221B Lecture Notes, Lecture notes of Quantum Mechanics

Download 221B Lecture Notes and more Lecture notes Quantum Mechanics in PDF only on Docsity! 221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content, yet applicable to wider varieties of systems than the conventional formulation of quantum mechanics. 1.1 Limitations of Multi-body Schrödinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schrödinger equation is a differential equation acting on the arguments of the Schrödinger wave function, and can never change the number of arguments. Similarly, it is useful to consider elementary excitations in multi-body systems, such as phonons, and they can be created or annihilated by putting energy into the system. Finally even the number of matter particles changes in relativistic elementary paticle physics by pair creations, pair annihilations etc. The limitaions of multi-body Schrödinger wave function mentioned here 1 call for a better formalism. The quantum field theory is designed specifically for that. 1.2 Aim of the Quantum Field Theory Quantum Field Theory is sometimes called “2nd quantization.” This is a very bad misnomer because of the reason I will explain later. But nonetheless, you are likely to come across this name, and you need to know it. The aim of the quantum field theory is to come up with a formalism which is completely equivalent to multi-body Schrödinger equations but just better: it allows you to consider a variable number of particles all within the same framework and can even describe the change in the number of particles. It also gives totally symmetric or anti-symmetric multi-body wave function au- tomatically, not as a consequence of symmetrization or anti-symmetrization done “by hand.” Moreover, once we go to relativistic systems, quantum field theory is the only known formalism to describe quantum mechanics consis- tent with Lorentz invariance and causality. It also allows a systematic way of organizing perturbation theory in terms of “Feynman diagrams,” a graphical representation of each term in perturbation theory. Even though the contents of multi-body quantum mechanics and quantum field theory are exactly the same in systems where the number of particles does not change, it turns out that the field theory is a far better formalism for many purposes and is widely used in condensed matter physics, elementary particle physics, quantum optics, and in some cases also in atomic/molecular physics and nuclear physics. It is particularly suited to multi-body problems. In systems where the number of particles changes, or where one needs super- position of states with different number of particles (sounds odd, but we will see examples later), quantum field theory is crucial. But it is clear that the formalism will confuse you (at least) once be- cause of too-close resemblance but conceptual difference from the ordinary Schrödinger equation. I will try to make it as clear as possible below. 1.3 Particle-Wave Duality It is interesting to note that the way quantum mechanics and quantum field theory work is a sort of the opposite. In quantum mechanics, you start with classical particle Hamiltonian mechanics, with no concept of wave or interfer- ence. After quantizing it, we introduce Schrödinger wave function and there 2 3 Quick Review of Quantization Procedure Let us briefly review how we quantize a particle mechanics system. Given an action on the phase space (pi, qi), S = ∫ dt L = ∫ dt ( ∑ i piq̇i −H(pi, qj)), (11) the variational principle gives the Hamilton equations of motion δS δpi = q̇i − ∂H ∂pi = 0, (12) δS δqi = −ṗi − ∂H ∂qi = 0. (13) You can again add a surface term and make it look symmetric between p and q as L = ∑ i 1 2 (piq̇i − qiṗi) − H(p, q). To quantize this system, we first identify the canonically conjugate momentum of the coordinate qi by pi = ∂L ∂q̇i . (14) Then we set up the canonical commutation relation [pi, pj] = [qi, qj] = 0, [pi, qj] = −ih̄δij. (15) This defines the quantum theory. The index i runs over all degrees of free- dom in the system. The quantum mechanical state is given by a ket |ψ(t)〉, which admits a “coordinate representation” 〈q|ψ〉 = ψ(q, t). The dynamical evolution of the system is given by the Schrödinger equation ih̄ ∂ ∂t |ψ(t)〉 = H(pi, qj)|ψ(t)〉. (16) It is also useful to recall the commutation relation between creation and annihilation operator of harmonic oscillators [ai, a † j] = δij, [a, a] = [a†, a†] = 0. (17) Here, I assumed there are many harmonic oscillators labeled by the subscript i or j. The Hilbert space is constructed from the ground state |0〉 which satisfies ai|0〉 = 0 (18) 5 for all i. All other states are given by acting creation operators on the ground state |n1, n2, · · · , nN〉 = 1√ n1!n2! · · ·nN ! (a†1) n1(a†2) n2 · · · (a†N)nN |0〉 (19) We will use the same method to construct Hilbert space for the quantized Schrödinger field. 4 Quantized Schrödinger Field Now we try to quantize the classical field theory given by the action Eq. (9). The procedure is the same as in particle mechanics, except that the index i that runs over all degrees of freedom is now replaced by ~x. What it means is that ~x is not an operator (!) but merely an index. This is probably one of the most confusing points about the field theory. φ(~x) at different positions are regarded as independent canonical coordinates exactly in the same way that qi with different i are regarded as independent canonical coordinates in particle mechanics. 4.1 Canonical Commutation Relation The canonically conjugate momenta π(~x) of the canonical coordinates ψ(~x) is obtained from Eq. (9) in the same way as in Eq. (14), namely π(~x) = ∂L ∂φ̇(~x) = iφ∗(~x). (20) We will use φ† below instead of φ∗ because it is more common notation for complex conjugates (hermitean conjugates) for operators. Note that we regard the integration over space ~x in the Lagrangian a generalization of the summation over the index i in Eq. (11). Given the canonical coordinates φ(~x) and canonical momenta π(~x), we introduce canonical commutation relation [π(~x), φ(~y)] = [iφ†(~x), φ(~y)] = −ih̄δ(~x− ~y). (21) The delta function is a generalization of the Kronecker’s delta in Eq. (15). 6 This defines the quantum theory of the Schrödinger field.1 The canonical commutation relation Eq. (21) is very similar to the case of the harmonic oscillator. We now go back to the normalization ψ(~x) = φ(~x)/h̄1/2, and we find [ψ(~x), ψ†(~y)] = δ(~x− ~y), [ψ, ψ] = [ψ†, ψ†] = 0, (22) which resembles the case of harmonic oscillator even better. Now you see that the use of ψ(~x) was more convenient (we are not afraid of having h̄ in the Lagrangian any more because we are discussing a quantum theory anyway). We now regard ψ(~x) as annihilation operator and ψ†(~x) creation operator of a particle at position ~x. The Hamiltonian of the system from the action Eq. (9) is read off in the same way as in Eq. (11): H = ∫ d~x ( φ†−∆ 2µ φ+ 1 2 γ(φ†φ)2 ) = ∫ d~x ( ψ†−h̄ 2∆ 2m ψ + 1 2 λψ†2ψ2 ) . (23) Here I recovered m = h̄µ, and introduced λ = h̄2γ to save space (and ink). Note that H|0〉 = 0 (24) because ψ(~x) in the Hamiltonian annihilates directly the vacuum. A general state must satisfy the Schrödinger equation in the same way as in particle mechanics ih̄ ∂ ∂t |Ψ(t)〉 = H|Ψ(t)〉 (25) It is useful for later purpose to calculate the commutator [H,ψ†(~x)]. [H,ψ†(~x)] = ∫ d~y [ ψ†(~y) −h̄2∆~y 2m ψ(~y) + 1 2 λψ†2(~y)ψ(~y)2, ψ†(~x) ] = ( −h̄2∆~x 2m ψ†(~x) + 1 2 λψ†2(~x)2ψ(~x) ) . (26) 1The commutation relation is given in the Schrödinger picture, where the opera- tors do not evolve in time while the states do. When switching to Heisenberg picture, we have to specify that the canonical commutation relation holds only at equal times: [π(~x, t), φ(~y, t)] = [iφ†(~x, t), φ(~y, t)] = −ih̄δ(~x− ~y). Commutation relation of operators at unequal times would depend on the dynamics of the system. 7 The first quantity we calculate is again its norm. Here we go: 〈~x1, ~x2|~y1, ~y2〉 = 1 2 〈0|ψ(~x2)ψ(~x1)ψ †(~y1)ψ †(~y2)|0〉 = 1 2 〈0|ψ(~x2)([ψ(~x1), ψ †(~y1)] + ψ†(~y1)ψ(~x1))ψ †(~y2)|0〉 = 1 2 〈0|ψ(~x2)δ(~x1 − ~y1)ψ †(~y2) + ψ(~x2)ψ †(~y1)ψ(~x1))ψ †(~y2)|0〉 = 1 2 〈0|δ(~x1 − ~y1)[ψ(~x2), ψ †(~y2)] + [ψ(~x2), ψ †(~y1)][ψ(~x1)), ψ †(~y2)]|0〉 = 1 2 (δ(~x1 − ~y1)δ(~x2 − ~y2) + δ(~x1 − ~y2)δ(~x2 − ~y1)) . (37) This normalization suggests that we are dealing with a two-particle state of identical particles, because the norm is non-vanishing when ~x1 = ~y1 and ~x2 = ~y2, but also when ~x1 = ~y2 and ~x2 = ~y1, i.e., two particles interchanged. A general two-particle state is given by |Ψ(t)〉 = ∫ d~x1d~x2Ψ(~x1, ~x2, t)|~x1, ~x2〉 = 1√ 2 ∫ d~x1d~x2Ψ(~x1, ~x2)ψ †(~x1)ψ †(~x2)|0〉. (38) Because [ψ†(~x1), ψ †(~x2)] = 0, the integration over ~x1 and ~x2 is symmetric under the interchange ~x1 ↔ ~x2, and hence Ψ(~x1, ~x2, t) = Ψ(~x2, ~x1, t). The symmetry under the exchange suggests that we are dealing with identical bosons. The fact that we see two identical particles becomes clearer by again working out the time evolution of a general state. Using the Schrödinger equation Eq. (25) with the Hamiltonian Eq. (23) and the state Eq. (36), the l.h.s of the Schrödinger equation is ih̄ ∂ ∂t |Ψ(t)〉 = ∫ d~x1d~x2 ( ih̄ ∂ ∂t Ψ(~x1, ~x2, t) ) |~x1, ~x2〉. (39) On the other hand, the r.h.s of the Schrödinger equation is √ 2H|Ψ(t)〉 = ∫ d~x1d~x2Ψ(~x1, ~x2, t)Hψ †(~x1)ψ †(~x2)|0〉 = ∫ d~x1d~x2Ψ(~x1, ~x2, t)([H,ψ †(~x1)] + ψ†(~x1)H)ψ†(~x2)|0〉 = ∫ d~x1d~x2 ( −h̄2∆~x1 2m ψ†(~x1) + 1 2 λψ†2(~x1)2ψ(~x1) ) Ψ(~x1, ~x2, t)ψ †(~x2)|0〉 10 + ∫ d~x1d~x2Ψ(~x1, ~x2, t)ψ †(~x1)[H,ψ †(~x2)]|0〉 = ∫ d~x ( ψ†(~x1) −h̄2∆~x1 2m + λψ†2(~x1)ψ(~x1) ) Ψ(~x1, ~x2, t)ψ †(~x2)|0〉 + ∫ d~x1d~x2ψ †(~x1) ( −h̄2∆~x2 2m ψ†(~x2) + 1 2 λψ†2(~x2)2ψ(~x2) ) Ψ(~x1, ~x2, t)|0〉 = √ 2 ∫ d~x1d~x2 ( −h̄2∆~x1 2m + −h̄2∆~x2 2m + λδ(~x1 − ~x2) ) Ψ(~x1, ~x2, t)|~x1, ~x2〉. (40) Comparing Eqs. (39,40), we find ih̄ ∂ ∂t Ψ(~x1, ~x2, t) = ( −h̄2∆~x1 2m + −h̄2∆~x2 2m + λδ(~x1 − ~x2) ) Ψ(~x1, ~x2, t) (41) which is nothing but a Schrödinger equation for two-particle wave func- tion Ψ(~x, t), with a delta function potential as an interaction between them. Therefore, the Fock space with two creation operators correctly describes the two-particle quantum mechanics. If we want a general interaction potential between them, the action Eq. (9) must be modified to S = ∫ dt [∫ d~x ( ψ∗ih̄ψ̇ + ψ∗ h̄ 2∆ 2m ψ ) − 1 2 ∫ d~xd~yψ∗(~x)ψ∗(~y)V (~x− ~y)ψ(~y)ψ(~x) ] . (42) The corresponding Hamiltonian is H = ∫ d~xψ∗−h̄ 2∆ 2m ψ + 1 2 ∫ d~xd~yψ∗(~x)ψ∗(~y)V (~x− ~y)ψ(~y)ψ(~x). (43) You can follow exactly the same steps as above and derive the equation ih̄ ∂ ∂t Ψ(~x1, ~x2, t) = ( −h̄2∆~x1 2m + −h̄2∆~x2 2m + V (~x1 − ~x2) ) Ψ(~x1, ~x2, t). (44) In summary, the quantized Schrödinger field correctly describes the two in- teracting identical bosons appropriately. 4.5 Three and More Particles Following the same analysis as two-particle case, we can consider the three- particle state |~x1, ~x2, ~x3〉 = 1√ 3! ψ†(~x1)ψ †(~x2)ψ †(~x3)|0〉, (45) 11 and its norm is given by 〈~x1, ~x2, ~x3|~y1, ~y2, ~y3〉 = 1 3! [δ(~x1 − ~y1)δ(~x2 − ~y2)δ(~x3 − ~y3) + δ(~x1 − ~y2)δ(~x2 − ~y3)δ(~x3 − ~y1) + δ(~x1 − ~y3)δ(~x2 − ~y1)δ(~x3 − ~y2) + δ(~x1 − ~y2)δ(~x2 − ~y1)δ(~x3 − ~y3) +δ(~x1 − ~y3)δ(~x2 − ~y2)δ(~x3 − ~y1) + δ(~x1 − ~y1)δ(~x2 − ~y3)δ(~x3 − ~y2)] . (46) A general three-particle state is |Ψ(t)〉 = ∫ d~x1d~x2d~x3Ψ(~x1, ~x2, ~x3, t)|~x1, ~x2, ~x3〉. (47) Using the Hamiltonian Eq. (43), the Schrod̈inger equation reduces to ih̄ ∂ ∂t Ψ(~x1, ~x2, ~x3) = [ − h̄ 2∆~x1 2m − h̄2∆~x2 2m − h̄2∆~x3 2m + V (~x1 − ~x2) + V (~x1 − ~x3) + V (~x2 − ~x3) ] Ψ(~x1, ~x2, ~x3). (48) In general, an n-particle state can be constructed as |Ψ〉 = 1√ n! ∫ d~x1 · · · d~xnΨ(~x1, · · · , ~xn)ψ†(~x1) · · ·ψ†(~xn)|0〉. (49) The symmetry of the wave function under interchange of any of the n particles is automatic, and hence it describes identical bosons. 4.6 The Number Operator The total number of particles is an eigenvalue of the operator N = ∫ d~xψ†(~x)ψ(~x). (50) This is probably obvious from the analogy to the harmonic oscillator, but nonetheless let us prove it. First consider the commutators [N,ψ(~x)] = ∫ d~y[ψ†(~y)ψ(~y), ψ(~x)] = ∫ d~y(−δ(~y − ~x))ψ(~y) = −ψ(~x), (51) 12 teraction Hamltonian is somewhat more complicated ∆H = ∫ d~xd~y 1 2 ψ∗(~x)ψ∗(~y)V (~x− ~y)ψ(~y)ψ(~x) = 1 2L6 ∑ ~p,~p′,~q,~q′ ∫ d~xd~ya†(~p)a†(~p′)V (~x− ~y)a(~q′)a(~q)e−i(~p−~q)·~x/h̄e−i(~p′−~q′)·~y/h̄ = 1 2L3 ∑ ~p,~p′,~q,~q′ ∫ d~xa†(~p)a†(~p′)V (~x− ~y)a(~q′)a(~q)δ~p+~p′,~q+~q′e−i(~p−~q)·~x/h̄ = 1 2 ∑ ~p,~p′,~q,~q′ V (~p− ~q)a†(~p)a†(~p′)a(~q′)a(~q)δ~p+~p′,~q+~q′ , (62) with V (~p− ~q) = 1 L3 ∫ d~x V (~x)e−i(~p−~q)·~x/h̄. (63) Kronecker’s delta represents the momentum conservation in the scattering process due to the potential V . Therefore the interpretation of the Hamiltonian is quite simple. The free part counts the number of particles and associates the kinetic energy ~p2/2m for each of them according to its momentum. The potential term causes scattering, by annihilating two particles in momentum states ~q, ~q′ and create them in different momentum states ~p, ~q′ with the amplitude V (~p − ~q) (note that ~p− ~q = ~q′ − ~q′ because of the momentum conservation). At this point, it is useful to note that one can introduce a term in the potential that can change the number of particles, such as a†a†a†aa for 2 → 3 body scattering if there is need for it. Indeed, when we discuss quantization of radiation field (namely, theory of photons by quantizating classical Maxwell field), we see a term in the Hamiltonian which creates or annihilates a photon. Such an interaction can never be described in conventional Schrödinger wave functions, but is possible in quantum field theory. 4.8 Background Potential The Hamiltonian Eq. (43) describes interacting bosons but we are often inter- ested in a system in a background potential. A good example is the electrons in an atom, where all of them are moving in the background Coulomb po- tential due to the nucleus. In this case, the correct field-theory Hamiltonian 15 is H = ∫ d~xψ∗ ( −h̄2∆ 2m − Ze2 |~x| ) ψ + ∫ d~xd~y 1 2 ψ∗(~x)ψ∗(~y) e2 |~x− ~y| ψ(~y)ψ(~x). (64) where the second term in the parantheses is the Coulomb pontential due to the nucleus, while the last term is the Coulomb repulsion among electrons. In this case, a more convenient expansion of the field operator would be ψ(~x) = ∑ nlm anlmunlm(~x), (65) where anlm is the annihilation operator and the c-number unlm(~x) is the complete basis for expanding the field operator with. Then the state with certain states filled can be written down as a†1sa † 2p,m=1|0〉 etc. (66) But this Hamiltonian gives bosons instead of fermions. How to obtain fermions out of quantized Schrödinger field is the issue in the next section. 5 Fermions We have seen that the quantized Schrödinger field gives multi-body states of identical bosons. But we also need fermions to describe electrons, protons, etc. How do we do that? The trick is to go back to the commutation relations Eq. (22). Instead of them, we can use anti-commutation relations {ψ(~x), ψ†(~y)} = δ(~x− ~y), {ψ, ψ} = {ψ†, ψ†} = 0. (67) The notation is {A,B} = AB+BA instead of [A,B] = AB−BA. This small change flips the statistics completely and the same Hamiltonian Eq. (43) describes a system of identical fermions. One noteworthy point is that ψ(~x)†2 = 1 2 {ψ(~x)†, ψ(~x)†} = 0. What this means is that one cannot create two particles at the same position, an ex- pression of Pauli’s exclusion principle for fermions. 16 Here are a few useful identities. Similarly to the identity of commutators [A,BC] = [A,B]C +B[A,C], we find [A,BC] = ABC −BCA = ABC +BAC −BAC −BCA = {A,B}C −B{A,C} (68) Similarly, [AB,C] = ABC − CAB = ABC + ACB − ACB − CAB = A{B,C} − {A,C}B. (69) Again it is useful to calculate the commutator [H,ψ†(~x)] for later pur- poses. [H,ψ†(~x)] = [∫ d~y ( ψ†(~y) −h̄2∆~y 2m ψ(~y) + ∫ d~z 1 2 ψ†(~y)ψ†(~z)V (~y − ~z)ψ(~z)ψ(~y) ) , ψ†(~x) ] = ∫ d~yψ†(~y) −h̄2∆~y 2m {ψ(~y), ψ†(~x)} + ∫ d~yd~z 1 2 ψ†(~y)ψ†(~z)V (~y − ~z) [ ψ(~z)ψ(~y), ψ†(~x) ] = −h̄2∆~x 2m ψ†(~x) + ∫ d~zψ†(~x)ψ†(~z)V (~x− ~z)ψ(~z). (70) Consider a two-particle state |Ψ〉 = 1√ 2 ∫ d~x1d~x2Ψ(~x1, ~x2)ψ †(~x1)ψ †(~x2)|0〉. (71) Because {ψ†(~x1), ψ †(~x2)} = 0, or more explicitly ψ†(~x1)ψ †(~x2) = −ψ†(~x2)ψ †(~x1), (72) the c-number function Ψ is hence anti-symmetric under the exchange of two positions Ψ(~x1, ~x2) = −Ψ(~x2, ~x1). (73) Such a state indeed describes identical fermions. 17