EE201/MSE207 Lecture 18 Density matrix (density operator) In this course we described a quantum state by a wavefunction. Wavefunction does not contain any randomness (entropy is zero, randomness only for measurement result). However, we often need to also describe a classical randomness (thermodynamics, decoherence, etc.) A possible way: list of states with probabilities State |휓1〉 with probability 푝1, state |휓2〉 with prob. 푝2, etc.

state |휓푖〉 with probability 푝푖, 푖 푝푖 = 1 However, this is a very lengthy description. Possible to use a shorter way. Instead of this list, let us define an operator 휌 = 푝 휓 〈휓 | (density 푖 푖 푖 푖 matrix) Somewhat surprisingly, this is a complete description of a quantum state (for different lists giving the same 휌 , all experimental predictions coincide). Some properties of 1. Hermitian (obvious, since a sum of projectors) density operator 휌 2. Positive semidefinite (all eigenvalues are non-negative) 3. Tr 휌 = 1 (proof later) Averages via density matrix

휌 = 푖 푝푖 휓푖 〈휓푖| for state |휓푖〉 with probability 푝푖, 푖 푝푖 = 1

Theorem For any observable 퐴 , its average (expectation) value is 퐴 = Tr(휌 퐴 ) (this is why 휌 is a complete description) Proof orthonormal basis

퐴 = 푖 푝푖 〈휓푖 퐴 휓푖〉 Tr 휌 퐴 = 푘〈푒푘 휌 퐴 푒푘〉

Tr 휌 퐴 = 푘〈푒푘 휌 퐴 푒푘〉 = 푘,푖 푒푘 휓푖 푝푖 휓푖 퐴 푒푘 = 휌 = 푝 휓 퐴 푒 〈푒 |휓 〉 = 푝 휓 퐴 휓 = 〈퐴 〉 푖 푖 푘 푖 푘 푘 푖 푖 푖 푖 푖 QED 1 Corollary 1 = Tr 휌 1 = Tr 휌 Therefore Tr 휌 = 1 Evolution of density matrix 휌 = 푖 푝푖 휓푖 〈휓푖| 푑 푑 휓 푑 휓 휌 = 푝 푖 휓 + |휓 〉 푖 = 푑푡 푖 푖 푑푡 푖 푖 푑푡 푖 푖 = − 푝 퐻 휓 〈휓 | − |휓 〉〈휓 |퐻 = − 퐻 , 휌 ℏ 푖 푖 푖 푖 푖 푖 ℏ 푑 푖 (Schrödinger equation 휌 = − 퐻 , 휌 푑푡 ℏ for density matrix) Pure and mixed states Pure state: a state, which can be represented by a wavefunction |휓〉 with probability 푝 = 1, so 휌 = 휓 〈휓| Then 휌 2 = 휓 휓 휓 〈휓| = 휌 휌 2 = 휌 Tr 휌 2 = 1 Mixed state: a state, which can not be represented by a wavefunction 2 2 2 2 Then 휌 ≠ 휌 Tr 휌 < 1 (proof via eigenbasis, 푝푖 < 푝푖 = 1) Thermal distribution (equilibrium d.m.)

휌 = 푒−퐻/푇 Tr(푒−퐻/푇) or 휌 = 푒−(퐻 −휇푁 )/푇 Tr(푒−(퐻 −휇푁 )/푇) Next subject: Schrödinger and Heisenberg pictures

What we considered in this course is called Schrödinger picture. 푑 푖 In this case Schrödinger equation for state: Ψ = − 퐻 |Ψ〉 푑푡 ℏ If 퐻 is time-independent, then formally Ψ(푡) = 푒−푖퐻 푡/ℏ|Ψ(0)〉 Then expectation value of an observable 퐴 at time 푡 is 푖퐻 푡/ℏ −푖퐻 푡/ℏ 퐴 푡 = Ψ 푡 퐴 Ψ 푡 = Ψ 0 푒 퐴 푒 |Ψ(0)〉

Heisenberg picture We could get the same 〈퐴 〉 is we assume that the state |Ψ〉 does not evolve, but instead the observable 퐴 evolves with time 푡:

퐴 푡 = 푒푖퐻 푡/ℏ퐴 푒−푖퐻 푡/ℏ 푑 푖 퐴 푡 = 퐻 , 퐴 푡 푑푡 ℏ Interaction picture (main practical approach) Interaction picture is a combination of both Schrödinger and Heisenberg pictures.

퐻 = 퐻 0 + 퐻 1 simple (solvable); assume time-independent

Heisenberg-picture idea for 퐻 0. For any observable 퐴 ,

푖퐻 푡/ℏ −푖퐻 푡/ℏ 푑 푖 퐴 푡 ≡ 푒 0 퐴 푒 0 퐴 푡 = 퐻 , 퐴 푡 푑푡 ℏ 0 푖퐻 0푡 Also introduce Ψ 푡 = 푒 ℏ |Ψ 푡 〉 (here usual Schrödinger |Ψ 푡 〉)

푖퐻 0푡 −푖퐻 0푡 So that 퐴 푡 = Ψ 푡 퐴 Ψ 푡 = Ψ 푡 푒 ℏ 퐴 푒 ℏ |Ψ (푡)〉

Then evolution for |Ψ 푡 〉 is 퐴 푡 푖퐻 푡 푖퐻 푡 푖퐻 푡 푑 푖퐻 0 0 0 −푖퐻 0 −푖퐻 1 Ψ = 푒 ℏ Ψ + 푒 ℏ Ψ = 푒 ℏ Ψ = 푑푡 ℏ ℏ ℏ 푖퐻 푡 −푖퐻 푡 SE 0 −푖퐻 1 0 ℏ ℏ 푑 푖 = 푒 푒 Ψ Ψ = − 퐻 푡 Ψ ℏ 푑푡 ℏ 1 Heisenberg Next subject: Methods for interacting electrons (terminology and ideas)

푉0(푟 ) “seed” potential

Unknown 푉 푟 = 푉0 푟 + Δ푉 푟

Problem: 푉(푟 ) changes because of electron-electron interaction, so need some self-consistent approach. Thomas-Fermi method (or approximation) Assume equilibrium Chemical potential 휇 (Fermi level)

휇 − 푉(푟 ) Unknown 푉 푟 = 푉0 푟 + Δ푉 푟

1 2푚 휇−푉 3/2 Idea: 휇 − 푉 푟 determines density of electrons, 푛(푟 ) = , 3휋2 ℏ2 then solve Poisson equation to find Δ푉(푟 ); self-consistency: 푉 → 푛 → 푉. Hartree method (or approximation) Non-equilibrium, but stationary case electron no chemical potential flow all electrons alike 푉 푟 = 푉0 푟 + Δ푉 푟 Idea: solve Schrödinger equation 퐻 휓 = 퐸휓 to find 휓(푟 ), then 푛 푟 ∝ 휓 푟 2, then solve Poisson equation to find Δ푉(푟 ); self-consistency 푉 → 휓 → 푛 → 푉.

Hartree-Fock method (or approximation) Idea: almost the same as Hartree, but excludes e-e interaction for an electron with itself, so that electron feels only field produced by other electrons

Density functional theory Even better (more accurate), uses functionals of electron density 푛(푟 ) Next subject: Language of second quantization This is a technique to describe states with variable number of particles. (Later it was found to be useful for a fixed number of particles as well.) Occupation number representation 푁 = |푁 , 푁 , 푁 , … 〉 State with 푁1 particles on level 1, 푁2 particles 1 2 3 on level 2, etc. We do not distinguish which particle is where (indistinguishable). This is now the basis, so that an arbitrary (pure) state is a superposition: 2 푐 푁 is probability 휓 = 푁 푐(푁) 푁 This wavefunction lives in the occupation number space (Fock space) Orthogonality: 〈푀 푁 = 0 if 푀 ≠ 푁, 〈푁 푁 = 1 Examples of (basis) states |0, 0, 0, … 〉 no particles, “vacuum”, |0〉 or |0〉 |0, 1, 0, … 〉 one particles in state 2 |0, 2, 1, 0, … 〉 two particles in state 2, 1 particle in state 3 Second quantization (cont.) Simple special case: one oscillator (main language in optics) Basis: 0 , |1〉, |2〉, |3〉, etc. Instead of the level number, we think about number of photons Wavefunction: 휓 = 푛 푐 푛 |푛〉 (Fock-space representation) Creation and annihilation operators † † 푎 2 0 = |0, 1, 0, 0, … 〉 푎 3 0 = |0, 0, 1, 0, … 〉 † For bosons 푎 푘 … 푁푘, … = 푁푘 + 1 | … 푁푘 + 1, … 〉 creates extra particle on level 푘 (factor 푁 + 1 as for an oscillator)

For bosons 푎 푘 … 푁푘, … = 푁푘 | … 푁푘 − 1, … 〉 annihilates (kills) one particle on level 푘 (factor 푁 as for an oscillator) If 푁푘 = 0, then 푎 푘 … 0푘, … = 0 (zero, not vacuum) † † In particular, 푎 푘푎 푘 … 푁푘, … = 푁푘 | … 푁푘, … 〉, so 푁푘 = 푎 푘푎 푘 Commutation relations † † † Sufficient for the 푎 푘, 푎 푙 = 훿푘푙 , 푎 푘, 푎 푙 = 푎 푘, 푎 푙 = 0 whole theory Second quantization (cont.) Operators can often be expressed in terms of 푎 † and 푎 † H = 푘 휀푘 푎 푘푎 푘 (non-interacting particles, basis of eigenstates) † If basis vectors are not eigenstates, then also terms 푘푙 퐻푘푙 푎 푘푎 푙 † † ∗ † Tight-binding model: 퐻 = 푗 휀푗 푎푗 푎푗 + 푗 (푇푗푎푗 푎푗+1 + 푇푗 푎푗+1푎푗) † † Coulomb interaction: 퐻 = 푘푙 퐻푘푙 푎 푘푎 푘 푎 푙 푎 푙 For fermions similar, but commutation relations are where † † † 푎 푘, 푎 푙 = 훿푘푙 , 푎 푘, 푎 푙 + = {푎 푘, 푎 푙 }+ = 0 퐴 , 퐵 ≡ 퐴 퐵 + 퐵 퐴 + + † † † † † † For example, this means that 푎 푘푎 푘 0 = −푎 푘푎 푘 0 , so 푎 푘푎 푘 0 = 0 (Pauli exclusion principle) For one particle it does not matter if it is fermion or boson, so boson rules are often used for electrons (in single-particle approaches)

Why called “second quantization”? 휓 푥 → 휓 = 푘 휓푘푎 푘