Chen C.J. Physics of Solar Energy

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Chapter 7

Quantum Transitions

As described in Chapter 2, sunlight embodies a wave–particle duality: it is both an elec-

tromagnetic wave and a stream of particles of indivisible energy packages, the photons.

The utilization of solar energy can be divided into two categories. The ﬁrst category

is related to and can be explained by its wave nature, such as generating heat and the

antireﬂection coatings for solar cells. The second category is related to the quantum

nature of light. The light quanta, or photons, excite atoms, molecules, and solids from

the ground state to an excited state, then convert part of the energy into electric power

or chemical energy, such as various types of solar cells and photosynthesis.

In this chapter, we present the conceptual basis of the eﬀect of sunlight as a stream

of photons, that is, quantum mechanics. It includes the quantum states of atomic

systems, its energy levels and wavefunctions, and transitions due to interactions with

radiation.

7.1 Basic Concepts of Quantum Mechanics

Quantum mechanics is a branch of physics which deals with the phenomena at the

atomic scale. It is a conceptual and mathematical description of how nuclei and elec-

trons come together to form condensed matter, how the system evolves over time, and

how the system interacts with radiation. The main diﬃculty of learning quantum me-

chanics is because the scale of atomic phenomena is below 1 nm, and quite diﬀerent

from the phenomena at the macroscopic scale. For example, Newtonian mechanics is

a conceptual and mathematical description of macroscopic phenomena. Using a tele-

scope, the motion of celestial bodies can be observed and measured directly for the

veriﬁcation of Newtonian mechanics. On the other hand, using optical microscopes,

micrometer-scale objects, such as cells in biology, can be observed, veriﬁed, and under-

stood directly. Only recently have the phenomena at the atomic scale, the entities of

quantum mechanics, become directly observable using scanning tunneling microscopy

(STM) and atomic force microscopy (AFM).

7.1.1 Quantum States: Energy Levels and Wavefunctions

In Newtonian mechanics, there are well-deﬁned concepts thato describe the physical

phenomena: position, velocity, acceleration, momentum, angular momentum, mass,

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Physics of Solar Energy C. Julian Chen

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144  Quantum Transitions

and force. Newton’s laws are the natural laws behind these phenomena. The motion

of the planets in the solar system and the motion of the moons of the planets were the

ﬁrst and still among the best demonstrations of Newtonian mechanics.

The structure of an atom is similar to a miniature solar system: a heavy, positively

charged proton surrounded by a number of light, negatively charged electrons. However,

classical mechanics is totally inadequate for describing the atomic systems. It fails

even to explain the fact that the atomic system can exist as a stable entity. Just

as Newtonian mechanics is used to describe experimental observations of the motions

of celestial bodies and macroscopic objects on Earth, quantum mechanics is used to

describe the motions of objects at the atomic scale. Instead of the concepts of position,

velocity, acceleration, momentum, angular momentum, mass, and force and so on, the

basic concept of quantum mechanics is the state. According to Dirac, a quantum state

is deﬁned as one of the various possible motions of the particles or bodies consistent

with the laws of force and is represented by either a bra vector, |,oraket vector, |.

For conﬁned systems, such as atoms and molecules, the states are discrete and can be

labeled by discrete quantum numbers, such as

n|, |n : n =0, 1, 2, ...; (7.1)

Quantum states in inﬁnite space are labeled by continuous variables. For example, a

plane-wave state can be labeled by a wave vector k,

k|, |k : k =(k

). (7.2)

The states can be represented either by a vector with complex elements of inﬁnite

rank or by a complex function of the space coordinates ψ(r), the wavefunction. For the

case of a vector, the elements or the values of a bra are the complex conjugates of the

elements or the values of a ket. For the case of a wavefunction, the bra is represented

by the complex conjugate of that of the ket wavefunction, ψ

∗

(r).

The terms bra and ket are taken from the word bracket. The expression of a complete

bracket represents the inner product of a vector,

a|b≡

∞



n=0

∗

, (7.3)

where a

∗

is the complex conjugate of a

, or the space integral of the wavefunctions,

ψ|χ≡



∗

(r)χ(r)d

(r). (7.4)

7.1.2 Dynamic Variables and Equation of Motion

The dynamic variables are expressed as operators to the states. If a state is represented

by a vector, then the operator is represented by a matrix. For clarity, we use the

convention that an operator is marked by a hat on a letter. For example,

|b =

V |a (7.5)

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7.1 Basic Concepts of Quantum Mechanics  145

means

. (7.6)

In order for the values of a dynamic variable are real, the matrix must be Hermitian,

which means that the matrix must have the properties

= V

∗

. (7.7)

Consequently, the diagonal elements of the operator must be real. If an operator is not

Hermitian, one can deﬁne an adjoint operator

†

such that

†

= A

∗

. (7.8)

It is easy to show that both

†

A and i(

†

−

A) are Hermitian.

According to the creators (Heisenberg, Schr¨odinger and Dirac), quantum mechan-

ics is formulated from the canonical form of classical mechanics with coordinate q, its

conjugate momentum p, and the Hamiltonian function of q and p, H(p, q). The quan-

tum mechanics of the system is constructed by considering the dynamic variables as

operators which satisfy the commutation relation

ˆq ˆp − ˆpˆq = i~, (7.9)

where ~ is Dirac’s constant, equal to the Planck constant h divided by 2π. For example,

in the wavefunction representation, the momentum ˆp

ˆp

|Ψi = −i~

∂

∂x

|Ψi. (7.10)

Then the commutation relation (Eq. 7.9), can be readily veriﬁed. A quantum state is

in general time dependent, the evolution of which follows the equation of motion

∂

∂t

|Ψi =

H|Ψi, (7.11)

where

H is the Hamiltonian of the system as a function of the coordinate ˆq and mo-

mentum ˆp. For a closed system, the Hamiltonian cannot contain time explicitly. In

this case, Eq. 7.11 has a series of stationary state solutions,

|Ψi = e

−iE

t/~

|ψi. (7.12)

Here we use a capital letter to denote a state with time variation and a lowercase letter

to denote the time-independent part of the state. The stationary quantum state of a

system |ψi is the solution of the Schr¨odinger’s equation

H|ψi = E|ψi. (7.13)

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146  Quantum Transitions

7.1.3 One-Dimensional Potential Well

An instructive problem in quantum mechanics is an electron in a one-dimensional

potential well; see Fig. 7.1. The Hamiltonian is

H =

ˆp

+ V (x), (7.14)

where

V (x)=0, 0 <x<L,

V (x)=∞,x<0andx>L.

(7.15)

In terms of coordinate x,Schr¨odinger’s equation 7.13 becomes



−



+ V (x)



ψ(x)=Eψ(x). (7.16)

Inside the box, 0 <x<L, V (x) = 0. Equation 7.16 is reduced to

−



ψ(x)

= Eψ(x). (7.17)

At x = 0 and x = L, the wavefunction should vanish because the electron cannot

penetrate an inﬁnitely high potential barrier, which implies two boundary conditions

ψ(0) = 0, (7.18)

ψ(L)=0. (7.19)

The solution satisfying the ﬁrst boundary condition (Eq. 7.18) is

ψ(x)=C sin(kx), (7.20)

where C is a constant, and

k =

√



. (7.21)

The second boundary condition (Eq. 7.19) requires

ψ(L)=C sin(kL)=0, (7.22)

which implies

k =

nπ

, (7.23)

where n is an integer. From Eq. 7.21, we obtain the energy levels

E =



. (7.24)

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7.1 Basic Concepts of Quantum Mechanics  147

Figure 7.1 One-dimensional potential well. Energy levels and wavefunctions of an electron in a

one-dimensional potential well.

The constant C is determined by the normalization condition

|ψ(x)|

dx =

sin

(kx)dx = 1, (7.25)

because the average value of sin

(x) is 1/2, which implies

C =

. (7.26)

By rewriting Eq. 7.21, one ﬁnds that the energy level E is a quadratic function of

wavevector k,

E =

. (7.27)

If mass is an unknown parameter, it can be obtained from the coeﬃcient of that

quadratic function,

∂

∂k

, (7.28)

which is an essential relation in semiconductor physics to determine the eﬀective mass

of an energy band.

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148  Quantum Transitions

7.1.4 Hydrogen Atom

The hydrogen atom, a miniature solar system composed of a positively changed proton

and a negatively charged electron, is the ﬁrst test case of quantum mechanics. To date,

it is still the best system in nature for the illustration and understanding of quantum

mechanics. By considering the proton immobile, the quantum state of a hydrogen atom

is the solution of the equation,



−



|ψ = E|ψ, (7.29)

where κ = q

/4π

=2.30 × 10

−28

J · m (see Chapter 2) m

isthemassofelectron,

and r is the distance from the proton to the electron. The mathematical form of

the Hamiltonian in quantum mechanics is identical to the Hamiltonian in classical

mechanics.

It is interesting to note that, before Schr¨odinger invented wave mechanics in 1927,

the equation of motion of hydrogen in matrix form was resolved by Pauli in 1926, see

Appendix C. Schr¨odinger’s solution in terms of diﬀerential equations is more popular,

and can be found in elementary quantum mechanics textbooks. Pauli’s treatment is

more compact and conceptually transparent.

The quantum states of the hydrogen atom can be described by a set of quantum

numbers. The most widely used set of quantum numbers includes the principal quantum

number n, the angular quantum number l, and the magnetic quantum number m.The

energy value of the hydrogen atom depends on only the principal quantum number,

H|n, l, m = E

|n, l, m, (7.30)

where

= −



. (7.31)

The energy of the ground state is E

= −2.17 × 10

−18

J=−13.53 eV. In a coordinate

representation, the quantum states are represented by wavefunctions. For example, the

wavefunction of the quantum state with the lowest energy level of a hydrogen atom is

r|1, 0, 0 = ψ

100

(r)=



πa

−r/a

, (7.32)

where a

is the Bohr radius,



∼

0.0529 nm = 52.9pm. (7.33)

Although the wavefunction is often a complex function, and not directly observable,

the probability of the charge density proportional to

ρ(r)=|ψ

100

(r)|

πa

−2r/a

(7.34)

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7.1 Basic Concepts of Quantum Mechanics  149

Figure 7.2 Quantum states of hydrogen atom. (a) The energy levels of the stationary states

of the hydrogen atom. The lowest energy level of the hydrogen atom, the ground level, is 13.53 eV

below the vacuum level. A series of energy levels lie between the vacuum level and the ground level.

(b) The probability distributions of the stationary states of the hydrogen atom, determined by the

wavefunctions. The numbers in the Dirac kets are, from left to right, the principal quantum number

n, the angular-momentum quantum number l, and the magnetic quantum number m.

is observable. In fact, both the energy level and the change density can be directly

observed using the scanning tunneling microscope. By deﬁnition, the probability is

normalized,

h1, 0, 0|1, 0, 0i =

ρ(r) d

r = 4π

∞

|ψ

100

(r)|

dr = 1. (7.35)

Figure 7.2 shows the stationary states of the hydrogen atom. Figure 7.2(a) shows

the energy levels. The lowest energy level of the hydrogen atom, the ground level, is

13.53 eV below the vacuum level. A series of energy levels lie between the vacuum

level and the ground level. Radiative transitions, that is, absorption or emission, can

occur between those levels. Figure 7.2(b) shows the charge density distributions of the

stationary states of the hydrogen atom.

The quantum state of hydrogen can be excited to a state with a higher energy level

by absorbing a photon, or de-excited to a state with a lower energy value by emitting

a photon.

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150  Quantum Transitions

7.2 Many-Electron Systems

For atomic systems in nature, in terms of quantum mechanics, only a handful have

analytic solutions: the hydrogen atom in both nonrelativistic and relativistic forms,

and the hydrogen molecular ion, which also has only one electron. For systems with

many electrons, an approximation is necessary. Because the nuclei are much heavier

than electrons, in searching for the solutions of the Schr¨odinger equation for a system

with N electrons, the coordinates of the nuclei r

are taken as predetermined ﬁxed

values. The kinetic energy terms of the Hamiltonian T only contains the coordinates

of the electrons,

T = −



j=N



j=1

∇

. (7.36)

The potential energy includes the following three terms: the attractive potential be-

tween the electrons and the nuclei,

= −

I=N



I=1

i=N



i=1

− r

; (7.37)

the repulsive potential among the electrons,

j=N



j=1

i=N



i=1

− r

; (7.38)

and the repulsive potential among the nuclei,

I=N



I=1

J=N



J=1

− r

. (7.39)

The Schr¨odinger equation is

[T + V

+ V

] |ψ = E|ψ, (7.40)

where |ψ depends on the coordinates of the electrons.

7.2.1 Single-Electron Approximation

Despite the complexity of the problem, a simple and very successful concept, single-

electron approximation, has been applied extensively in condensed-matter physics for

many decades. An earlier version of this approach is the Hartree–Fock method or the

self-consistent ﬁeld (SCF) method. A later version, density-functional theory (DFT),

makes the numerical computation more eﬃcient and in general more accurate. For

both methods, the problem of the many-electron system can be reduced to a number of

single-electron problems, each electron moving in the mean potential φ formed by the

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7.2 Many-Electron Systems  151

nuclei and the rest of the electrons. Due to advances in computers and computational

algorithms, quantum chemistry has reached a status where the predictions are reliable

and useful.

In the case of a molecule, the solution of the SCF problem generates a series of

stationary states, each with its own wavefunction and energy eigenvalue, which can be

labeled by a positive integer n,



−



∇ + qφ



|n = E

|n. (7.41)

Suppose the molecule is made of nuclei with order number Z

. The total positive

charge is

N =

nuclei



I=1

. (7.42)

Charge neutrality requires that for the normal state of the molecule there must be N

electrons. following the Pauli exclusion principle, the electrons ﬁll up the states one by

one, starting with the state with the lowest energy value. Once the N-th state is ﬁlled,

the molecule becomes neutral.

For solid states, the number of nuclei is eﬀectively inﬁnite, and the labeling of the

single-electron states needs a continuous variable, or a combination of an integer and

a continuous variable. Because the lattice has translational symmetry, a wavevector k

is a good quantum number.

The solutions to the resulting nonlinear equations behave as if each electron is

subjected to the mean ﬁeld generated by all other particles. Those equations can only

be resolved numerically.

7.2.2 Direct Observation of Quantum States

Until the 1980s, the quantum states of atomic systems were considered merely as math-

ematical conveniences to explain macroscopic physical quantities. After the invention

of the scanning tunneling microscope by Binnig and Rohrer in 1981, the quantum states

of atomic systems, including the molecular orbitals, became directly observable [18].

Figure 7.3 shows a schematic of a scanning tunneling microscope. A probe tip,

usually made of W or Pt–Ir alloy, is attached to a piezodrive, which consists of three

mutually perpendicular piezoelectric transducers: x− piezo, y− piezo, and z-piezo.

Upon applying a voltage, a piezoelectric transducer expands or contracts. By applying

a sawtooth voltage on the x-piezo and a voltage ramp on the y-piezo, the tip scans on

the xy-plane. Using the coarse positioner and the z-piezo, the tip and the sample are

brought to within a fraction of a nanometer from each other. The electron wavefunc-

tions in the tip overlap electron wavefunctions in the sample surface. A ﬁnite tunneling

conductance is generated. By applying a bias voltage between the tip and the sample,

a tunneling current is generated.

The tunneling current is converted to a voltage by the current ampliﬁer, which is

then compared with a reference value. The diﬀerence is ampliﬁed to drive the z-piezo.

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152  Quantum Transitions

Figure 7.3 Scanning tunneling microscope. The scanning waveforms, on the x- and y-piezos,

make the tip raster scan on the sample surface. A bias voltage is applied between the sample and

the tip to induce a tunneling current. The z-piezo is controlled by a feedback system to maintain the

tunneling current constant. The voltage on the z-piezo represents the local height of the topography.

To ensure stable operation, vibration isolation is essential [18].

The phase of the ampliﬁer is chosen to provide a negative feedback: if the absolute

value of the tunneling current is larger than the reference value, then the voltage

applied to the z-piezo tends to withdraw the tip from the sample surface, and vice

versa. Therefore, an equilibrium z-position is established. As the tip scans over the

xy-plane, a two-dimensional array of equilibrium z-positions, representing a contour

plot of the equal tunneling-current surface, is obtained, displayed, and stored in the

computer memory.

Therefore, the STM image can display simultaneously the energy level (by varying

the bias voltage) and the wavefunction (by displaying the electron charge density).

7.2.3 Quantum States of Molecules: HOMO and LUMO

Consider a molecule with N positive charges; see Eq. 7.42. The Pauli exclusion princi-

ple requires that each state (with diﬀerent spin as diﬀerent states) can only be occupied

by one electron. By adding electrons one after another starting from the lowest state,

with N electrons, the molecule becomes neutral. The state or molecular orbital with

highest energy level of a neutral molecule is the highest occupied molecular orbital

(HOMO). The state with energy level above that HOMO is unoccupied for a neutral

molecule, and is called the lowest unoccupied molecular orbital (LUMO). For organic

molecules having a connection to a piece of conductor, HOMO is below the Fermi level.

Otherwise the electron in that state can escape to form a positive molecular ion. On

the other hand, the LUMO is above the Fermi level. Otherwise an electron from the

connected conductor can occupy it to form a negative molecular ion.

Using STM, the electron charge density and the energy level of the HOMO or LUMO

can be directly observed. Figure 7.4 illustrates the tunneling current and tunneling