introduction solid hamiltonian

1 Introduction and Motivation

know what a block of iron is made of in order to understand ferromagnetism:Ferromagnetism occurs when many constituents work together.

Also, while knowing the fundamental Hamiltonian of solid sate physics givesus a recipe to solve for the behavior that we are interested in (namely put theHamiltonian for sufficiently many constituents on a computer and evaluate theeigenenergies and eigenfunctions numerically), this is often only of limited usefor understanding the behavior that we are interested in. Due to the exponentialgrowth of the dimension of the Hilbert space with the number of particles, onecan nowadays solve problems with about 20 particles (spin-1/2), but 1000 parti-cles are even in principle out of reach since dim H becomes of order the numberof particles in the universe.

The strategy in solid state theory is therefore to perform suitable systematicapproximations in our fundamental Hamiltonian that render the resulting modelsolvable using analytical or numerical methods. A key role in this strategyis played by symmetries, which allow us to simplify models without makingapproximations. This strategy determines the structure of this course.

1.2 Reminder: Quantum mechanics of more than

one particle

Let us first look at the case of N = 2 particles. The following fundamentalaxiom of quantum mechanics sets the stage:

The total Hilbert space H of two particles (1 and 2) is the tenor product of theirindividual Hilbert spaces,

H = H1 H2 . (1.1)

For indistinguishable fermions (e.g. electrons) we further need to restrict Hto only those states that are antisymmetric under exchange of the particles.Likewise, for indistinguishable bosons (e.g. photons) we need to restrict H toonly those states that are symmetric under exchange of the particles.

So for two distinguishable particles with quantum numbers q (particle 1) and q

(particle 2) the total state is

| = |q |q (1.2)

But if these particles are indistinguishable bosons we have for q 6= q

| = 12(|q |q+ |q |q) (1.3)

and for q = q

| = |q |q (1.4)

Introduction to Solid State Theory (Kehrein) 2 University of Goettingen


For indistinguishable fermions we find

| = 12(|q |q |q |q) (1.5)

Notice that this state does not exist when q = q, which just expresses the Pauliprinciple: Two indistinguishable fermions cannot occupy the same quantumstate.

Also notice that all the above states are normalized | = 1 if we assume thatthe single particle states are orthonormal, q|q = qq .As an explicit example we consider two particles (position coordinates x1, x2)in a one-dimensional harmonic oscillator. The Hamiltonian in position repre-sentation is given by

H = H(1) +H(2) (1.6)

= ~2

2m

2

x21+m

22 x21

~2

2m

2

x22+m

22 x22 (1.7)

1. We consider the particles as distinguishable: Particle 1 has quantum num-ber n, particle 2 has quantum number n. One can easily verify that| = |n |n is an eigenstate of the above H with eigenenergy

E = ~ (n+ n + 1) (1.8)

and wavefunction

(x1, x2) = (x1| x2|) | (1.9)= n(x1)n(x2) (1.10)

Here n(x) is the well-known eigenfunction corresponding to eigenenergy~(n+ 1/2) of one particle in a harmonic oscillator:

n(x) =(m~

)1/4 12n n!

Hn() e2/2 (1.11)

where Hn() is the n-th order Hermite polynomial and def= x

m/~.

2. We consider identical bosons with spin 0. Then the normalized eigenfunc-tions are

for n 6= n : (x1, x2) = 12(n(x1)n(x2) + n(x1)n(x2))

for n 6= n : (x1, x2) = n(x1)n(x2) (1.12)



3. We consider electrons (spin-1/2). Then

x1, s1;x2, s2| = (x1, s1;x2, s2) (1.13)= (x1, x2)(s1, s2) (1.14)

where (x1, x2) is the orbital wavefunction and (s1, s2) is the spin wave-function. From the antisymmetrization postulate we know that the totalwavefunction is antisymmetric under exchange of the particles

(x1, s1;x2, s2) = (x2, s2;x1, s1) (1.15)and therefore we find the following identity

(x1, s1;x2, s2) =1

2((x1, x2)(s1, s2) (x2, x1)(s2, s1))

=1

4

[((x1, x2) + (x2, x1)) ((s1, s2) (s2, s1))

+((x1, x2) + (x2, x1)) ((s1, s2) (s2, s1))]

(1.16)

This means that the total wavefunction of two electrons can always bewritten as a superposition of

a) symmetric orbital wavefunction, antisymmetric spin wavefunctionand

b) antisymmetric orbital wavefunction, symmetric spin wavefunction.

Notice that this cannot be generalized to N > 2.

a)

for n 6= n : (x1, x2) = 12(n(x1)n(x2) + n(x1)n(x2))

for n 6= n : (x1, x2) = n(x1)n(x2) (1.17)with the spin wavefunction

=12(| | | | ) (1.18)

that is total spin S = 0 (spin singlet).

b) (x1, x2) =12(n(x1)n(x2) n(x1)n(x2)) (1.19)

The possible spin wavefunctions are

= | | , | | , 12(| | + | | ) (1.20)

that is total spin S = 1 and therefore threefold degenerate. Noticethat this state does not exist for n = n, which again just exemplifiesthe Pauli principle. The Pauli principle turns out to be the keyingredient for understanding the structure of matter.



We now generalize to N > 2 particles. The total Hilbert space is

H = H1 . . .HN . (1.21)

For identical bosons the total state vector must be symmetric under exchangeof any two particles, for identical fermions the total state vector must be anti-symmetric under exchange of any two particles.

In the sequel we will only discuss the case of identical fermions since this courseis mainly concerned with the electronic structure. The normalized basis of the1-particle Hilbert space for the electrons is denoted by |, where can be somemulti-index (like |n, l,m, s for the bound states of the hydrogen atom). Onecan show that a normalized basis of the N -fermion Hilbert space is given by

|1, . . . , N def= 1N !

piSN

(sgn) |pi(1) |pi(2) . . . |pi(N) (1.22)

where the sum runs over all elements of the permutation group SN . The signof the permutation is defined as +1 for an even number of transpositions, and-1 for an odd number of transpositions making up . One can show that themany-particle wavefunction corresponding to these states can be written as adeterminant

(r1, s1; . . . ; rN , sN ) = r1, s1; . . . ; rN , sN |1, . . . , N (1.23)

=1N !

1(r1, s1) 1(r2, s2) . . . 1(rN , sN )2(r1, s1) 2(r2, s2) . . . 2(rN , sN )...

...N (r1, s1) N (r2, s2) . . . N (rN , sN )

Here (r, s) = r, s| is the 1-particle wavefunction with respect to the basis.The above determinant is called Slater determinant. From its structure onecan see immediately that the state |1, . . . , N only exists if all the quantumnumbers are different, otherwise two rows in the determinant are identical and itvanishes. This is the general expression of the Pauli principle that two identicalfermions cannot be in the same quantum state, which follows from the generalantisymmetrization postulate.

1.3 Fundamental Hamiltonian and Born-Oppenheimer

approximation

For simplicity we consider a monoatomic solid consisting ofNn nuclei of massM ,nuclear charge Ze, and Ne = Z Nn electrons. The position coordinates of the



electrons will always be denoted with ri, the position coordinates of the nucleiwith Ri.

1

Then we can write down the fundamental Hamiltonian that we need to solve:

H = He +Hn +Hen (1.24)

Its various parts are

Electronic contribution:

He = Te + Vee (1.25)

with the total kinetic energy of the electrons

Te = ~2

2m

Nei=1

~2ri (1.26)

and the Coulomb interaction energy of the electrons

Vee =1

2

i 6=j

e2

|~ri ~rj | (1.27)

Contribution from the nuclei:

Hn = Tn + Vnn (1.28)

with the total kinetic energy of the nuclei

Tn = ~2

2m

Nei=1

~2Ri (1.29)

and the Coulomb interaction energy of the nuclei

Vnn =1

2

i 6=j

(Ze)2

|~Ri ~Rj |(1.30)

Coulomb attraction between nuclei and electrons:

Hen =i 6=j

Ze2|~ri ~Rj |

(1.31)

Essentially there are only two approximations that we have made in writingdown our fundamental Hamiltonian:

1This section closely follows lecture notes by Thomas Pruschke.



1. We have neglected relativistic effects. This is usually a very good approx-imation because the relevant velocities in a solid are typically at most oforder cvac/100.

2. Related to this we have neglected all spin effects, especially spin-orbitinteractions. This is not always a good approximation, especially for heavynuclei. However, one can easily build in the spin-orbit interaction into theHamiltonian, but it makes the notation more cumbersome. That is whywe ignore it for the time being.

It is an empirical observation that (usually) in a solid the nuclei move only

slightly around their equilibrium positions ~R(0)i . We will give a theoretical rea-

son underlying this approximation further below (Born-Oppenheimer approxi-mation). For now we want to plug it into our fundamental Hamiltonian (1.24)because it motivates the outline of this course:

H = E(0)n + Te + Vee + V(0)en +Hph +Heph (1.32)

1. E(0)n =1

2

i 6=q

(Ze)2

|~R(0)i ~R(0)j |(1.33)

is the potential energy of the equilibrium configuration of the nuclei, thatis a constant without further importance for the dynamics.

2. V (0)en =i 6=j

Ze2|~ri ~R(0)j |

(1.34)

is the Coulomb interaction between the electrons and the nuclei at theirequilibrium positions. Notice that this is simply a potential term for theelectrons.

3. Hph = Hn E(0)n (1.35)describes the dynamics of the nuclei around their equilibrium positions,that is phonons.

4. Heph = Hen V (0)en (1.36)describes the interplay of electron and lattice dynamics.

The outline of this course follows the various terms above, taking more andmore of them into account to arrive a more and more realistic picture of a solid:

1. Chapter 2, Homogeneous electron gas: We study only the kinetic energyof the electrons, Te. This is a minimal model for the electrons in a solidthat already introduces important concepts like Fermi surface, etc.

2. Chapter 3, Crystal structure: The crystal structure determines E(0)n .



3. Chapter 4, Band structure: We put the above pieces together and inves-tigate the following terms in the fundamental Hamiltonian

H0 = E(0)n + Te + V

(0)en (1.37)

This is the most importance approximation for understanding solids andexplains a multitude of phenomena. A thorough understanding of (1.37)is the key goal of this course.

4. Chapter 5, Lattice dynamics: Hph is very important for describing e.g.the specific heat of solids.

5. Chapter 6, Transport: In addition to H0 we need to study deviations fromthe perfect crystal structure and scattering processes due to Heph.

6. Chapter 7, Magnetism: We will study magnetic phenomena based on avery simplified treatment of Vee.

7. Chapter 8, Superconductivity: In most materials this is due to electron-phonon scattering, Heph.

8. Chapter 9, Outlook: Quasiparticles and Fermi liquid theory: This gives afirst glimpse at correlations effects due to Vee. A thorough investigationof Vee is the content of the Masters course on solid state theory since itrequires many-body techniques.

Next we want to theoretically motivate the above treatment of the nuclei dynam-ics as a (small) perturbation. This is a consequence of the Born-Oppenheimerapproximation, which is also one of the key approximations in quantum chem-istry. The Born-Oppenheimer approximation starts from the observation thatfor typical nuclei one has mM = O(10

4), the nuclei are much heavier than theelectrons. If we would set M =, then Tn 0 and every eigenfunction of thefull Hamiltonian can be written in the following way:

({~r}, {~R}) = ({~r})({~R}) (1.38)

Here ({~R}) is a wave function describing the spatial structure of the nucleiand ({~r}) is an eigenfunction of

(He +

d{~R}

Ven(~ri ~Rj)({~R})

)({~r}) = Ee ({~r}) (1.39)

For mM > 0 the ansatz (1.38) becomes an approximation, the so called Born-Oppenheimer approximation. Another way of viewing (1.38) is the time scaleseparation between fast electron dynamics and slow nuclei dynamics: thereforethe electrons move in a quasistatic background provided by the instantaneouspositions of the nuclei, which motivates the ansatz (1.38). The full problem


introduction solid hamiltonian

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