Chapter 17

Semi-classical Theory of Radiations

17.1   INTRODUCTION

In Chapter 16, we discussed the time-dependent perturbation theory and studied the time-development of states of (one-particle) system. In particular we studied the time-dependence of states in cases when perturbation is switched on adiabatically or suddenly In this chapter, we shall study time dependence of states of (one-electron) atom or ion under the time-dependent harmonic potential.

In particular we shall study the effect of electromagnetic field on the states of (one electron) atom or ion. We shall study the transition probability of atoms from one state to the other. Thus we shall study the phenomenon of stimulated emission and absorption. In this study, the electromagnetic field shall be treated at classical level and, of course, the atomic system shall be treated quantum mechanically. That is why the treatment is known as semi-classical theory of radiation. In Section 17.2, we shall discuss the interaction of one-electron atom with the (classical) electromagnetic field. In next sections, we shall study the process of absorption and emission of radiations alongwith the selection rules.

17.2   INTERACTION OF ONE-ELECTRON ATOM WITH ELECTROMAGNETIC FIELD

17.2.1   The Classical Electromagnetic Field

The classical electromagnetic field is described by the electric and magnetic field vectors Ε and B, which satisfy Maxwell’s equations

where ρ(r, t) and J(r, t) are the charge and current densities that produce electromagnetic fields Ε and B. ∊₀ and µ₀ are the permittivity and permeability of free space, and µ₀∊₀ = (1/c²); c is velocity of light in vacuum. For given charge and current densities, we may find out the scalar and vector potentials ϕ and A and from these, the electric and magnetic fields may be obtained as (see Griffiths ‘Introduction to Classical Electrodynamics’)

 

It can be easily seen that the potentials are not completely defined, as Ε and Β determined through Eqs (17.2) remain unchanged by the substitution

 

where χ is an arbitrary function. Choosing the so-called Coulomb gauge, which suits in regions with no sources, we may have

 

 

 

The vector potential A then satisfies the wave equation

whose plane wave solutions are

 

 

where k is the wave vector, e is a unit vector, and ω the angular frequency. With the choice of Coulomb gauge and the condition ϕ = 0, the electric and magnetic fields may be obtained from Eqs (17.2) and (17.6) as

 

where E₀(ω) = – ωΑ₀(ω) and e_k is unit vector along k.

The energy density of the e.m. field is given by

As the average of harmonic function sin² (k · r – ωt) over a period Τ = (2π/ω) is given by

the average energy density ρ_av(ω) is

In e.m. field, the energy is flowing at a speed of c. Therefore, the intensity Ι(ω), defined as the average rate of energy flow through a unit cross-sectional area, perpendicular to the direction of flow, is given as

17.2.2   A Charged Particle in the Electromagnetic Field

The non-relativistic Hamiltonian of a particle of mass m and charge q, moving in an electromagnetic field described by vector potential A(r, t) and scalar potential ϕ(r, t) may be obtained (see Goldstein ‘Classical Mechanics’) by making the substitutions

 

 

 

in the field–free energy expression

We obtain the resulting Hamiltonian as

The corresponding (time-dependent) Schrodinger equation of the particle is

Now in Coulomb gauge, ∇ · A = 0 and ϕ = 0.

Therefore,

 

So, Eq. (7.15) gives

17.2.3   An Atom in the Electromagnetic Field

Let us now consider a hydrogen-like atom (an electron of charge –e and mass m_e in the field of a nucleus of charge Ze and mass M) in the presence of an electromagnetic field (Figure 17.1). The Hamiltonian in this case will be [as in Eq. (17.17)] the Hamiltonian of an electron of charge –e in the e.m. field but with additional interaction energy [–Ze²/(4π ∊₀)r] of the electron with the nucleus. The nuclear mass Μ being very large in comparison to the electron mass, we neglect the effect of e.m. field on its motion. Therefore, the time-dependent Schrondinger equation in this case can be written as

Figure 17.1 Coordinate system for one-electron atom and the electromagnetic wave. Various unit vectors are shown. Unit vector e_k points in the direction of propagation of e.m. wave. The e_r points in the radial direction. e is the polarization vector (i.e., a unit vector in direction of electric field E or vector potential A)

Where

and μ = [m_eM/(m_e + Μ)] is the reduced mass of the electron.

If we consider a weak e.m. field, the term containing A² may be neglected in comparison to the linear term in A. Therefore, the perturbation term becomes

17.2.4   The Dipole Approximation

If we look at the order of magnitude of the wavelengths of the e.m. waves that matter here (i.e., the e.m. waves which are emitted by the atom during its de-excitation process or the waves which are absorbed by the atom in its excitation process), the wavelengths are very large in comparison to the atomic size. The atomic size is of order ~10^–10 m (~ 1 Å) and optical wavelength is of order ~ 5000Å. Therefore, we may safely take the e.m. field to be uniform over the size of the atom, that is, we may neglect the r-variation of A in the perturbation term (17.19c). This approximation is called as dipole-approximation (the terminology will become clear shortly). Under this approximation, Ĥ′ may be written as

17.3   HARMONIC PERTURBATION THEORY

When a single-electron atom is put in an e.m. field of angular frequency ω, the total Hamiltonian of the system under dipole approximation is

 

where

It may be pointed out here that the Hamiltonian (17.21) consists of two terms: (i) Hamiltonian Ĥ₀ of the (isolated) atom and (ii) Hamiltonian Ĥ′ describing the interaction energy of atom and the e.m. field. And there is no term in the Hamiltonian (17.21) which describes total energy of the e.m. field. In fact, in this semi-classical treatment, we are only interested in studying the change in the states of atom as a result of atom – field interaction. We are not interested in this study to know what is happening to the state of e.m. field as a result of atom–field interaction. Only when we treat the e.m. field quantum mechanically and then study the quantum theory of atom–field, we may study both the change in states of atom as well as the change in states of the e.m. field, as a result of atom–field interaction. This quantum mechanical treatment is a part of the topic known as quantum electrodynamics and will not be discussed here.

Referring to the time-dependent perturbation theory discussed in Chapter 16, we start with the Schrodinger equation for the unperturbed system (atom here) in terms of unperturbed states

The objective is to solve time-dependent Schrodinger equation corresponding to the total Hamiltonian [Ĥ₀ + Ĥ′ (r, t)]

As discussed in Chapter 16, the time-dependent eigenstates of the total Hamiltonian are expanded in terms of the complete set of unperturbed states

and the time dependence of expansion coefficients C_b(t) is written as

where

and

Now following the same convention as in Chapter 16, let the system be initially in a well-defined state of energy and let the e.m. field be switched on at time t = 0. These initial conditions may be expressed as

 

 

Then to first order in the perturbation Ĥ′, we get [as in Eq. (16.22b)]

or

Now we know that

so

Similarly

and

Combining these three equations, we have

Therefore,

The integral appearing in Eq. (17.25) may be evaluated as

Substituting Eqs (17.28) and (17.29) in Eq. (17.25), we get

Where

 

 

and

At this stage, let us find the Hamiltonian Ĥ′ for the interaction between the electromagnetic field and an atom in a more simple way. We consider a one-electron atom (as shown in Figure 17.1) with a nucleus of charge Ze at the origin of co-ordinate system, in presence of the plane polarized e.m. plane wave propagating in the direction of wave vector k as shown in the figure. The interaction energy of the electron (at r) with the electric field Ε is

 

 

As mentioned in Section 17.2.4, if we consider the fact that the wavelength λ in the optical range is ~ 5000 Å, whereas the atomic size is ~ 1Å, we may neglect r-variation of the electric field over the size of the atom. This is called the dipole approximation, as mentioned earlier. So the interaction term becomes

 

 

The name dipole approximation has been given as the interaction Hamiltonian, which may now be written in terms of the electric dipole of the atom D = (–e) r as

 

 

It can be easily checked that if we use expression (17.34) as perturbation in place of expression (17.20), we get similar expression for amplitude C_b(t) as obtained in expression (17.30).

17.3.1   Transition Rate for Absorption

It may be re-emphasized here that writing Eq. (17.30), we have considered two states of the atom and with energies and respectively (see Figure 17.2). We have assumed that at t = 0, this atom is in state , that is

For time t > 0, C_b(t) starts increasing (from its initial value zero at t = 0). It can be seen that for large values of t, the function

Figure 17.2 Two states of atom. While considering absorption process, the atom is in state at t = 0. For considering emission process the atom is in state at t = 0

is simply peaked around (see Figure 17.3). Therefore, we take (ω_ba/ω) ≈ 1 in the expression of C_b(t) [Eq. (17.30)]. Obviously, the second term in Eq. (17.30) corresponds to the process of absorption. The probability of finding the atom (initially in state ) in state when exposed for a time t to electromagnetic field of angular frequency ω, is

where θ is the angle between the direction of polarization e and the electric dipole vector D = –er (see Figure 17.1). Using Eq. (17.11), the expression (17.36) may be written in terms of the intensity I(ω) of the e.m. field

The transition probability corresponding to the transition (i.e., the probability that at time t, the atom has made transition from its initial state to the state ) in the presence of monochromatic radiations of frequency ω with intensity I(ω) is

 

 

The incident e.m. radiations are generally not monochromatic (except in case of lasers). So, assuming the incident radiations to be incoherent and distributed over a range of frequencies, we can add the transition probabilities corresponding to different frequencies to find total transition probability. Of course, in case of coherent (laser) incident radiations, we should add amplitudes (corresponding to different frequencies) to find the total amplitude, from which comes the total transition probability. So the total transition probability P_ba(t) at time t, of exciting the atom from state to state , in presence of radiation of intensity distribution Ι(ω) is

Figure 17.3 Variation of (at fixed t) with (ω_ba – ω).

Here, denotes the average value of cos²θ over different orientations. Now as the intensity distribution I(ω) varies slowly with ω over the range where the function F(t, ω_ba – ω) has appreciable values, Ι(ω) may be replaced by I(ω_ba) and may be taken out of the integral. Then Eq. (17.39) gives

The limits of integration are extended to ω = ±∞, as the integrand F²(t, ω_ba – ω) has negligible values except when ω has values around ω_ba. Using the result from table of definite integrals (Appendix C6) for the integral in Eq. (17.40) and evaluating the average value of cos²θ over different orientations

we obtain

We see from the above expression that the transition probability increases linearly with time t. It must be remembered here that we have used the first-order time-dependent perturbation theory of Chapter 16, to arrive at the result [Eq. (17.42)]. The first-order perturbation theory and the result [Eq. (17.42)] is valid only for small times t, such that P_ba(t) << 1.

The transition probability or the probability of absorption (in the dipole approximation) within first-order perturbation theory is given by Eq. (17.42). From this, we can easily find the transition rate for absorption W_ba as

17.3.2   Transition Rate for Stimulated Emission

In Section 17.3.1, we considered an atom in the ground state at time t = 0 when the e.m. radiations of intensity distribution I(ω) were switched on. We studied the transition rate of absorption W_ba (i.e., the rate at which atom, by absorbing energy from the e.m. field, makes transition to the excited state ). Now we want to study the reverse process, that is, we want to study the case where at time t = 0, the atom is in the excited state when the e.m. field of intensity distribution I(ω) is switched on. The atom goes through the process of stimulated emission [i.e., the atom shall make transition from (excited) state to the ground state in presence of the e.m. field].

To find the transition rate for stimulated emission, we shall have to repeat similar algebra as done in Section 17.3.1. We shall have to return to Eq. (17.30) (with indices b and a interchanged). It is now the first term in the bracket on R.H.S. of Eq. (17.30) (obtained after interchanging the indices b and a) that will correspond to a transition in which the atom loses energy in going from initial (excited) state to (ground) state .

After repeating the whole algebra, the transition rate for stimulated emission W_ab is found to be

In fact, we know

So, we have

 

17.4   SPONTANEOUS EMISSION: EINSTEIN A AND Β COEFFICIENTS

If we consider an atom in excited state, we know it will make spontaneous transition to a state of lower energy with the emission of a photon. This process is called the spontaneous emission and takes place in the absence of external electromagnetic field. This is a quantum mechanical process as the atom makes transition from one quantum state to the other quantum state and emits a photon which is the quanta of energy of the electromagnetic field. But at this stage of semi-classical treatment (where we are describing the atom quantum mechanically and the e.m. field classically), the process may be considered the quantum analogue of the classical phenomenon of emission of radiations predicted by classical electrodynamics. We know from classical electrodynamics that an accelerated charged particle emits electromagnetic radiations.

However, if the atom is in an excited state in the presence of external electromagnetic field, it has an additional process of making transition to a state of lower energy (along with the process of spontaneous transition, in which the external e.m. field has no role to play). This additional process of transition is nothing but the process of stimulated emission discussed in Section 17.3. This is a purely quantum mechanical phenomenon as discussed in Section 17.3. Further, if the atom is in its ground state in presence of external e.m. field, it will absorb energy from the field and shall make transition to a state of higher energy. The process called as (stimulated) absorption was discussed in Section 17.3. So, if we consider a number of atoms in an external e.m. field, three processes are simultaneously taking place: (i) (stimulated) absorption process, (ii) stimulated emission process, and (iii) spontaneous emission process.

In 1916, when Einstein studied the phenomenon of emission and absorption of radiation by atoms, the quantum mechanics, was not developed to describe the time dependence of quantum states of atom due to atom –e.m. field interaction. And, therefore, there was no theoretical evidence for the phenomenon of stimulated emission. It was ingenious of Einstein who imagined that a process like stimulated emission should be there. He was, in fact, studying the Planck’s law by considering (single kind of) atoms and radiation in an enclosure in thermal equilibrium at temperature T. Assuming that a stimulated emission process is present there, along with the absorption and spontaneous emission processes, Einstein was able to derive Planck’s law (in an alternative way). In fact, through this study, he established that the stimulated emission is an integral part of the emission process; spontaneous emission being the other part.

Let us consider an enclosure having a number of (identical) atoms and radiations in thermal equilibrium at temperature T. As shown in Figure 17.2 we consider two (non-degenerate) energy levels a and b of the atom with energies and Let N_a and N_b represent the number of atoms (per unit volume) in levels a and b, respectively.

The number of atoms per unit time, N_ba, making a transition from level a to level b by absorbing radiation of angular frequency is proportional to the total number of atom (N_a) in state and also to the energy density of the radiation per unit frequency range u(ω_ba). So, we have

 

 

where the proportionality constant B_ba is known as the Einstein Coefficient for absorption.

On the other hand, the number of atoms making the transitions from (excited) level b to level a per unit time N_ab is the sum of (i) the number of spontaneous transitions per unit time which is proportional to N_b, the number of atoms in state b and is independent of energy u(ω_ba) and (ii) the number of stimulated transitions per unit time, which is proportional to both the number N_b and the energy density u(ω_ba). Thus, we have

 

 

where the proportionality constant A_ab is known as Einstein coefficient for spontaneous emission and B_ab is Einstein coefficient for stimulated emission.

In thermal equilibrium, the number of upward transitions N_ba should be equal to that of downward transitions N_ab. Thus, we have

 

 

or

In thermal equilibrium, the ratio of the populations of two levels (N_b/N_a) is given by the Boltzmann factor

where k is Boltzmann constant. From Eqs (17.48) and (17.49), we get

Now this expression of energy density per unit angular frequency at temperature T is the same as the one given by Planck’s law

if

 

 

and

We can easily check that

17.5   SELECTION RULES FOR ELECTRIC DIPOLE TRANSITIONS

In Section 17.3, we found expressions of the transition rate of emission and absorption of radiation by an atom put in an electromagnetic field of frequency ω. Let us consider a hydrogen-like atom with two states and having corresponding energy eigenvalues and put in an e.m. field. Within the dipole approximation, the transition rate is found to be directly proportional to |G_ba|² = |e · D_ba|², where the matrix element G_ba is given by

If the initial state of the atom is characterized by the quantum numbers n_a, l_a, m_a and the final state by quantum numbers, n_b, l_b, m_b, then we need to evaluate the matrix element

Here e_r is a unit vector along position vector r. The unit vector e_r has x-, y-, z-components as sinθ cosϕ, sinθ sinϕ, and cosθ, respectively. Therefore, x, y, z components of the matrix element may be written as

where radial integral is

and angular integrals are

Now, the radial integral [Eq. (17.56)] is always non-zero, but the angular integrals B_i are non-zero only for certain set of values of (l_a, m_a) and (l_b, m_b), thus leading to selection rules which are found below.

17.5.1   Magnetic Quantum Numbers

Let us firstly consider the integral over ϕ in B_x and B_y. Using the expression (9.53a) for Y_{l, m}(θ, ϕ), the ϕ-dependent part of B_x contains

This integral vanishes unless m_a – m_b = ± 1. Similar result is obtained from the integral over ϕ in B_y. The integral over ϕ in B_z is

which vanishes unless m_a = m_b. Therefore, the matrix element G_ba shall be non-zero only if one of the following three conditions is satisfied by magnetic quantum numbers

 

17.5.2   Angular Momentum Quantum Numbers

We now consider the integral over θ in B_x, B_y, and B_z. But, firstly let us have a look at how the dipole matrix element behaves under the reflection or parity operation in which position vector r is replaced by –r. In spherical polar co-ordinates, this corresponds to replacing r, θ, ϕ by r, π – θ, ϕ + π. (In Cartesian co-ordinate system, this corresponds to replacing x, y, z by –x, –y, –z). We discussed in Chapter 9 that under reflection, the spherical harmonics have the parity (–1)^l, that is,

 

 

With this relation, all the integrals in Eq. (17.57) have the property

 

 

which dictates that all the integrals B_x, B_y, B_z vanish unless (l_a + l_b) is odd. That means the electric dipole induces transitions in between two states of different parity.

Let us consider only θ-dependent part of integrals in Eqs (17.57). With the selection rules for magnetic quantum numbers, the conditions m_b – m_a = 0, ± 1 are to be satisfied and as a result θ-dependent part of Eqs (17.57a) and (17.57b) is

and θ-dependent part of (17.57c) is

Now using the recurrence relation

integral (17.62a) may be written as

and

With the orthogonality condition (9.51), Eqs (17.64) give

 

 

Using a different recurrence relation

integral (17.62b) may be written as

Again using orthogonality condition (9.51), the above equation gives

 

 

To summarize, an electric dipole induces transitions in between two states of an atom, which satisfy

 

 

and

 

 

The condition (17.69a) ensures that the transition takes place between two atomic states and only if their parities are different. It may be noted that the dipole operator does not have any effect on the spin state of the electron. Therefore, in an electric dipole transition spin-component along the direction of quantization (z-axis, usually) remains unchanged.

17.6   LIFETIME AND LINE-WIDTH

When an atom, initially (t = 0) in state makes a transition to state in the presence of an electromagnetic field, the transition probability P_ba(t) is determined by the quantity |D _ba |² [Eq. (17.36)]. This is stimulated transition. For example, in a two-level system, this transition may be described through the time-dependent amplitudes C_a(t) and C_b(t). |C_a(t)|² is simply the probability of finding the system in state at time t. The amplitudes C_a(t) and C_b(t) are connected through the differential equations [like Eq. (16.12)]. If initially the system is in state [i.e. C_a(0) = 1, C_b(0) = 0], the probability |C_a(t)|² will decrease with time because of its coupling to state . As a consequence |C_b(t)|² will increase with time (|C_a(t)|² + |C_b(t)|² = 1). As |C_b(t)|² increases, the back coupling between and will make the reverse transition process; |C_b(t)|² now decreases and |C_a(t)|² increases. In such a two-level system, the probabilities |C_a(t)|² and |C_b(t)|² have oscillatory behaviour with time (see Section 19.6, case III).

Let us now come to the spontaneous transition in an atom from state to state (with corresponding energy eigenvalues and It may be emphasized here that this emission process is not really a two-level process. The atom makes transition (from state ) to state along-with a photon, which may be emitted in any direction with any angular frequency ω. The quantum state of a photon depends on its direction of propagation and its frequency ω. Therefore, the system (atom + e.m. field) makes a transition from one state to a continuum of states. However, the probability of ω having a value outside a very small region about is very small. Now final continuum of states are incoherent and are not able to act cooperatively to build up reverse transition. As a result, the probability of finding the atom in initial state decreases steadily with time. We will show that under these conditions, the decay rate comes out to be exponential.

We had seen in Section 16.3 that the probability P_b(t), that in time t the system (initially in state has made a transition to a group of final states, is proportional to t [Eq. (16.40)]. Therefore, the transition probability per unit time or the transition rate, given as [dP_b(t)/dt] comes out to be constant. Let this be denoted by W_ab. Now the probability P_b(t + dt) of finding the atom in state at time t + d t is equal to the probability P_b(t) of finding the atom in state at time t multiplied by the probability that no transition takes place during time dt, that is

 

or

With the initial condition P_b(0) = 1, above equation gives

 

where

is called the lifetime or half-life of state . From Eq. (17.71), we may have expression of the amplitude C_b(t) (taking it to be real),

 

Now when state is not decaying (i.e., the coupling with e.m. field is absent), its time dependence is given as

and the energy has well-defined value . But in the case when state is decaying according to the amplitude C_b(t) given by Eq. (17.73) (as a result of its coupling with the radiation field), its time dependence will be given as

In this case, the energy does not have well-defined real value. It has complex value For seeking physical meaning of the complex value of energy, let us express the wave function as a superposition of energy eigenstates;

or

Multiplying Eq. (17.75) by e^i(E″t/ħ) on both sides and integrating over time t from 0 to ∞ (we assume that for t < 0), we get

Now,

So, Eq. (17.77) gives

Integrating it, we get

which gives

The quantity |A(E)|² is a measure of the probability of finding the initial state with definite energy E. If the initial state (with energy Ε) makes a transition to a final state (of well-defined energy ) and emits a photon of energy ħω, then energy conservation requires

Therefore, the quantity |A(E)|² also determines the intensity distribution as a function of angular frequency ω of the emitted spectral line

This distribution is called Lorentzian distribution (shown schematically in Figure 17.4). It may be re-written as

where

Figure 17.4 Schematic plot of Lorentzian intensity distribution |A(ω)|²

We see from Figure 17.4 that the width of the distribution at half maximum is where (having dimensions of energy) is called natural width of the line, that is;

which gives

So is also the energy width of state . From Eq. (17.85), we have

 

 

which is consistent with the uncertainty relation

EXERCISES

Exercise 17.1

Find out the ratio of the number of stimulated emissions to that of the spontaneous emissions in thermal equilibrium at temperature (i) T = 300 Κ and (ii) T = 1000 Κ for

• visible region λ ~ 5000 Å
• microwave region λ ~ 1 mm

Exercise 17.2

In continuation to Exercise 17.1, show that at room temperature (T = 300 K) the thermally stimulated emissions dominate well below the frequency 5 × 10¹² Hz, whereas the spontaneous emissions dominate well above this frequency.

Exercise 17.3

Consider the three commutation relations

Find out the matrix elements of L.H.S. of these equations in between states to derive selection rule for m, ∆m = 0, ± 1.

Exercise 17.4

Derive the following commutation relation

Exercise 17.5

Find out the matrix elements of L.H.S. and R.H.S. of the operator equation of above exercise in between states to derive selection rule for l, ∆l = ± 1.

SOLUTIONS

Solution 17.1

We know

•  
  T = 300 K
  

  for

   

  λ = 500 Å = 5 × 10^–7 m
  

  for

   

  λ = 1 mm
  
•  
  T = 1000 K
  

  for

   

  λ = 500 Å
  

  for

   

  λ = 1 mm,         R ≈ 68.8

 

We observe that in the visible range the thermally stimulated emissions (stimulated emissions for which black body radiation is the agency) are negligible both at 300 Κ as well as at 1000 K. However, in microwave range the thermally stimulated emissions dominate over spontaneous emissions at both the temperatures.

Solution 17.3

or

or

so

 

So unless m = m′, the matrix elements of z in between any two atomic states are zero.

Next, let us consider

or

or

From the commutation

we may, similarly, get

Combining Eqs (17.90) and (17.91), we have

So

from Eqs (17.89) and (17.92), we get selection rule for m:

 

Solution 17.4

Let us evaluate

So,

Similarly, other components may be evaluated to get the general result.

Solution 17.5

Taking matrix element of R.H.S. of Eq. (17.88)

So, either

 

 

or

 

Equation (17.95a) gives

 

 

or

 

 

The first factor [(l′ + l + 1)² – 1] = 0 gives trivial solution l = l′ = 0. The second factor gives

 

 

or

 

 

So no transition occurs unless ∆l = ±1.

REFERENCES

1. Griffths, D.J. 1999. Introduction to Classical Electrodynamics. New Jersey: Prentice Hall.
2. Born, M. and Wolf, E. 1970. Principles of Optics. Oxford: Pergamon.
3. Loudon, R. 1983. The Quantum Theory of Light. Oxford: Clarendon Press.
4. Bransden, B.H. and Joachain, C.J. 2000. Quantum Mechanics, Singapore, Pearson Education Ltd.
5. Liboff, R.L. 1992. Introductory Quantum Mechanics. Mass: Addison-Wesley.