−10

m, for both

types of interaction. The magnitudes of the corresponding magnetic moments

are

,µ

= 2.8

, (6.59)

where m

= 1.67 × 10

−27

kg is the proton mass. Thus, the energy separations

between the sublevels for the two cases are deﬁned by the expressions

E

≈

4π

≈ 1.1 × 10

−5

eV, (6.60)

E

≈ 1.4

4π

≈ 1.25 × 10

−8

eV. (6.61)

The estimated values agree to within an order of magnitude with the experimental

data as well as with the theoretical results which were obtained on the basis of

exact quantum-mechanical calculations. For example, the exact value of E

is 4.5 ×10

−5

eV. We see that E

caused by the spin–orbital coupling is three

orders of magnitude larger than E

. Therefore, the splitting of the spectral