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QUANTUM THEORY of SOLIDS Here A+ Is Simply the Hermitian Conjugate of A 1 QUANTUM THEORY OF SOLIDS Here a+ is simply the Hermitian conjugate of a. Using these de¯nitions we can rewrite the Hamiltonian as: Version (02/12/07) ~! H = (aa+ + a+a): (6) 2 SECOND QUANTIZATION We now calculate (aa+ ¡ a+a)f(x) = [a; a+]f(x), f any function In this section we introduce the concept of second quantization. Historically, the ¯rst quantization is the = 1 ¢ f(x) , since p = ¡i~@x: (7) quantization of particles due to the commutation rela- Hence tion between position and momentum, i.e., [a; a+] = 1; (8) [x; p] = i~: (1) which is the Boson commutation relation. In addition, Second quantization was introduced to describe cases, [a; a] = [a+; a+] = 0. Using this commutation relation we where the number of particles can vary. A quantum ¯eld can rewrite the Hamiltonian in second quantized form as such as the electromagnetic ¯eld is such an example, since the number of photons can vary. Here it is necessary H = ~!(a+a + 1=2): (9) to quantize a ¯eld ª(r), which can also be expressed in terms of commutation relations In order to solve the Hamiltonian in this form we can write the wave function as [ª(r); ª+(r0)] = ±(r ¡ r0): (2) (a+)n Ãn = p j0i; (10) However, there is only one quantum theory, hence n! nowadays ¯rst quantization refers to using eigenval- and aj0i = 0, where j0i is the vacuum level of the system. ues and eigenfunctions to solve SchrÄodingerequation, We now show that Ãn de¯ned in this way is indeed the whereas in second quantization one uses creation and an- solution to SchrÄodinger'sequation HÃn = EnÃn. To hilation operators to solve SchrÄodinger'sequation. Both show this we ¯rst calculate methods are consistent which each other and the choice p + n+1 p of technique is dictated by the type of problems consid- + (a ) n + 1 + n+1 a Ãn = p j0i = p (a ) j0i = n + 1Ãn+1 ered. In general, when considering many particles second n! (n + 1)! quantization is the preferred technique and will be intro- (11) duced next. and a(a+)n aa+(a+)n¡1 aÃn = p j0i = p p j0i The Harmonic oscillator n! n (n ¡ 1)! (a+a + 1)(a+)n¡1 p = p p j0i = nÃn¡1; (12) The Hamiltonian of the harmonic oscillator is given by n (n ¡ 1)! p2 K + + n¡1 + n¡1 H = + x2; (3) where we used that a a(a ) j0i = (n ¡ 1)(a ) j0i 2m 2 by using the commutation relation (n-1) times. Coming p back to the Hamiltonian we then have where ! = K=m. The eigenvalues of this Hamiltonian + +p are then obtained by solving the SchrÄodingereigenvalue HÃn = ~!(a a + 1=2)Ãn = ~!(a nÃn¡1 + Ãn=2) equation Hà = EÃ, where p = ¡i~@=@x is used and one = ~!(nÃn + Ãn=2) obtains for the energy = ~!(n + 1=2) à (13) | {z } n En = ~!(n + 1=2); (4) En where n is an integer. In the second quantized form An important di®erence between the ¯rst quantization we introduce the following operators: method and the second quantization method is that we don't need the expression for the wave functions in order ³m! ´1=2 to obtain the spectrum, which simply follows from the a = (x + ip=m!) "anhilation operator" 2~ form (9). The form of the Hamiltonian in (9) is diagonal ³m! ´1=2 since the energy is simply given by counting the number a+ = (x ¡ ip=m!) "creation operator": (5) + 2~ of states, since a a is just the number operator. Indeed, 2 a+a(a+)nj0i = nj0i, since we have n levels above vac- The expectation value of the operator O then becomes: uum. Hence, the whole task in solving problems with X ¤ + the second quantization method is to obtain an expres- Oà = hà OÃi = O®¯a® a¯: (17) sion of the Hamiltonian in a diagonal form, like (9). The ®;¯ BCS theory of superconductivity in the next section is + a nice example of this diagonalization procedure. How- If a and a are de¯ned as the creation and anhilation ever, before that we will see how we can transform any operators we then have O written in second quantized Hamiltonian in ¯rst quantized form into second quan- form. tized form. We now generalize this to more particles: In this case the wave-function is written as Ã(¢ ¢ ¢ ; ri; ¢ ¢ ¢ ; rj; ¢ ¢ ¢) = Ã(¢ ¢ ¢ ; rj; ¢ ¢ ¢ ; ri; ¢ ¢ ¢) bosons General form Ã(¢ ¢ ¢ ; ri; ¢ ¢ ¢ ; rj; ¢ ¢ ¢) = ¡Ã(¢ ¢ ¢ ; rj; ¢ ¢ ¢ ; ri; ¢ ¢ ¢) fermions ¡iθ Ã(¢ ¢ ¢ ; ri; ¢ ¢ ¢ ; rj; ¢ ¢ ¢) = e Ã(¢ ¢ ¢ ; rj; ¢ ¢ ¢ ; ri; ¢ ¢ ¢) anyons The general idea to second quantize a Hamiltonian or X = A Á (r ) ¢ ¢ ¢ Á (r ): (18) operatorP is to start by choosing a complete basis, jÁ¸(r)i ®1;¢¢¢;®N ®1 1 ®N N ® ;¢¢¢;® with ¸ jÁ¸(r)ihÁ¸(r)j = 1. Here we are only interested 1 N in operators of the form O(r) or V (r ¡r ). For example, P i j Here A are simply the coe±cients. A simple prod- O(r ) can be a single particle potential like a con¯ne- ®1;¢¢¢;®N i i P uct like Á (r ) ¢ ¢ ¢ Á (r ) has the correct symmetry for ment potential and V (r ¡r ) can be a two-particle ®1 1 ®N N i6=j i j Bosons but not for Fermions, hence it is useful to intro- interaction potential, such as the Coulomb potential. For duce an antisymmetrization operator for Fermions, de- N particles we write O in our complete basis as: ¯ned as ¯ ¯ ¯ Á (r ) ¢ ¢ ¢ Á (r ) ¯ ¯ ®1 1 ®N 1 ¯ XN Y 1 ¯ . ¯ S¡ Á® (ri) = p ¯ . ¯ ; (19) O = O(ri) i N! ¯ . ¢ ¢ ¢ . ¯ i ¯ Á (r ) ¢ ¢ ¢ Á (r ) ¯ i=1 à ! ®1 N ®N N X X X = jÁ®(ri)ihÁ®(ri)j O(ri) jÁ¯(ri)ihÁ¯(ri)j which is the Slater determinant. Then we can write for i ® ¯ Fermions X X Y = jÁ®(ri)i hÁ®(ri)jO(ri)jÁ¯(ri)i jhÁ¯(ri)j Ã(¢ ¢ ¢ ; r ; ¢ ¢ ¢ ; r ; ¢ ¢ ¢) = A S Á (r ): | {z } i j ®1;¢¢¢;®N ¡ ®i i i;®;¯ ®1;¢¢¢;®N i O®¯ X X (20) = O®¯ jÁ®(ri)ihÁ¯(ri)j; (14) Now we don't have to worry about symmetry anymore. ®;¯ i The next step is to introduce the occupation number R so that we can rewrite the wave function as where hf(r)i = f(r)dr: X Similarly, we can write: à = jn®1 ; ¢ ¢ ¢ ; n®N i with N = n®i ; (21) i where n® is the number of particles in state ®i. The 1 XN i V = V (r ¡ r ) occupation number operator is then de¯ned as 2 i j i6=j n jn i = n jn i (22) 1 X X |{z}®i ®i |{z}®i ®i = V jÁ (r )ijÁ (r )ihÁ (r )jhÁ (r )j 2 ®¯γ± ® i ¯ j γ i ± j operator number ®;¯,γ;± i6=j and V®¯γ± = hÁ®(ri)jhÁ¯(rj)jV (ri ¡ rj)jÁγ (ri)ijÁ±(rj)i The number of states for fermions is limited to 0 and 1, whereas for bosons, n can take any integer value ¸ 0. (15) ®i In similarity with the harmonic oscillator described in the previous section, the Boson creation and anhilation This is all very general. Now the idea is to write (14,15) operators are then de¯ned as in second quantized form. If we had only one particle we p could write the wave-function as + b®i j ¢ ¢ ¢ ; n®i ; ¢ ¢ ¢i = n®i + 1j ¢ ¢ ¢ ; n®i + 1; ¢ ¢ ¢i X p b®i j ¢ ¢ ¢ ; n®i ; ¢ ¢ ¢i = n®i j ¢ ¢ ¢ ; n®i ¡ 1; ¢ ¢ ¢i (23) à = a¯jÁ¯(r)i and ¯ X and ¤ + à = hÁ®(r)ja : (16) ® [b ; b+ ] = ± : (24) ® ®i ®j ®i;®j 3 + + + + + n + n All other commutators ([bi; bj] and [bi ; bj ]) are zero. where we used [b; b ] = 1 and b b(b ) j0i = n(b ) j0i For fermions we have (the occupation number operator).P The lastP term in (33) + is obtained p times from the i, hence i ±¯;®i = p. c j ¢ ¢ ¢ ; n® ; ¢ ¢ ¢i = j ¢ ¢ ¢ ; n® + 1; ¢ ¢ ¢i if n® = 0 ®i i i i Therefore, (33) leads to = 0 otherwise X + c®i j ¢ ¢ ¢ ; n®i ; ¢ ¢ ¢i = j ¢ ¢ ¢ ; n®i ¡ 1; ¢ ¢ ¢i if n®i = 1 O = O®¯b® b¯ (34) = 0 otherwise: (25) ®;¯ + for bosons. For fermions the derivation is quite similar Hence, c®i creates an electron in state ®i if state ®i is as well as for V . empty and c®i anhilates an electron from state ®i if ®i was occupied. It is quite straightforward to check that for symmetry reasons the Fermion creation and anhila- tion operators have to obey the following commutation Typical Hamiltonian relations: In ¯rst quantization we consider + + + fc®i ; c®j g = c®i c®j + c®j c®i = ±®i;®j (26) X 2 2 X + + ~ @i 1 and all others (fc ; c g and fc ; c g) are zero. Also, H = ¡ + U(ri) + V (ri ¡ rj) : (35) i j i j 2m 2 c j0i = 0. The N-particle fermionic wave-function is i i6=j ®i | {z } | {z } now written as H0 V Y Y à = S jÁ (r )i = c+ j0i; (27) ¡ ®i i ®i In order to write this Hamiltonian in second quantization i i we ¯rst need a basis. We chose as basis the free particle where the product of creation operators is antisymmetric basis: by construction of the f,g. Similarly, we have for bosons: 1 ik®r Y Y Á®(r) = p e ; (36) + V à = jÁ®i (ri)i = b®i j0i: (28) i i Rwhich is a complete orthogonal basis with Hence, with these de¯nitions we obtain Á¤ (r)Á (r)dr = ± .
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