《高等量子力学》课程参考教材：An Introduction to Advanced Quantum Physics（Hans P. Paar）.pdf_P16-P20

8 FLECTROMAGNETIC RADIATION AND MATTER The e(k)are called polarization vectors.One can interpret this solu- tion as a four-dimensional version of the Fourier transform which is familiar from the solution of a wave equation in one dimension.The one- dimensional Fourier transform has a pre-factor 1/v2,hence the(1/2)2 in Equation (1.22).This is the inte ral ver n of the Fo rier transfo of Equation (1.22)in the wave (1.21)shows th Equation (1)is indeeda solution if the condition)0issatisfied for each term separately.This condition is satisfied for the Dirac-8 function as in generalx8(x)=0 because the Dirac-8(x)function is even in x and it is multiplied by the odd function x.The general solution in Equation(1.22) satisfies the condition in Equation (1.19)by construction.We know that A and E are parallel,see Equati (1.17 ,so the directi of A as specified by ai(k,)e(k)and a2(k,)e2(k)is the direction of polarization We can simplify Equation(1.22)by integrating over using the constraint provided by the factor 8(k2)=0.The condition k2 =k2-02=0 leads to the requirement that=kl.This is also expressed by the property of the Dirac-8 function a)=s2-=a[6+o+8- (1.23) We introduce @k =lkl and obtain A(x,t) 会尝2ak++水 2∫兴ak-w+ak内 (1.24) We call the first term in Equation (1.24)with the negative value of the tion,while we call the second t m in Equation(1.24 of the positive. to remove ar and a lactor nd tadacrr Equation(1.24).This merely redefines the coefficients ax.The reason for the first change is that the Fourier integral over dsk ought to be accompanied by a pre-factor 1/v2 for each integration variable.The reason for the other two changes will become clear in the next subsection. We v the)Equation (1.24).The positive energy term is already of this form but the negative energy term is not because the terms k.x and @t in the exponential do not have opposite signs.We can arrange for that by introducing a new integration variable k'=-k in the negative energy term only.This gives

8 ELECTROMAGNETIC RADIATION AND MATTER The ελ(k) are called polarization vectors. One can interpret this solution as a four-dimensional version of the Fourier transform which is familiar from the solution of a wave equation in one dimension. The onedimensional Fourier transform has a pre-factor 1/ √ 2π, hence the (1/2π) 2 in Equation (1.22). This is the integral version of the Fourier transform. Substitution of Equation (1.22) in the wave Equation (1.21) shows that Equation (1.22) is indeed a solution if the condition k2δ(k2) = 0 is satisfied for each term separately. This condition is satisfied for the Dirac-δ function as in general x δ(x) = 0 because the Dirac-δ(x) function is even in x and it is multiplied by the odd function x. The general solution in Equation (1.22) satisfies the condition in Equation (1.19) by construction. We know that A and E are parallel, see Equation (1.17), so the direction of A as specified by a1(k, ω)ε1(k) and a2(k, ω)ε2(k) is the direction of polarization. We can simplify Equation (1.22) by integrating over ω using the constraint provided by the factor δ(k2) = 0. The condition k2 = k2 − ω2 = 0 leads to the requirement that ω = ±|k|. This is also expressed by the property of the Dirac-δ function δ(k2) = δ(k2 − ω2) = 1 2ωk δ(|k| + ω) + δ(|k| − ω) (1.23) We introduce ωk = |k| and obtain A(x, t) = 1 2π 2 d3kdω 2ωk 2 λ=1 aλ(k, ω) ei(k·x−ωt) δ(ωk + ω) + δ(ωk − ω) ελ(k) = 1 2π 2 d3k 2ωk 2 λ=1 aλ(k, −ωk) ei(k·x+ωkt) + aλ(k, ωk) ei(k·x−ωkt) ελ(k) (1.24) We call the first term in Equation (1.24) with the negative value of ω the negative energy solution, while we call the second term in Equation (1.24) with the positive value of ω the positive energy solution. It is conventional to remove a factor 1/ √ 2π and a factor 1/ 2ωk and to add a factor √4π in Equation (1.24). This merely redefines the coefficients aλ. The reason for the first change is that the Fourier integral over d3k ought to be accompanied by a pre-factor 1/ √2π for each integration variable. The reason for the other two changes will become clear in the next subsection. We would like to reinstate the compact notation exp(ikx) in Equation (1.24). The positive energy term is already of this form but the negative energy term is not because the terms k · x and ωkt in the exponential do not have opposite signs. We can arrange for that by introducing a new integration variable k = −k in the negative energy term only. This gives

HAMILTONIAN AND VECTOR POTENTIAL 9 three minus signs from d3k = −d3k and changes the signs of the integration limits in each of the three integrals. Exchanging the new upper and lower integration limits in each of the three integrals gives another three minus signs. The net result is that all minus signs cancel. We drop the prime in k . The argument of aλ in the negative energy term will now be −k. Another way to see this is to consider that we are looking for solutions of the wave equation in a large box with periodic boundary conditions at its surface. Solutions with +k and −k have the same physical properties. The expression for A becomes A(x) = 1 2π 3 2 √ 4π d3k 2ωk 2 λ=1 aλ(−k, −ωk) e−ikxελ(−k) + aλ(k, ωk) eikxελ(k) (1.25) A is used to calculate E and B according to Equation (1.17). In the next section A, E and B become operators. The eigenvalues of E and B are observable, so the operators E and B and therefore A must be Hermitian operators. This means that A must be real quantity and we require that A = A∗. We find that A∗(x) = 1 2π 3 2 √ 4π d3k 2ωk 2 λ=1 a∗ λ(−k, −ωk) eikxε∗ λ(−k) + a∗ λ(k, ωk) e−ikxε∗ λ(k) (1.26) Comparing Equation (1.26) with Equation (1.25) and using that the exponentials exp(±ikx) are orthogonal to each other, we find that λ aλ(−k, −ωk)ελ(−k) = λ a∗ λ(k, ωk)ε∗ λ(k) (1.27) λ aλ(k, ωk)ελ(k) = λ a∗ λ(−k, −ωk)ε∗ λ(−k) (1.28) The two relations are each other’s complex conjugate, so there is just one condition on aλ and its complex conjugate. We will not attempt to formulate relations between individual terms in the sums over λ because we do not need those, and if we try it would lead to a left-handed set of three unit vectors ε or the appearance of minus signs in nasty places. Using Equation (1.27) in Equation (1.25) we have the result A(x) = 1 2π 3 2 √ 4π d3k 2ωk 2 λ=1 aλ(k) eikxελ(k) + a∗ λ(k) e−ikxε∗ λ(k) (1.29)

0 FLECTROMAGNETIC RADIATION AND MATTER We shall also use the form where a discrete sum over k with an infinite number of terms is used.This corresponds to quantization in a box with length L.The prefactor in a one-dimensional Fourier series becomes 1/vL instead of 1/v2 so in three dimensions we get a pre-factor 1/V.One can consider the integral form of the Fourier transform as the limiting case where L (and V)go to he use a Fourier series instead of a Fourier integral we A=VF∑ae产e+kee (1.30) The summation over is understood to be from 1 to 2.We do not explicitly show the dependence of ax (and a)upon wk because wk=lkl so it is specified by k.We allow for complex ex(k)for applications with circular polarized electromagnetic radiation The ex(k) a be and can in that case outide the sauare bracke ar o uo (1.29)and Equanor (1.30).The expressions Equation (1.29)and Equation (1.30)explicitly show that A is real because A is the sum of two terms that are each other's complex conjugate.We will sometimes write h.c.'(Hermitian conjugate) for the second term. We have discussed how to qu uantize the part of the classical Hamiltonian given in E 1(1.1.wWe now quantize the cla Equation (1.0)after which Aand hall see that the operator A and thus Hem create and annihilate photons.Very interesting stuff indeed! 1.2 SECOND QUANTIZATION 1.2.1 Commutation Relations Planck's 1901 treatment of black-body radiation and Einstein's 1905 dis cussion of the photo-electric effect assume that electromagnetic energy Equation(1.20)is quantized.Likewise,Compton scattering shows that the momentum of the electromagnetic field is quantized as well.Besides the fact that quantization itself needs to be'explained',both quantizations have spe cific values for their quanta whose val es need to be explained'as well.The quantiz ons of en rgy and momentum of the electr magne tic field and the numerical values of their quanta are introduced at the start of most Quan- tum Physics courses,but they are not derived there.This was also the case historically:Second Quantization,which provides this derivation,was not developed until about 1930.The quantization of the electromagnetic field contains many subtleties that have given rise to a variety of sophisticated treatments.We will not do that here rather follow a more her ristic path

10 ELECTROMAGNETIC RADIATION AND MATTER We shall also use the form where a discrete sum over k with an infinite number of terms is used. This corresponds to quantization in a box with length L. The prefactor in a one-dimensional Fourier series becomes 1/ √ L instead of 1/ √ 2π so in three dimensions we get a pre-factor 1/ √V. One can consider the integral form of the Fourier transform as the limiting case where L (and V) go to infinity and the sum becomes an integral. If we use a Fourier series instead of a Fourier integral we get A(x) = 4π V k,λ 1 2ωk aλ(k) eikxελ(k) + a∗ λ(k) e−ikxε∗ λ(k) (1.30) The summation over λ is understood to be from 1 to 2. We do not explicitly show the dependence of aλ (and a∗ λ) upon ωk because ωk = |k| so it is specified by k. We allow for complex ελ(k) for applications with circular polarized electromagnetic radiation. The ελ(k) are real for linearly polarized radiation, and can in that case be taken outside the square bracket part of Equation (1.29) and Equation (1.30). The expressions Equation (1.29) and Equation (1.30) explicitly show that A is real because A is the sum of two terms that are each other’s complex conjugate. We will sometimes write ‘h.c.’ (Hermitian conjugate) for the second term. We have discussed how to quantize the part of the classical Hamiltonian given in Equation (1.1). We will now quantize the classical Hamiltonian Equation (1.20) after which A and Hem are operators. We shall see that the operator A and thus Hem create and annihilate photons. Very interesting stuff indeed! 1.2 SECOND QUANTIZATION 1.2.1 Commutation Relations Planck’s 1901 treatment of black-body radiation and Einstein’s 1905 discussion of the photo-electric effect assume that electromagnetic energy Equation (1.20) is quantized. Likewise, Compton scattering shows that the momentum of the electromagnetic field is quantized as well. Besides the fact that quantization itself needs to be ‘explained’, both quantizations have specific values for their quanta whose values need to be ‘explained’ as well. The quantizations of energy and momentum of the electromagnetic field and the numerical values of their quanta are introduced at the start of most Quantum Physics courses, but they are not derived there. This was also the case historically: Second Quantization, which provides this derivation, was not developed until about 1930. The quantization of the electromagnetic field contains many subtleties that have given rise to a variety of sophisticated treatments. We will not do that here but rather follow a more heuristic path

SECOND OUANTIZATION 11 We know from the treatment of the Harmonic Oscillator,using the operator formalism as opposed tothe Hermite polynomils that we can lower and raise the energy of the system with the operators A and At,one quantum of energy ho at a time.This is the time to review that material in your favorite textbook.There appears to be some similarity with a quantized electromagnetic field whose energy can be increased or decreased by the creation or annihilation of photons.This similarity will be explo ited.The A and At opera of the Ha rmonic Oscill ato commutation relations that were derived e co nmutat on relation between momentum and position in Equation(1.5).In Second Quantization we postulate commutation relations between the coefficients ax and a*in Equation(1.29)and Equation(1.30)that are the same as those for A and At This means of course that the a and a*become operators,so we write Hermitian conjugate at instead of complex conjugate a.Because the(k) and are operators,so is A and therefore are E and B!The formalism of second quantization starts with postulating commutation relations a(k),a,(k】=dd (1.31) [a(k,a(k】=ak,a(K】=0 (1.32) The second relation in Equation(1.32)is the Hermitian conjugate of the first one in Equation(1.32),so it provides no additional information.These invo enta k and polarizations A but are ewise identical toth r the H armonic The same commutation relations lead to the same results so there is no need to derive these results again here.This is one reason why it is important to treat the Harmonic Oscillator with the operator formalism, even though analytic solutions of the Schodinger equation exist that have the same physical content.Thus we have the following relations ak)(k)=√m(k)+1n(k)+1) (1.33) a2.(k)n(k)=√(k)m(k)-1 (1.34) where the kets form an orthonormal set.We call the at and a creation and annihilation operators instead of raising and lowering operators because they are thought to create and annihilate quanta(photons).Also in analogy with the Harmonic Oscillator,we have an operator N defined by Ni(k)a:(k)a(k) (1.35) called the number operator.It satisfies the commutation relations [a(k,N(k】=88'a(k) (1.36) [a(k,N(k月=-drda(k) (1.37)

SECOND QUANTIZATION 11 We know from the treatment of the Harmonic Oscillator, using the operator formalism as opposed to the solutions using Hermite polynomials, that we can lower and raise the energy of the system with the operators A and A†, one quantum of energy ω at a time. This is the time to review that material in your favorite textbook. There appears to be some similarity with a quantized electromagnetic field whose energy can be increased or decreased by the creation or annihilation of photons. This similarity will be exploited. The A and A† operators of the Harmonic Oscillator satisfy commutation relations that were derived from the commutation relations between momentum and position in Equation (1.5). In Second Quantization we postulate commutation relations between the coefficients aλ and a∗ λ in Equation (1.29) and Equation (1.30) that are the same as those for A and A†. This means of course that the a and a∗ become operators, so we write Hermitian conjugate a† instead of complex conjugate a∗. Because the aλ(k) and a† λ(k) are operators, so is A and therefore are E and B! The formalism of second quantization starts with postulating commutation relations [aλ(k), a† λ(k )] = δkkδλλ (1.31) [aλ(k), aλ(k )] = [a† λ(k), a† λ(k )] = 0 (1.32) The second relation in Equation (1.32) is the Hermitian conjugate of the first one in Equation (1.32), so it provides no additional information. These commutation relations involve momenta k and polarizations λ but are otherwise identical to those for the Harmonic Oscillator. The same commutation relations lead to the same results so there is no need to derive these results again here. This is one reason why it is important to treat the Harmonic Oscillator with the operator formalism, even though analytic solutions of the Schodinger equation exist that have ¨ the same physical content. Thus we have the following relations a† λ(k) nλ(k) = nλ(k) + 1 nλ(k) + 1 (1.33) aλ(k) nλ(k) = nλ(k) nλ(k) − 1 (1.34) where the kets form an orthonormal set. We call the a† and a creation and annihilation operators instead of raising and lowering operators because they are thought to create and annihilate quanta (photons). Also in analogy with the Harmonic Oscillator, we have an operator N defined by Nλ(k) = a† λ(k) aλ(k) (1.35) called the number operator. It satisfies the commutation relations [aλ(k), Nλ(k )] = δkkδλλ aλ(k) (1.36) [a† λ(k), Nλ(k )] = −δkkδλλ a† λ(k) (1.37)