Set of supplementary notes.

More documents

Recommendations

Info

56 CHAPTER 4. ELECTRONIC STRUCTURE |ψ〉 = ∑ k c k |k〉 When we apply the Hamiltonian Ĥ to this, we find [ ] [ ] ∑ ∑ ∑ Ekc 0 k e ikr + V G e iGr c k e ikr = E ∑ k G k k c k e ikr (where E 0 k = h¯2k 2 2m ), which can be rewritten as ∑ Ekc 0 k |k〉 + ∑ V G c k |G + k〉 = E ∑ k G,k k c k |k〉 We now relabel the k’s in the middle sum, G + k → k, to obtain: ∑ Ekc 0 k |k〉 + ∑ V G c k−G |k〉 = E ∑ k G,k k c k |k〉 From this, we can extract an equation for the coefficients c k by left multiplying with a single plane wave state. This gives the eigenvalue equation ( E 0 k − E ) c k + ∑ G V G c k−G = 0 (4.3) This is a key result. Here, k can be anywhere in reciprocal space. We can go one step further and relate the general wavevector k to a vector q which lies in the first Brillouin zone, by shifting through a reciprocal lattice vector G ′ : q = k + G ′ , where q lies in the first Brillouin zone. If we now replace the sum over all G by one over all G ′′ = G + G ′ , then we find ( h¯ 2 2m (q − G′ ) 2 − E ) c q−G ′ + ∑ V G ′′ −G ′c q−G ′′ = 0 (4.4) G ′′ Bloch’s theorem from considering a plane wave basis Although Eqn. (4.4) appears like a single equation, it is really an infinite set of simultaneous equations. For a given wavevector in the first Brillouin zone, q, we need to consider all the coefficients c q−G ′, which are associated with plane wave states that can be connected with |q〉 via a reciprocal lattice vector, because they contribute to the sum in the second term of (4.4). It is an eigenvector/eigenvalue problem. We can in principle solve (4.4) to find the set of coefficients c q−G . This set of coefficients is a distinct sub-set of all c k . It allows us to find a particular eigenfunction of Ĥ: ψ q (r) = ∑ G c q−G e i(q−G)r
4.2. BLOCH’S THEOREM FROM DISCRETE TRANSLATIONAL SYMMETRY 57 By taking out a factor e iqr , this can be rewritten as ψ q (r) = e iqr ∑ G c q−G e −iGr = e iqr u j,q (r) where the function u j,q (r) is built from periodic function e −iGr , and must therefore have the same periodicity as the lattice. This is Bloch’s theorem: Eigenstates of the one-electron Hamiltonian can be chosen to be a plane wave multiplied by a function with the periodicity of the Bravais lattice. 4.2 Bloch’s theorem from discrete translational symmetry Another way of thinking about Bloch’s theorem is to consider what happens to an eigenstate of the Hamiltonian (kinetic energy plus periodic lattice potential), if it is translated in space. An arbitrary translation operation will not necessarily produce an eigenstate, because the new state, generated by this translation, may not match the lattice potential correctly. If the translation operation is matched to the lattice, however, so if it is a translation by a Bravais lattice vector, then the resulting state is also an eigenstate of the Hamiltonian. The reason for this lies in the connection between symmetry and quantum mechanics, which is discussed in quantum mechanics courses: if an operator (such as the Hamiltonian) is unchanged under a change of coordinate system (i.e. a symmetry operation such as translation, rotation, etc.), then applying a symmetry operation on the eigenstate of such an operator produces another eigenstate of the operator, with the same eigenvalue as the original one. Either the two eigenstates produced by the symmetry operation are actually the same and differ only by a complex prefactor, or they are different, in which case we are dealing with a set of degenerate eigenstates. In the first case, it is clear that the original eigenstate is also an eigenstate of the symmetry operation. In the second case, it can be shown that we can always choose from the subspace of degenerate eigenstates a set of eigenstates which are also eigenstates of the symmetry operation. Lattices are symmetric under discrete translation of the coordinate system by lattice vectors. Accordingly, the eigenstates of the Hamiltonian can be chosen to be eigenstates of the discrete lattice translation operation. This is the underlying origin of Bloch’s theorem, which we will now explore in more detail. 4.2.1 Symmetry in quantum mechanics – applied to the lattice Consider a symmetry operator ˆT, e.g. translation 〈〉 r|ˆTψ = ψ(r + a). If the Hamiltonian Ĥ commutes with the symmetry operator ˆT, then this implies that ˆT maps one eigenstate of the Hamiltonian Ĥ onto another eigenstate of Ĥ with the same energy:
Page 1:
Quantum Condensed Matter Physics Pa
Page 4 and 5:
4 CONTENTS 5.1 Bands and Brillouin
Page 6 and 7: 6 CONTENTS also serves to give impo
Page 8 and 9: 8 CONTENTS
Page 10 and 11: 10 CHAPTER 1. CLASSICAL MODELS FOR
Page 24 and 25: 24 CHAPTER 2. SOMMERFELD THEORY Fig
Page 26 and 27: 26 CHAPTER 2. SOMMERFELD THEORY 2.4
Page 28 and 29: 28 CHAPTER 2. SOMMERFELD THEORY Res
Page 30 and 31: 30 CHAPTER 2. SOMMERFELD THEORY Thi
Page 32 and 33: 32 CHAPTER 2. SOMMERFELD THEORY
Page 34 and 35: 34 CHAPTER 3. FROM ATOMS TO SOLIDS
Page 58 and 59: 58 CHAPTER 4. ELECTRONIC STRUCTURE
Page 74 and 75: 74 CHAPTER 5. BANDSTRUCTURE OF REAL
Page 88 and 89: 88 CHAPTER 6. EXPERIMENTAL PROBES O
Page 100 and 101: 100 CHAPTER 7. SEMICONDUCTORS Figur
Page 102 and 103: 102 CHAPTER 7. SEMICONDUCTORS is fo
Page 104 and 105: 104 CHAPTER 7. SEMICONDUCTORS Figur
Page 106 and 107:
106 CHAPTER 8. SEMICONDUCTOR DEVICE
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
ecessitates variances in circuit th
Page 118 and 119:
nc... Freescale Semiconductor, I
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
124 CHAPTER 9. ELECTRONIC INSTABILI
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
146 CHAPTER 10. FERMI LIQUID THEORY
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158:
show all

Set of supplementary notes.

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?