Light - Atom Interaction

Transcription

1 1 Light - Atom Interaction PHYS261 and PHYS381 - revised december 2005 Contents 1 Introduction 3 2 Time dependent Q.M. illustrated on the two-well problem 6 3 Fermi Golden Rule for quantal transitions Time-Dependent Schrödinger Equation Perturbation Theory for the Time-dependent Schrödinger Equation Approach to Dirac delta-function Fermi Golden Rule

2 2 3.5 Constant rate and exponential decay Line width from exponential decay Generally on quantum treatment of extended systems - fields Normal Coordinates for coupled harmonic vibrations Algebraic Method for Harmonic Oscillator Electromagnetic fields 35 8 Eigenmodes of Radiative Electromagnetic Field 43 9 The Quantum Theory of Electromagnetic Field Density of States Charged Particles In an Electromagnetic Field 52

3 The Hamiltonian of Interaction Emission of Radiation (Light) By an Excited Atom: Spontaneous and Stimulated Emission Dipole Approximation Detailed Evaluation of Emission rate Einstein Coefficients Formula Collection 69 Based in part on Nazila Yavari s Thesis and Older Notes 1 Introduction In beginning courses of Quantum Mechanics one learns about quantum jumps and similar intuitive ideas. Quantum mechanics of time developement draws

4 4 a slightly different picture, that of probability density gradually moving from certain possible situations (configurations) to other configurations of systems in question. For understanding the emission of light by excited atoms we need to consider at least a system consisting of one atom and the space with electromagnetic field. To give an idea in the world of everyday life, our system looks very much like a microwave oven, where the 2.45 GHz oscillator source is replaced by our excited atom, and no electricity is connected, i.e. no other Emission of light means that the atom + electromagnetic field system transfers from a state where the atom is excited and the electromagnetic field is in a certain state (in a simplest model, the ground state of the elmag. field) to a state where the atom is in a lower energetic state and the electromagnetic field is excited. The excitation of the field is called a photon, and our model is also aimed to understand what a photon is. The first task is describe such transfer process, also descibed by terms as decay, transition, etc. These processes are treated in texts on quantum mechanics, as well as in our B and J book under the Time-dependent perturbation theory, while the time development itself would be generally treated in quite different places. Therefore we discuss these matters in the beginning of our text.

5 5 The features of our model can be characterized by the following items: 1. Time dependent quantum mechanics: for a sysstem with two isolated states, contrasted to situation with one isolated level in one part energetically embedded in (quasi-) continuum of states in the other part. 2. We work with the basic formulae for the derivation and understanding of the transition rate (probability change per time unit) 3. Time dependent perturbation theory 4. Fermi Golden rule derivation ( we have also an alternative treatment with Fermi Golden Rule Simulator - see the www-pages; not done here) 5. The delta function (often mentioned as energy conservation; that is not precise). 6. In contrast to the delta function, the Lorentzian shaped finite line width (also known as Breit-Wigner formula or shape) 7. Electromagnetic field is an extended (actually continuous) system, therefore we must learn how to understeand eigenmodes of large system.

6 6 When they are linear (harmonic), they behave as a limiting case of a large system of coupled oscillators. Therefore we show that coupled oscillators can be transformed to a system of independent, de-coupled eigenmodes, via so called normal coordinates. 8. Any harmonic oscillator can be described by so called creation and anihilation operators. This can be called harmonic oscillator via the algebraic method. 9. More about the radiation field 10. Evaluation of the transmissin rate for the emission proces 11. Einstein coefficients 2 Time dependent Q.M. illustrated on the two-well problem The two eigenstates in each of the isolated wells are ϕ 1 and ϕ 2.

7 7 The static eigenstates of the particle confined to both of the wells are approximated by ψ + = 1 2 (ϕ 1 + ϕ 2 ) ψ = 1 2 (ϕ 1 ϕ 2 ) clearly, ϕ 1 = 1 2 (ψ + + ψ ) ϕ 2 = 1 2 (ψ + ψ ) The states ψ +, ψ are eigenstates of the Hamiltonian, but ϕ 1, ϕ 2 are not, they are eigenstates for each isolated well.

8 8 Figure 1: One potential well (file potwell.eps) If at t = 0 the system is brought into the state ϕ 1, then at any other later time Ψ(t = 0) = ϕ 1 = 1 2 (ψ + + ψ ) Ψ(t) = 1 2 ( ψ+ e ie +t/hbar + ψ e ie t/hbar ) This can be rewritten schematically as Ψ(t) = C(t) ( ψ + + ψ e iωt)

9 9 so that we can see immediately that for t n = n π ω the Ψ(t n ) will be a multiple of ϕ 1,i.e.concentrated to the left, for even n, and a multiple of ϕ 2,i.e.concentrated to the right, for odd n. At general times there would be a continuously changing distribution between the two regions.

10 Figure 2: System (particle) placed in a state which is not an eigenstate 10

11 Figure 3: States in two potential wells (file PotWells.eps) 11

12 12 3 Fermi Golden Rule for quantal transitions 3.1 Time-Dependent Schrödinger Equation The time-dependent Schrödinger equation i ψ(t) = Ĥ(t) ψ(t) (1) t is very often solved via a transformation to the matrix formulation. The matrix formulation arises from expansion of the unknown wavefunction ψ(t) in a set of basis functions φ i, much in analogy with Fourier series or expansions using orthogonal polynomials ψ(t) = α i (t) φ i (2) The unknown quantities to be found are the expansion coefficients, which form a vector. In this formulation, the time-dependent Schrödinger equation (eq. (1) above) is replaced by a set of coupled differential equations, which are

13 13 conveniently expressed by matrix notation i d dt α 1 α 2. α n = H 11 H H 1n H 21 H H 2n H n1 H n2... H nn α 1 α 2. α n 3.2 Perturbation Theory for the Time-dependent Schrödinge Equation The time-dependent perturbation theory can be used to derive the so called Golden Rule formula, known as Fermi s Golden Rule. The expression perturbation theory is used in many fields, it denotes a situation when the equations for a given system differ only slightly from another system, the solutions for which are well known. The difference is then assumed to disturb (or perturb) the well known solutions. For example, if we have a pendulum which passes every period through a small region of water, it will still most of the time behave as a pendulum (the known system), and the difference, the passage through the water, is only a small disturbance - or using an other word - perturbation.

14 14 In quantum mechanics the perturbation theory is used very often, since there are very few systems which can be solved exactly. In most cases the perturbation theory method consists in finding which parts of equations can be neglected. To apply the perturbation theory, we must identify the small difference. It is usually in the Hamiltonian H(t), which differs only slightly from a well known Hamiltonian H 0, usually the difference contains all the time depandence: H(t) = H 0 + H (t) (3) If the H (t) were not present, and chosing then as φ k the eigensolutions of H 0 φ k = E k φ k (4) the matrix for H(t) H 0 would be diagonal with the eigenenergies E k on the diagonal. The solutions for the the α k (t) would be then very simply α k (t) = e ie kt/ h (5) and for usual stationary states only one of the α k (t) would be different from

15 15 zero. Since the probability P a of the system being in a state a is P a (t) = α a (t) 2 it means that the system with probability 1 is in that state a, and with 0 in any other state, i.e. α k (t) = δ ka (6) This is the unperturbed situation. Perturbation theory is the prescription to obtain solution where there is a small H (t). If the perturbation really is small, the α a (t) ) remains close to 1, while or others remain close to zero. Setting this assumption into the above matrix system of coupled equations, they decouple and we obtain independent equations for each α k (t) i d dt c k(t) = H ka (t) exp(i (E k E a )t ) (7) h where we made a substitution α k (t) c k (t) exp( i E kt h ). For each of the pair of states a,k ω ak = E a E k h (8)

16 16 We say that the original assumption about the coefficients is a zero-th order approximation, thus the superscript (0), c (0) k = δ ka (δ (k a) ) (9) while the above equation is the first order, thus we rewrite it ċ (1) b = 1 i h H e (iωt) (10) A detailed description of perturbation theory pictures it as an iterative process where each order is obtained by applying the preceeding order of the approximation. c (1) b (t) = 1 t H e (iωt ) dt H (t) H (11) i h t b So the transition probability for going from a state a to a state b is defined like

17 17 P ba (t) = c (1) b (t) 2 (12) Now we evaluate the integral in eq.(11), which is elementary for a constant potential: b (t) = H ba 1 e iωt dt ; t 0 = 0 i h t 0 = H ba 1 1 i h iω (eiωt 1) c (1) t = H ba hω (eiωt 1) (13) and rearrange (e iωt 1) e (iωt) ωt (i 1 = e [ 2 ) ωt (i e 2 ) ωt ( i e )] 2 ωt (i = 2ie 2 ) sin ωt 2

18 18 Inserting this back into eq.(11) c (1) b (t) = 1 t H e iωt dt (14) i h t 0 c (1) b (t) 2 = 1 h 2 H ba 2 F (t, ω) (15) where the phase factor with absolute value one vanishes. 3.3 Approach to Dirac delta-function It is easy to find that the F is a function of t and ω, and for each time t it is equal to F (t, ω) = 4 sin2 ωt 2 ω 2 = 4 1/4 sin2 ωt 2 ωt 2 4

19 19 = t 2 sin2 x ; x = ωt x 2 2 (16) It can be seen that the function F (ω, t) for larger and larger t approaches the shape of the Dirac delta-function, see the (animated on the www) drawing. Trying to integrate the function F over ω shows this t 2 sin 2 x dω = 2t x 2 = 2t = 2πt sin 2 x x 2 sin 2 x dx x 2 d( ωt 2 ) (17) that for large t it really behaves as F (t, ω) = sin2 ωt 2 ( ω 2 )2 2πtδ(ω) (18)

20 20 The summation over all the states used for the expansion can be assumed to be replaced by the integration over ω or the energy E = ω. This however needs to determine the factor ρ(e) which accounts for the density of states. Since the F approaches delta-function, we only need the density of states close to the final state b: and also integration over dω gives P ba (t) = 1 h Eb +η 2 H ba 2 F (t, ω)ρ(e)de (19) E b η δ(ω)dω = δ(e)de δ(ω) = δ(e) de dω = hδ(e E b ) (20)

21 Fermi Golden Rule The final result is P ba (t) = 1 h Eb +η E b η 2πt H ba 2 δ(e E b )ρ(e)de = 2π h t < b H a > 2 ρ(e b ) (21) The derivation of the above formula has been based on the assumption of a small perturbation. It shows that the probability of transition to the state b or states close to b increases linearly with time. Thus the rate of probability change per time is given by dp ba dt = W ba = 2π h < b H a > 2 ρ(e b ) (22) This result is known as Golden Rule formula or Fermi Golden Rule.

22 Constant rate and exponential decay Fermi golden rule gives a constant rate dp f dt = w (23) or if we consider the probability decrease to find the system in the original state, dp i = w (24) dt If this, imstead of a definition, is taken as a differential equation P i = P 0 wt (25) can quickly become negative.however, one can quite easily realize that the correct differential equation must be dp i dt = wp i (26)

23 23 since the loss of probability must be proportional to how much is left, i.e. the P i itself.it can also be guessed from the differential equations of Q.M. (leading to the dp f ), since they contained the amplitude, which we replaced by 1. dt The last differential equation leads to the well known exponential decay, since its solution is since P i (t = 0) = 1 P i = e wt (27) 3.6 Line width from exponential decay In the equation (7) the expansion coefficient of the initial state α 0 (t) 1 for all times leads to the delta function for energy (frequency). In order to take into account the flow of probability from the initial state 0, the relation α k (t) = δ ka

24 24 when working with the time dependent problem must be changed to α k (t) = δ ka exp( w 2 t) where w is the constant rate factor obtained in the perturbation theory. If we take into account the result of previous section 3.5, we have c 0 (t) exp( wt/2). Inserting this into equation (7), the integrals can still be performed, but the expressions which were found to converge to the δ-function in frequency (energy) for time going to infinity are not obtained. The term V ba e iω bat dt leading to V ba 1 ω ba (e iω bat 1) which has been shown to lead to δ-function like behaviour when integrated over ω is now replaced by e i(ω ω ba)t e wt/2 1 i(ω ω ba ) w 2

25 25 at large times t this leads to 1 2 i(ω ω ba ) w = 2 1 (ω ω ba ) w2 The rate, which has the dimension of frequency becomes the energy width when multiplied by h. Γ = hw This leads to the so called Lorentz shape of the line, I(E) I 0 ( Γ 2 ) 2 (E E 0 ) 2 + ( ) 2 (28) Γ 2 Thus a more realistic treatment than the perturbation theory with constant decay rate leads to the natural width of the energy spectrum of the ejeced photons as observed. This behaviour has been illustrated in our Golden Rule Simulator program.

26 26 4 Generally on quantum treatment of extended systems - fields. The system to be considered consists of an atom and the electromagnetic field. The field cannot be described in the same way as a single particle or a certain small number of particles, as e.g. the many-electron atom. The field has in principle infinitely many degrees of freedom, i.e. the values of field variables in each point of space. The electromagnetic field can be treated by wave equations, it is thus a sort of harmonic system. We shall thus show how the energy of a finite harmonic system can always be transformed to a sum of independent harmonic oscillators, each of them is in fact the eigenmode. Each harmonic oscillator can be then described using the equations for harmonic oscillator, with the algebraic method and number of quanta states. The creation and annihilation operators are treated in the section 6. The normal modes or eigenmodes for a general harmonic system are discussed in the section 5.

27 27 The method outlined in these two sections is applied in the quantal theory of harmonic crystals. The quanta are there called phonons, which can be observed in many solid state physics experiments. The prescription 1. identify the eigenmodes 2. quantize each of the modes as independent harmonic oscillator 3. use quanta of eigenmodes 4. express the general coordinates of the system using the eigenmode coordinates (inverse transformations to those discussed in section 5. Each coordinate χ i will then be replaced by its combination of the creation and annihilation operators as discussed in section 6.

28 28 5 Normal Coordinates for coupled harmonic vibrations. Transformation to normal coordinates can be described as follows: Take the total energy, x represents the vector of all coordinates, x T represents the transposed vctor, M is the mass matrix, V is the matrix which gives the potential energy, including the couplings. Note that the mass matrix is written in a very general form, most often this matrix would be diagonal. E = 1 2ẋT Mẋ xt Vx First we transform the kinetic energy to a spherical form 1 2ẋT Mẋ = 1 2 ηt S T MS η = 1 2 ηt η

29 29 This transformation does not conserve the lengths. The N-dimensional ellipsoid is transformed to an N-dim. sphere. S T MS = 1 If M ij = m j δ ij, it is quite easy to find S ij = m j 1/2 δ ij, Transformation S simplifies the kinetic energy, but the potential remains complicated. Therefore we use one more, from η to χ, 1 2 xt Vx = 1 2 ηt S T VSη = 1 2 χt U 1 S T VSUχ Transformation η = Uχ is a simple rotation obtained by diagonalizing the matrix of potential energy: ( U 1 S T VSU ) ij = Ω2 j δ ij = W ij Therefore we took U 1 = U T. We can see that the length is conserved. 1 2 ηt η = 1 2 χt U 1 U χ = 1 2 χt χ

30 30 E = 1 2ẋT Mẋ xt Vx = 1 2 χt χ χt Wχ This expression however is a sum over the new mutually independent degrees of freedom, since W is diagonal. N i=1 1 2 χt χ χt Wχ = 1 2 m iẋ 2 i N i,j=1 N i=1 V ij x i x j ( 1 2 χ2 i Ω2 i χ 2 i N i=1 ) ( 1 2 χ2 i Ω2 i χ 2 i ) 6 Algebraic Method for Harmonic Oscillator. Basis for Second Quantization In this section we show how one can work with the harmonic oscillator introducing so called ladder operators which move from state to state, the states

31 31 being separated by the same quantum of energy. Yhe classical hamiltnian can be transformed: 1 2m p2 + mω2 2 q2 transforms to 1 P = h m ω p With this the commutator [ q, p ] = i h becomes [ Q, P ] = i hω 2 Q = m ω h ( P 2 + Q 2) This simplifies the equation and brings it into a form where the energy is expressed in hω. The main transformation, however, is to go over to linear combinations of P and Q a = 1 2 Q + i P and a + = 1 Q 2 2 q i 2 P

32 32 By a very simple algebra we find that for their commutator [ a, a + ] = 1 and the energy is transferred to hω 2 Using the commutator a a + ( P 2 + Q 2) hω 2 ( a a + + a + a ) a + a = 1 we finally get H = hω a + a + hω 2 = hω N + hω 2 where we already include that will be a number operator. N = a + a Why number? Let us play with [ a, a + ] = 1 or alternatively a a + a + a = 1 and the operator N. We quickly derive that [ N, a + ] = a + and [ N, a ] = a

33 33 This is done simply by writing out [ N, a + ] = Na + a + N = (a + a) a + a + (a + a) = a + (a a + a + a) = a + [ a, a + ] = Since H = hω N + hω 2 we also have [ H, a + ] = hω a + and [ H, a ] = hω a These are the last equations which bring the physical interpretation. If there is a Q.M. state ψ(q) ψ such that then H ψ = E ψ H (a ψ ) = a H ψ hω a ψ = a E ψ hω a ψ = (E hω ) (a ψ ) that means that (a ψ ) is also an eigenstate. It has energy lower by hω. This we can continue again and again, getting eigenstates for E hω, E 2 hω, E 3 hω, E 4 hω, E 5 hω...

34 34 Since this cannot go on infinitely, we get finally a state such that We quickly verify that this state has a ψ 0 = 0 E 0 = hω 2 further that the same thing is possible with a + only with opposite sign, so that we get energies E 0, E 0 + hω, E hω, E hω, E hω, E hω... Each of the eigenstates is an eigenstate of both H and N with obvious number of quanta. Therefore we call a + and a creation and annihilation operators. They make states with one more ore one less particle or quantum

35 35 7 Electromagnetic fields This text is taken from N. Yavari, who insisted on using the SI-units. It is useful to know both SI units, since they are widely used, but for physical understanding, the Gaussian units are more suitable. In other parts of the text we use most often the Gaussian units. The classical electromagnetic field is described by electric and magnetic field vectors E and B, which satisfy Maxwell s equations. We shall express these and other electromagnetic quantities in rationalized MKS units, which form part of the standard SI system. In atomic physics it is otherwise often more advantageous to work in Gaussian units, where the strengths of the fields have the same physical dimension. Maxwell s equations that shows the relationship between the electric and magnetic fields D = 4πρ (29) B = 0 (30) E = 1 c B t (31)

36 36 H = 1 E c t + 4π J (32) Here, the current density and the charge density have been showed by J and ρ. D stands for the dielectric displacement vector and H for the magnetic intensity vector. These vectors are dependent on the external field and describe the induced internal fields, inside a material medium. They are given by D = E + 4π P (33) H = B 4π M (34) where P and M are the polarization and the magnetization vectors. In vacuum, these two vectors vanish. The electric field E and magnetic field B can be generated from scalar and vector potentials φ and A by E( r, t) = φ( r, t) t A( r, t) (35) B( r, t) = A( r, t) (36)

37 37 The potentials are not uniquely determined by the above equations. In particular, the physically observable field strengths E( r, t) and B( r, t) remain the same when the potentials are changed by the substitutions A( r, t) A( r, t) + λ( r, t) λ( r, t) φ( r, t) φ( r, t) t where λ( r, t) is any scalar field. This property is called gauge invariance. This property allows us to impose a further condition on A, which we shall choose to be A = 0 (37) When A satisfies this condition, we are said to be using the Coulomb Gauge. From Maxwell s equations (without sources) we can show that A satisfies the wave equation (as do φ, B and E) 2 A 1 c A t 2 = 0 (38)

38 38 In what follows we shall set the scalar potential φ = 0 since in empty space (vacuum) the most general solution of Maxwell s equations for a radiation field can always be expressed in terms of potentials such that A = 0 and φ = 0. A monochromatic plane wave solution of equations (37) and (38), corresponding to the angular frequency ω (i.e. the frequency ν = ω/2π) is one that represents a real vector potential A as A(ω; r, t) = 2 A 0 (ω) cos(k r ωt + δ ω ) = A 0 (ω) [exp[i(k r ωt + δ ω )] + c.c.] (39) Here A 0 is a vector which, as shall see shortly, describes both the intensity and the polarization of the radiation, K is the propagation vector, δ ω is a real phase and c.c. denotes the complex conjugate. We note that A = 0 is satisfied if k A0 (ω) = 0, so that A 0 (ω) is perpendicular to K and the wave is transverse. Eq.(38) is satisfied provided that ω = kc, where K is the magnitude of the propagation vector K. The electric and magnetic fields associated with the vector potential A(ω; r, t), are given respectively from E(r, t), B(r, t) equations by

39 39 E = 2ω A 0 (ω)ˆε sin( k r ωt + δ ω ) B = 2 A 0 (ω)( k ˆε) sin( k r ω + δ) (40) where have written A 0 (ω) = A 0 (ω)ˆε. The direction of the electric field E is long the real unit vector ˆε, which specifies the polarization of the radiation, and is called the polarization vector. From k A 0 (ω) = 0, we note that ˆε must lies in a plane perpendicular to the propagation vector K and can therefore be specified by giving its components along two linearly independent vectors lying in this plane. We also see from eqs.(40) that both E and B are perpendicular to the direction of propagation K, and to each other. The expression eqs.(40) describe a linearly polarized plane wave, namely a plane wave with its electric field vector E always in the direction of the polarization vector ˆε. In quantum description of the electromagnetic field the energy in each mode of angular frequency ω, in some region of volume V, is carried by a number N(ω) of photons, each of energy hω. The total energy in the mode is therefore given by N(ω) hω and the energy density by N(ω) hω/v.

40 40 In order to relate this quantum description with the classical approach we are using here, we first construct the energy density of the field, which is given by 1/2(ε 0 E 2 + B 2 /µ 0 ) = 4ε 0 ω 2 A 2 o (ω) sin 2 ( k r ωt + δ ω ) (41) where ɛ 0 and µ 0 are respectively the permitivity and permeability of free space. The average energy density over a period 2π/ω is ρ(ω) = 2ε 0 ω 2 A 2 0 (ω) (42) Equating this result with the energy density N(ω) hω/v, we have A 2 0(ω) = Similarly, the magnitude of the Poynting Vector h N(ω) (43) 2ε 0 ωv ( E B)/µ 0 is the rate of energy flow through a unit cross-sectional area normal to the direction of propagation K. Average over a period, this quantity denotes the intensity of the radiation, which is given by

41 41 [ ] N(ω) hω I(ω) = 2ε 0 ω 2 ca 2 0(ω) = c V = ρ(ω)c (44) what we have got here, is the intensity of radiation per unit angular frequency. A general pulse of radiation can by taking φ = 0 and representing A(r, t) as a superposition of the plane waves A(ω; r, t). Taking each plane wave component to have the same direction of propagation ˆk and adopting a given direction of linear polarization ˆε, so that we write A(r, t) = ω A 0 (ω) = A 0 (ω)ˆε A 0 (ω)ˆε [ exp[i( k r ωt + δ ω )] + c.c. ] (45) We shall be concerned with the case in which the radiation is nearly monochromatic, so that the amplitude A 0 (ω) is peaked about some angular frequency

42 ω 0, differing from zero in a region of width ω. In a naturally occurring pulse the radiation arises from many atoms emitting photons independently, which implies that the phases δ ω are distributed at random,as a function of ω ; in other words, the radiation is incoherent. It follows that the average energy density in a pulse of the form eq.(45) is ρ = ω 2ε 0 ω 2 A 2 0 (ω)d(ω) = ω ρ(ω)d(ω) (46) [This remark dose not apply to radiation from lasers, which exhibits a high degree of coherence.] As seen by comparing with eq. (42), the contribution from each mode are summed, with no interference term. Similarly the average intensity is Ī = ω I(ω)dω (47) where the intensity per unit angular frequency range I(ω) is given by I(ω) = ρ(ω)c (48) 42

43 43 The energy stored or contained in the electromagnetic field is given by the formula H field = 1 d 3 r (E.E + B.B) (49) 8π This formula is given in Gaussian units, where both the electric and magnetic field strengths have the same dimension. This energy formula is very important in determining the quantization of electromagnetic field operators. 8 Eigenmodes of Radiative Electromagnetic Field Here we use the Gaussian system of units. The two components of the electromagnetic field, E as the electric, and B as the magnetic component, that can be derived from φ and A as the scalar and vector potentials, by using the eqs.(35) and B = A We use Coulomb gauge with it s condition that s A = 0. The propagating waves are in space where there is no current distribution and no charges.

44 44 In this case the field equations can be simplified to ρ = J = P = M = 0 H B D E (50) To visualize this situation, we can think about the inner part of a microwave oven, where the air is in its behaviour close to the vacuum. Now we consider the electromagnetic field, enclosed in a cubic box with length of L and volume V = L 3, with the boundary conditions of A(0, y, z, t) = A(L, y, z, t) A(x, 0, z, t) = A(x, L, z, t) A(x, y, 0, t) = A(x, y, L, t) Such harmonic system (it obeys wave equations) must have a set of eigenmodes. In this case, unlike the case of a finite number of coupled oscillators in section 5, the number of eigenmodes is infinite, but discrete. These eigenmodes are plane waves.

45 45 9 The Quantum Theory of Electromagnetic Field The energy stored or contained in the electromagnetic field is given by the formula H field = 1 d 3 r (E.E + B.B) (51) 8π This formula is given in Gaussian units, where both the electric and magnetic field strengths have the same dimension. This energy formula is very important in determining the quantization of electromagnetic field operators. Using the results of sections 8 and 9.1 we can write the vector potential A in the form of Fourier series A( r, t) = kλ 2π hc 2 V ω k ê kλ ( a kλ e i( k r ωt) + a kλ e i( k r ωt) ) (52) while in the eq.(52), all the possible values of the propagation vector k has been considered, also the two possible polarizations λ = 1, 2.Now we should input

46 46 the boundary conditions and to do this, it s necessary that the components of k be quantized; while the possible values for them is (nx, n y, n z ) 2π and n L i is integer. The normalization factor 2π hc 2 V ω k has been chosen so that the following equation can give us the total energy of the inside electromagnetic field of the box. (E 2 + B 2 ) H rad = H dv = dv (53) V V 8π by using the eq.(35) and the relation B = A in the eq.(53), the total energy inside the box is achieved H rad = H kλ = hω k N kλ = hω k a kλ a kλ (54) kλ kλ kλ The operator a kλ is called annihilation and a kλ is called creation operators. The number operator N kλ that it s familiar for us, and is defined as

47 47 N kλ a kλ a kλ, by its expectation value shows the number of photons in each mode. To obtain the eq.(54), we ignored the term 1 2 kλ hω k that refers to zero point energy of the field as it would be infinite if the summation had done over the infinite number of modes. We should here mention the orthogonality of some of the properties of the eq.(52) by ê kλ ê kλ = δ λ,λ (λ, λ = 1, 2) (55) and [a kλ, a k λ ] = δ k k δ λλ, [a kλ, a k λ ] = [a kλ, a k λ ] = 0 (56) As the collection of the harmonic oscillator Hamiltonians of each state, with different frequency, can give us the Hamiltonian of the field; the state of the field can be written as a direct product of the vector states for each of the oscillators n kλ n k λ = n kλ n k λ (57) So for the state vectors, we can show the effect of the different operators as a kλ n kλ n k λ = n kλ n kλ 1 n k λ (58) a kλ n kλ n k λ = n kλ + 1 n kλ + 1 n k λ (59)

48 48 while N kλ n kλ n k λ = n kλ n kλ n k λ H kλ n kλ n k λ = hω k n kλ n kλ n k λ p kλ n kλ n k λ = h kn kλ n kλ n k λ n kλ = 0, 1, 2, 3,...,. (60) (61) (62) In Physics,we most of the time measure the average or the expectation value of quantities, and the quantities that are most important in this chapter are energy and momentum that we can measure their expectation values in this way E = H rad = H kλ = hω k N kλ = hω k n kλ (63) kλ kλ kλ p = kλ h k N kλ = kλ h kn kλ (64)

49 Density of States We consider the electromagnetic field enclosed in a box with volume V. For a finite volume there is a discrete number of modes satisfying the imposed boundary conditions. Therefore the sum over final states is an ordinary sum. As this volume approaches infinite size, the summation over k, will be approaching an integral. The density of states factor is found from performing such a limiting process, using following relations. The summation will be replaced by an integral ρ( k) is the density of states in the K-space. ρ( k) d k (65) k The allowed discrete values of k are obtained by combinations of components k (nx) x = 2π L n x k (ny) y = 2π L n y k (nz) z = 2π L n z (66)

50 50 where the numbers n x,n y,n z are positive and negative integers. It means that each of the allowed vectors k occupies a small volume of the K-space ( ) 2π 3 k x k y k z = = (2π)3 L V (67) The density of states in the K-space is thus a constant (i.e. one per the above small k-space volume), and the above relation can be written as V (2π) 3 k d k (68) Since the derivation of the golden rule assumes integration over frequencies, or energies, we shall transform this integral over momenta (i.e. wave numbers k) to integral over energy, V d (2π) k ρ(e)de 3 V (2π) 3 d k = V dω (2π) 3 k Ω k k 2 dk = Ω k [ ] V dk k2 (2π) 3 de dω k de (69)

51 51 so that the ρ(e) can be identified as V dk k2 dω (2π) 3 k (70) de Ω k If the processes depend on the direction of the wave vector, which is true in photon emission case, we must keep the angular information inside of the density of states, and perform the angular integration afterwards. We must now evaluate the above density of states in terms of energy only, using k = ω E = hkc c Setting these relations of k and E, we obtain the expression for ρ(e) ρ(e) = V (2π) E 2 3 ( hc) dω 3 k = V (2π) 1 3 h ω2 c dω 3 k (71)

52 52 10 Charged Particles In an Electromagnetic Field The Hamiltonian of a spinless particle q and mass m in an electromagnetic field is H = 1 2m (p q A) 2 + qφ (72) where p is the generalized momentum of the particle. Ignoring for the present purpose small spin-dependent terms, the Hamiltonian of an electron of mass m in an electromagnetic field is giving by the above equation, with q = e. In order to describe a hydrogenic atom, as an example, in an electromagnetic field we must also take into account the presence of the nucleus, of charge Ze and mass M. Since M is very large compared to the electron mass m (we are considering here ordinary one-electron systems such as H, He +,...) the interaction between the radiation field and the nucleus can be ignored to a high degree of accuracy. In the same spirit we shall also neglect reduces mass effects and take the nucleus to be the origin of the coordinates. However, we must include in the Hamiltonian the electrostatic Coulomb potential

53 53 Ze 2 /(4πε 0 )r between the electron and the nucleus. It is convenient to regard this electrostatic interaction as an additional potential energy term, while the radiation field which perturbs the atom is described in terms of a vector potential A alone, as discussed above. The time dependent Schrödinger equation for a hydrogenic atom in an electromagnetic field then reads i h [ ] t ψ(r, t) = 1/2m( i h + ea) 2 Ze2 ψ(r, t) (73) (4πε 0 )r where we have written p = i h. Because of the gauge condition A = 0, we have ( Aψ) = A ( ψ) + ( A) ψ }{{} 0 = A ( ψ) (74) so that and A commute. Schrödinger equation as Making use of this fact, we may rewrite the

54 54 i h [ h 2 t ψ(r, t) = 2m 2 Ze2 4πε 0 r i he ] A m + e2 A 2m 2 ψ(r, t) (75) We shall treat the weak field case in which the term in A 2 is small compared with the term linear in A The Hamiltonian of Interaction Now, we study the interaction between radiation and atomic system. In this case the total Hamiltonian of the system is written as H = H atom + H rad + H I = H 0 + H I (76) H atom is the Hamiltonian of the atomic system and H rad is the Hamiltonian of the free radiation field. H I shows the Hamiltonian of the interaction effect between two previous systems. By replacing H I, we replace p by p + e/c A in

55 55 H atom we can construct H I as H I = e mc p A + e2 2mc 2 A 2 H I e mc p A (77) As we see the first term of the eq.(77), is proportional to A that contributes in the transitions involving the emission or absorption of a single photon and the next term that s proportional to A 2 contributes in the transitions involving the emission or absorption of two photons. We can also see that the first term; A; contains just one creation or annihilation operator, but A 2 contains the terms as : a kλ a kλ, a kλ a kλ, a kλ a kλ, a kλ a kλ which correspond to absorption of two photons; emission of two photons. Here we study the first term. The probability for a transition involving two photons contains e 4 while this probability for one photon is proportional to e 2. This extra power of e 2 will lead to one more power of α 1/137. The eigenfunctions of the unperturbed Hamiltonian H are direct products of the atomic and radiation wave functions ψ atom ψ rad = ψ a ; n 1, n 2,..., n i,... (78) where ψ a is the wave function of the atomic Hamiltonian, n i n ki λ i is the

56 56 wave function describing the mode k i λ i in the radiation field. 11 Emission of Radiation (Light) By an Excited Atom: Spontaneous and Stimulated Emission Here we study the emission of a single photon by an excited atom, and we use the time-dependent perturbation theory to find the transition probability for atom.if we consider i as the initial state and f as the final state, then the probability per unit time for a transition by the emission of a single photon from the initial to final state is given by W i f ( k, λ) = 2π h f H p i 2 ρ(e f ) (79) where h k denotes the momentum and λ denotes the polarization, and H p is the perturbing Hamiltonian. This equation is known as Fermi s Golden Rule.

57 57 The perturbing Hamiltonian for this process is H p = e mc p A (80) by referring to the eq.(52), we have H p = e 2π hc 2 ( p ê kλ a kλ e i( k r ωt) + a mc V ω kλ e ) i( k r ωt) (81) k kλ The initial and final states i ; f ;, i.e. the unperturbed wavefunctions before and after the emission of one photon is i = a n kλ (82) f = b n kλ + 1 The energy difference of these two states is (83) E f E i = ( E b + n kλ hω k + hω k ) ( Ea + n kλ hω k ) = Eb E a + hω k (84) Comparing the last part of the eq.(79) with the eq.(84), we find out that δ(e f E i emphasis on the energy conservation in the transition processes. Inserting

58 58 the final and initial states and the Hamiltonian into the matrix element, we obtain f H p i = e 2π hc 2 b p ê kλ e i k r a n kλ + 1 (85) mc V ω k By considering the eq.(85) we can transform the Golden Rule formula eq.(79), into the form W i f ( k, λ) = 2π h ( ( ) e 2 2π hc 2 (n kλ + 1) b p mc) ê kλ e i k r 2 a ρ(ef ) (86) V ω k From the last equation, we see that it can be considered as a sum of two terms; one is proportional to the number of photons in the field n kλ, so it s field dependent; and the other one (expressed by number one in the paranthesis) is independent of the field. The first term, if different from zero, leads to what is called stimulated emission or induced emission, while the other term is responsible for spontaneous emission. It should be mentioned that in emission of a photon, the lifetime of the excited state, against this emission is given by ( 1 τ ) a b = kλ W i f ( k, λ) (87)

59 Dipole Approximation An important simplification is obtained by applying the so called dipole approximation, considering the case kr 1. This means that the wavelength of the emitted photon is much larger than atomic dimension. The electric dipole approximation is obtained by replacing, to a good approximation, i.e. e i k r 1 b pe i k r a b p a This approximation is limited by these requirements: λ 1Å and hω 10 kev. Here we also derive the relation between matrix element of momentum and coordinate using the following commutation relations [ r, p 2 ] = p [r, p] + [r, p] p = 2i hp (88)

60 60 [ r, H atom ] = i h p (89) m we change the matrix element of the momentum to a matrix element of the special coordinate of the atomic electron b p a = m i h b [ r, H atom] a = im h (E a E b ) b r a = imω ab b r a (90) 11.2 Detailed Evaluation of Emission rate For spontaneous emissionis that, we first perform the summation over the two possible polarizations by considering the following figure, fig.4. The figure shows that placing for a moment the z-axis in the direction of the wavevector k, the directions of the polarization vectors ê kλ, for λ = 1, 2 can then define the x- and y-axes, so that p b ê kλ e i k r a 2 pe = b i k r 2 a (sin 2 θcos 2 φ + sin 2 θsin 2 φ) λ=1,2

61 61 = b pe i k r a 2 (sin 2 θ) (91) where the angle θ is the angle between the wave number k and the momentum vector of the electron (if it were classically defined). Conversely, after performing this reduction, we can think that our z-axis is always placed along the direction of p, so that we can identify the angle θ with the polar angle of the photon emission to be integrated over.

62 62 By using the eq.(70) and dipole approximation discussed above, we can evaluate the integral over all emission angles Ω k, ( 1 τ ) a b = e2 ω k 2πm 2 hc 3 where the integration angular variables are b p a 2 sin 2 θ dω k (92) dω k = sin θ dθ dφ (93) Note that in the above evaluations the volume V in ρ[e f ] cancels the volume in the field normalization factor 2π hc 2 V ω k in equation (52). The last equation eq.(92) then becomes ( 1 τ ) = e2 a b ω k b p 2πm 2 a 2 hc3 dφ sin 2 θ sin θ dθ (94)

63 63 The angular integration is easily seen to yield a factor 8π/3, so that finally ( 1 τ ) a b = 4ω abe 2 3 hm 2 c 3 b p a 2 (95) Further we use the relation between matrix element of momentum and coordinate eq. (90) in section 11.1 b p a = imω ab b r a and thus eq.(95) becomes W sp. em. = ( 1 τ ) a b (96) = 4e2 ωab 3 b r a 2 (97) 3 hc 3 = 4 ( ) ( hω ) 3 ε ab α 3 r 2 b 3 h a (98) while we have a = h 2 /me 2 ε = 2 Ryd = e 2 /a = ev and α as the fine structure constant is, α = e 2 / hc = 1/ ε a

64 64 For one-electron hydrogenlike atoms (or ions), the life time of the transition of 2p 1s of one photon is given by ( 1 τ ) 2p 1s = Z 4 sec 1 (99) It can be noted that if the upper state cannot decay to lower state by the emission of one photon, because the matrix element vanishes, it can decay by emitting two photons. For such processes the life time is much longer, i.e. the simultaneous emission of two photons is much less probable. The two photon lifetimes are of the magnitude τ 2 = 0.12/Z 6 sec. For the case of stimulated emission, that is dependent on the field, we can say W st. em. = ( 1 τ ) a b (100) = 4e2 ωab 3 3 hc n b r 3 a 2 (101) = 4 ( ) ( hω ) 3 ε ab α 3 r 2 n ω b 3 h a (102) ε a

65 65 where n ω represents the average number of photons in the field, with the energy of E ω = hω. 12 Einstein Coefficients Consider an enclosure containing atoms (of a single kind) and radiation in equilibrium at absolute temperature T, and let a and b denote two non-degenerate atomic states, with energy value E a and E b such that E b E a. We denote by ρ the energy density of the radiation at the angular frequency ω = (E b E a )/ h. The number of atoms making the transition from a to b per unit time by absorbing radiation, N b, is proportional to the total number N a of atoms in the state a and to the energy density ρ. N b = B N a ρ (103)

66 66 B is called the Einstein Coefficient for the absorption. ρ is defined as ρ = I/c i.e. energy/(length)2 time length/time = energy volume (104) On the other hand, the number of atoms making the transition b a per unit time, N a, is the sum of the number of spontaneous transition per unit time, which is independent of ρ.thus

67 67 N a = AN b + B N b ρ (105) A is the Einstein Coefficient for spontaneous emission. So, atoms can do stimulated emission by the field present. At equilibrium those two changes add to zero So we deduce that ( N b ) + ( N b ) = 0 = ( N b ) = ( N b ) (106) and at thermal equilibrium it s given N a = A + B ρ N b B ρ (107) N a N b = e (Ea E b)/kt = e E/kT = e hω/kt (108) Since the atoms are in equilibrium with the radiation at temperature T, the energy density ρ(ω) is given by the Planck distribution law ρ(ω) = hω3 π 2 c 3 1 e hω/kt 1 (109)

68 68 Note that it builds on the boson assumption. Now by putting (108) into (106) and eliminating N b, we will have gives Note that hω3 as I/c, i.e. Planck s law is We already had e hω/kt B B ρ A = 0 (110) ρ = A e hω/kt B B (111) is not a number! it s energydensity time, because ρ is defined π 2 c 3 energy/(length) 2 time length/time energy volume.frequency ρ = = energy volume. This means, while A is naturally dim.less time. But ρ in ρ = ρ ω (112) A e hω/kt B B = A B 1 e hω/kt 1 Consistency between the present considerations and Plank s formula requires thus B = B

69 69 Now we have ρ ω = A B 1 e hω/kt 1 = ω hω3 π 2 c 3 1 e hω/kt 1 (113) and this gives us A = hω3 ω (114) B π 2 c3 13 Formula Collection This section contains simplified formulae used as a transparency for overview lectures..a = 0 k.a k = 0 B = A B k = k A k

70 70 ω k = c k Vector Potential Quantized H field = 1 8π d 3 r (E.E + B.B) A = 2π hc 2 k,ε k ωv ε ( k a + e ik.r+iωt k,εk + a k,εk e ik.r iωt) w = 2π h f V i 2 ρ(e f ) w = 1 2π ( ) 3 E a.u. hω α 3 h E a.u. dω f ε k. r 2 i ε k a 0 τ 2p 1s (Z) sec Z 4

71 71

72 72 Z K P e K1 Y e K2 X Figure 4: As the z-axis is for the moment in the direction of the wavevector k, the directions of the polarization vectors ê kλ (perpendicular to each other and to the vector k, can then define the x- and y-axes. The angle θ is the angle between the wave number k and the momentum vector of the electron p.

73 73 b B a b b a A a B