III. Quantization of electromagnetic field

Transcription

1 III. Quantization of electroagnetic field Using the fraework presented in the previous chapter, this chapter describes lightwave in ters of quantu echanics. First, how to write a physical quantity operator in general is introduced. Net, a haronics oscillator is quantized according to this ethod. Then, light is quantized, following the way of quantizing a haronic oscillator. Canonical quantization (the procedure of transforation fro a classical to a quantu worlds) In classical echanics, position and oentu p are the two fundaental physical quantities for epressing physical systes. An arbitrary physical quantity A can be epressed by and p. A = A(, p) A typical eaple is the energy of a physical syste H. H = H(, p) The canonical quantization is a ethod of quantizing physical quantities as; First, a physical quantity is epressed by {, p} in a classical way, and then{, p} are replaced by operators { } that satisfy, p = iħ, p (This relationship is presented in the previous section.) Haronic oscillator A haronics oscillator is a physical syste that oves in a sinusoidal anner, which can be quantizes as follows. We consider a spring oscillation as an eaple of a haronic oscillator. k :position :ass k:spring const. d k dt cos k t Haronics oscillation with an angular frequency of k kinetic energy: potential energy: v: velocity p v p = v: oentu k k k' d' The classical energy is p quantization p p

2 energy operator: p H Eigenvalues of this energy operator, i.e., possible energy values, is derived as follows. First, the following operators are introduced for the derivation. a = (ω + i p) a = (ω i p) = p = i ħ ω ( a + a) ( a a) ( and are herite conjugate) (, thus and are not Heritian operators not observable) The energy operator is rewritten by H = a a {, a } as a a + ω ħ ω a + a a + a = 4 a a a a a a + a a + 4 a a + a a + a a + a a = a a + a a = a a + a a a a + a a = a a + [ a, a ] + a a = a a + [, ] aa = { ω + i p ω i p ω i p ω + i p } = i ω p p = i, p = ħ uncertainty relationship Here, we introduce an operator defined as Using is an Heritian operator. Thus, an arbitrary state can be epressed by a linear cobination of eigenstates of., the energy operator is rewritten as H = n + This epression indicates that it is good enough to derive eigenvalues of Thus, we will discuss eigenvalue/eigenstate of in the following. n n n n n: eigenvalue n>: eigenstate The inner product with n> left: n n n n a n n right: n n n n n n n n a n to have energy eigenvalues. non-negative

3 Operate onto the eigenvalue/eigenstate equation. left: a n aa n ( aa ) n ( ) n ( ) n right: n n 3 n n n n n n n ( n ) n a n is an eigenstate of, with an eigenvalue of (n ). Fro the definition, an eigenstate with an eigenvalue of (n ) is n > n cn n c n :proportional const. (inner product with itself) n n n is an operator that decreases n by annihilation operator n a a n n cn cn n n n n cn n n n cn c n n Phase can be without loss of generality. Fro the above, Eigenvalue of, n, is non-negative. n n n n is a natural nuber. Consecutively operate onto n> a n n n n n n( n ) n c n If n is not a natural nuber.5.5 (.5) If n is a natural nuber n becoes negative. NG n is a natural nuber for n to be non-negative Siilarly for left: n n n n a n n ( ) n ( aa ) n ( na ) n right: n n

4 ( n ) n n n n n ( n ) n a n is an eigenstate of, with an eigenvalue of (n + ). 4 a n n a n n n is an operator that increases n by creation operator n aa n ( n a ) n n ( ) n d n n n [, ] aa = Utilizing the above results, eigenvalues of the energy operator are obtained as; H = n + Hn >= n + n >= n + n > eigenvalue E n = n + (n =,,, 3,,,,) Energy eigenvalue of a haronic oscillator. Energy of electroagnetic wave discrete in the quantu world Now, let us describe an electroagnetic wave quantu-echanically, based on the previous section. First, we will see that the structure of the energy epression for an electroagnetic wave is siilar to that for a haronic oscillator. Then, electroagnetic wave will be quantized, following the quantization of a haronic oscillator. An electroagnetic wave is classically epressed as i( kz i( kz E( z, Ee Ee i( kz i( kz By( z, Be Be where E B c (rewrite) i( kz i( kz E( z, A{ iae ia e } A i( kz i( kz By ( z, { iae ia e } c a: noralized aplitude A = ε V constant for a to be nondiensional.

5 The classical energy of an electroagnetic wave is 5 H ( E H y ) dv V R ( E By ) R E A R { a e i( kz ( a ) e i( kz A R { a cos[( kz ] aa R: cross section area in the -y plain : length along the z direction aa a a a} a} A R { aa A R( aa a oscillating with k a} a a) = (aa + a a) << linearly increasing (aa a a) Here, we decopose a into the real part q and the iaginary part p as a = (ωq + ip) a = (ωq ip) cos[ ( kz ] z R Substitute the into energy H. H ( p q ) On the other hand, the energy of a haronic oscillator is classically written as H p (p: oentu, : position, : ass) The sae structure with =. An electroagnetic wave can be quantized siilarly to a haronic oscillator.

6 Quantization of electroagnetic wave 6 Coparison of the energy epressions of an electroagnetic wave and a haronic oscillator EM wave haronic p p q Relying on this relationship, {p, q} of electroagnetic wave are quantized as {p, } of an haronic oscillator as: p p q q with q, p = q p p q = iħ According to the quantization of {p, q}, {a, a } are also quantized as: a a = (ω q + i p) a a = (ω q i p) annihilation operator satisfying creation operator satisfying n n n a n n n et us confir the coutation relation. a, a = a a a a = { ω q + i p ω q i p ω q i p ω q + i p } = ω q iω q p + iω p q + p ω q iω q p + iω p q p = i ħ q, p = [, ] According to the above quantization, the energy of electroagnetic wave is quantized as: H ( p q ) ( H p q ) = [ ( a a ) + ω ħ ω ( a + a ) = ( a a + a a) = a a + Siilarly to a haronic oscillator, we introduce an operator defined as: which is an Heritian operator, and has eigenvalue/eigenstate as: n n n n - n is a easured value of physical quantity. - n is a natural nuber (n =,,, 3,,,, )

7 Using, the energy operator is rewritten as 7 H = n + Then, the energy operator of an electroagnetic wave has eigenvalue/eigenstate as Hn >= n + n > This equation indicates that energy values of an electroagnetic wave are discrete, and is written as E n = n + (n =,,, 3,,,, ) Here, we should recall that the energy of lightwave has a iniu unit (i.e., photon) and thus is discrete. Also, the energy of one photon is hn or. The above eigenvalues of the energy operator indicate these postulates. Eigenstate n> represents the state of light whose energy value is (n + ) or whose photon nuber is n. This state is called photon nuber state or Fock state. Note in the epression of eigenvalues that light has an energy of even when it has no photon. This is called zero-point energy originating fro the vacuu fluctuation or the quantu fluctuation. The details will appear in the net chapter. In the end, an electroagnetic wave is quantized, based on the above discussion. E(z, = i ε V { aei(kz ω a e i(kz ω } This operator is an Heritian operator. E = i ε V a e i kz ωt ae i kz ωt = E This epression will be used in the following chapters.