Density matrix (density operator) In this course we described a - PowerPoint PPT Presentation

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

Averages via density matrix 𝜍 = 𝑗 𝑞 𝑗 𝜔 𝑗 〈𝜔 𝑗 | for state |𝜔 𝑗 〉 with probability 𝑞 𝑗 , 𝑗 𝑞 𝑗 = 1 Theorem 𝜍 For any observable 𝐵 = Tr( 𝐵) 𝐵 , its average (expectation) value is (this is why 𝜍 is a complete description) Proof orthonormal basis 𝐵 = 𝑗 𝑞 𝑗 〈𝜔 𝑗 𝜍 𝜍 𝐵 𝜔 𝑗 〉 𝐵 = 𝑙 〈𝑓 𝑙 Tr 𝐵 𝑓 𝑙 〉 𝜍 𝜍 𝐵 𝑓 𝑙 〉 = 𝑙,𝑗 𝑓 𝑙 𝜔 𝑗 𝑞 𝑗 𝜔 𝑗 𝐵 = 𝑙 〈𝑓 𝑙 Tr 𝐵 𝑓 𝑙 = 𝜍 = 𝑗 𝑞 𝑗 𝑙 𝜔 𝑗 𝐵 𝑓 𝑙 〈𝑓 𝑙 |𝜔 𝑗 〉 = 𝑗 𝑞 𝑗 𝜔 𝑗 𝐵 𝜔 𝑗 = 〈 𝐵〉 QED 1 Corollary 𝜍 1 = Tr 1 = Tr 𝜍 Therefore Tr 𝜍 = 1

𝜍 = 𝑗 𝑞 𝑗 𝜔 𝑗 〈𝜔 𝑗 | Evolution of density matrix 𝑒 𝑒 𝜔 𝑗 𝜔 𝑗 + |𝜔 𝑗 〉 𝑒 𝜔 𝑗 𝑒𝑢 𝜍 = 𝑗 𝑞 𝑗 = 𝑒𝑢 𝑒𝑢 = − 𝑗 𝐼 = − 𝑗 𝐼 𝜔 𝑗 〈𝜔 𝑗 | − |𝜔 𝑗 〉〈𝜔 𝑗 | ℏ ℏ 𝑗 𝑞 𝑗 𝐼, 𝜍 𝑒 𝜍 = − 𝑗 (Schrödinger equation 𝑒𝑢 𝐼, 𝜍 ℏ for density matrix) Pure and mixed states Pure state: a state, which can be represented by a wavefunction |𝜔〉 with probability 𝑞 = 1 , so 𝜍 = 𝜔 〈𝜔| 𝜍 2 = 𝜔 𝜔 𝜔 〈𝜔| = 𝜍 2 = 𝜍 2 = 1 Then 𝜍 𝜍 Tr Mixed state: a state, which can not be represented by a wavefunction 𝜍 2 ≠ 𝜍 2 < 1 (proof via eigenbasis, 𝑞 𝑗 2 < 𝑞 𝑗 2 = 1 ) Then 𝜍 Tr Thermal distribution (equilibrium d.m.) 𝐼/𝑈 Tr(𝑓 − 𝑂)/𝑈 Tr(𝑓 −( 𝑓 − 𝑓 −( 𝐼−𝜈 𝐼−𝜈 𝐼/𝑈 ) 𝑂)/𝑈 ) 𝜍 = 𝜍 = or

Next subject: Schrödinger and Heisenberg pictures What we considered in this course is called Schrödinger picture . 𝑒𝑢 Ψ = − 𝑗 𝑒 𝐼|Ψ〉 In this case Schrödinger equation for state: ℏ Ψ(𝑢) = 𝑓 −𝑗 If 𝐼𝑢/ℏ |Ψ(0)〉 𝐼 is time-independent, then formally Then expectation value of an observable 𝐵 at time 𝑢 is 𝐼𝑢/ℏ 𝑓 𝑗 𝐵 𝑓 −𝑗 𝐼𝑢/ℏ |Ψ(0)〉 𝐵 𝑢 = Ψ 𝑢 𝐵 Ψ 𝑢 = Ψ 0 Heisenberg picture We could get the same 〈 𝐵〉 is we assume that the state |Ψ〉 does not evolve , but instead the observable 𝐵 evolves with time 𝑢 : 𝐼𝑢/ℏ 𝐵 𝑢 = 𝑓 𝑗 𝐵 𝑓 −𝑗 𝐼𝑢/ℏ 𝑒 𝐵 𝑢 = 𝑗 𝐼, 𝐵 𝑢 𝑒𝑢 ℏ

Interaction picture (main practical approach) Interaction picture is a combination of both Schrödinger and Heisenberg pictures. 𝐼 = 𝐼 0 + 𝐼 1 simple (solvable); assume time-independent Heisenberg-picture idea for 𝐼 0 . For any observable 𝐵 , 𝑒 𝐵 𝑢 = 𝑗 𝐼 0 𝑢/ℏ 𝐵 𝑢 ≡ 𝑓 𝑗 𝐵 𝑓 −𝑗 𝐼 0 𝑢/ℏ 𝐼 0 , 𝐵 𝑢 𝑒𝑢 ℏ 𝑗 𝐼 0 𝑢 ℏ |Ψ 𝑢 〉 Ψ 𝑢 = 𝑓 (here usual Schrödinger |Ψ 𝑢 〉 ) Also introduce 𝑗 −𝑗 𝐼 0 𝑢 𝐼 0 𝑢 = | 𝐵 𝑢 = Ψ 𝑢 𝐵 Ψ 𝑢 Ψ 𝑢 𝑓 𝐵 𝑓 Ψ(𝑢)〉 ℏ ℏ So that 𝐵 𝑢 Then evolution for | Ψ 𝑢 〉 is Ψ = 𝑗 −𝑗 −𝑗 𝑗 𝑗 𝑗 𝑒 𝐼 0 𝐼 𝐼 1 𝐼 0 𝑢 𝐼 0 𝑢 𝐼 0 𝑢 ℏ 𝑓 Ψ + 𝑓 Ψ = 𝑓 Ψ = ℏ ℏ ℏ 𝑒𝑢 ℏ ℏ −𝑗 𝑗 −𝑗 𝐼 1 SE 𝐼 0 𝑢 𝐼 0 𝑢 𝑒 Ψ = − 𝑗 = 𝑓 𝑓 Ψ ℏ ℏ 𝐼 1 𝑢 Ψ ℏ 𝑒𝑢 ℏ Heisenberg

Next subject: Methods for interacting electrons (terminology and ideas) 𝑊 0 ( 𝑠) “seed” potential Unknown 𝑊 𝑠 = 𝑊 0 𝑠 + Δ𝑊 𝑠 Problem: 𝑊( 𝑠) changes because of electron-electron interaction, so need some self-consistent approach. Thomas-Fermi method (or approximation) Assume equilibrium Chemical potential 𝜈 (Fermi level) 𝜈 − 𝑊( 𝑠) Unknown 𝑊 𝑠 = 𝑊 0 𝑠 + Δ𝑊 𝑠 3/2 1 2𝑛 𝜈−𝑊 𝑠 determines density of electrons, 𝑜( 𝑠 ) = Idea: 𝜈 − 𝑊 , 3𝜌 2 ℏ 2 then solve Poisson equation to find Δ𝑊( 𝑠) ; self-consistency: 𝑊 → 𝑜 → 𝑊 .

Hartree method (or approximation) Non-equilibrium, but stationary case electron no chemical potential flow all electrons alike 𝑊 𝑠 = 𝑊 0 𝑠 + Δ𝑊 𝑠 Idea: solve Schrödinger equation 2 , 𝐼𝜔 = 𝐹𝜔 to find 𝜔( 𝑠) , then 𝑜 𝑠 ∝ 𝜔 𝑠 then solve Poisson equation to find Δ𝑊( 𝑠) ; self-consistency 𝑊 → 𝜔 → 𝑜 → 𝑊 . 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 𝑜( 𝑠)

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 State with 𝑂 1 particles on level 1, 𝑂 2 particles 𝑂 = |𝑂 1 , 𝑂 2 , 𝑂 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 𝜔 = 𝑂 𝑑(𝑂) 𝑂 𝑑 𝑂 is probability This wavefunction lives in the occupation number space (Fock space) 〈𝑂 𝑂 = 1 〈𝑁 𝑂 = 0 if 𝑁 ≠ 𝑂 , Orthogonality: 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: 𝜔 = 𝑜 𝑑 𝑜 |𝑜〉 (Fock-space representation) Creation and annihilation operators † 0 = |0, 1, 0, 0, … 〉 † 0 = |0, 0, 1, 0, … 〉 𝑏 2 𝑏 3 † … 𝑂 𝑙 , … = For bosons 𝑏 𝑙 𝑂 𝑙 + 1 | … 𝑂 𝑙 + 1, … 〉 creates extra particle on level 𝑙 (factor 𝑂 + 1 as for an oscillator) For bosons 𝑏 𝑙 … 𝑂 𝑙 , … = 𝑂 𝑙 | … 𝑂 𝑙 − 1, … 〉 annihilates (kills) one particle on level 𝑙 (factor 𝑂 as for an oscillator) If 𝑂 𝑙 = 0 , then 𝑏 𝑙 … 0 𝑙 , … = 0 (zero, not vacuum) † † 𝑏 𝑙 … 𝑂 𝑙 , … = 𝑂 𝑙 | … 𝑂 𝑙 , … 〉 , so In particular, 𝑏 𝑙 𝑂 𝑙 = 𝑏 𝑙 𝑏 𝑙 Commutation relations † = 𝜀 𝑙𝑚 , † = 0 Sufficient for the † , 𝑏 𝑙 , 𝑏 𝑚 𝑏 𝑙 , 𝑏 𝑚 = 𝑏 𝑙 𝑏 𝑚 whole theory

Second quantization (cont.) 𝑏 † and Operators can often be expressed in terms of 𝑏 † H = 𝑙 𝜁 𝑙 𝑏 𝑙 𝑏 𝑙 (non-interacting particles, basis of eigenstates) † If basis vectors are not eigenstates, then also terms 𝑙𝑚 𝐼 𝑙𝑚 𝑏 𝑙 𝑏 𝑚 † 𝑏 𝑘 + 𝑘 (𝑈 † 𝑏 𝑘+1 + 𝑈 † Tight-binding model: ∗ 𝑏 𝑘+1 𝐼 = 𝑘 𝜁 𝑘 𝑏 𝑘 𝑘 𝑏 𝑘 𝑏 𝑘 ) 𝑘 † † Coulomb interaction: 𝐼 = 𝑙𝑚 𝐼 𝑙𝑚 𝑏 𝑙 𝑏 𝑙 𝑏 𝑚 𝑏 𝑚 For fermions similar, but commutation relations are where † † , † } + = 0 𝑏 𝑙 , 𝑏 𝑚 + = 𝜀 𝑙𝑚 , 𝑏 𝑙 , 𝑏 𝑚 + = { 𝑏 𝑙 𝑏 𝑚 𝐶 + ≡ 𝐶 𝐵, 𝐵 𝐶 + 𝐵 † † 0 = − † † 0 , so † † 0 = 0 For example, this means that 𝑏 𝑙 𝑏 𝑙 𝑏 𝑙 𝑏 𝑙 𝑏 𝑙 𝑏 𝑙 (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”? 𝜔 𝑦 → 𝜔 = 𝑙 𝜔 𝑙 𝑏 𝑙