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8 apr 2021 density matrices . L13–1

Pure and Mixed States in Quantum

Review of the Basic Formalism and Pure States • Definition: A pure is a vector Ψ = |ψi in a H, a complex with an inner product hφ|ψi. This defines a norm in the space, kψk := hψ|ψi1/2, and we will usually assume that all vectors are normalized, so that kψk = 1. For a moving in a region R of space these vectors are commonly taken to be square-integrable functions ψ(r) in H = L2(R, d3x), with inner product defined by Z hφ|ψi := d3x φ∗(x) ψ(x) . R • Choice of and interpretation: Any state can be written as a |ψi = P c |φ i of √α α α i k·x elements of a complete orthonormal set {|φαi | hφα|φα0 i = δαα0 } (for example φk(x) = e / V ), where iθα the coefficients cα = |cα| e are the probability amplitudes for the system to be found in the corresponding states. • : A quantum is an Aˆ : H → H that is self-adjoint (if the corresponding ˆ classical observable is real). The possible outcomes of a measurement of A are its eigenvalues λα, satisfying ˆ ˆ A φα = λαφα, where φα are its eigenvectors. The expectation value of A in a given state Ψ is Z ˆ ˆ 3 3 ∗ ˆ P ∗ hAiψ = hψ|A|ψi = d q1... d qN ψ (q) A ψ(q) = α,α0 cαcα0 Aαα0 .

ˆ • evolution: It is governed by the Hamiltonian operator H. If {φα} is a basis of eigenfunctions of the ˆ Hamiltonian, with Hφα = Eα φα, the Schr¨odingerequation and the time evolution of a state ψ are given by

∂ψ(x, t) i R ˆ = − Hψˆ (x, t) , ψ(x, t) = Uˆ(t, t ) ψ(x, t ) = e−i Hdt/h¯ ψ(x, t ) = P c φ (x) e−iEαt/h¯ . ∂t ¯h 0 0 0 α α α • Density- notation: Given any state Ψ, define the operator ρ = |ψi hψ|. If the vector Ψ is normalized this operator satisfies ρ2 = ρ, ρ† = ρ (it is a projection operator), and tr ρ = 1, and we can rewrite expectation ˆ ˆ P values as hAiψ = tr ρA. Then, if Ψ = α cα Φα, in terms of a complete orthonormal set

P ∗ ρ = αα0 ραα0 |φαihφα0 | , with ραα0 = cαcα0 ,

Mixed Quantum States • Idea: A mixed state gives us the probability of finding the system in any given element of a basis for the

Hilbert space, but does not contain information on all phases θα. We represent it as an operator ρ : H → H satisfying ρ† = ρ and tr ρ = 1, but not necessarily ρ2 = ρ. The space of density matrices is Liouville space.

• Interpretation: Since we can use any basis to calculate a , we choose as basis {φα} one whose elements ˆ are eigenvectors of the observable A, with corresponding eigenvalues λα; in this basis, Aαα0 = λαδαα0 . Then, for any mixed state ρ, ραα is the probability of finding the system in eigenstate α, as with a pure state, and ˆ ˆ P hAiρ = tr ρA = α ρααλα .

• Mixed states from averaged-out phase information: In quantum mixed states often 2 arise as follows. If the probabilities |aα| that a quantum system will be found in each of the |φαis are known while the phases θα are not, assume that all values are equally likely and average the matrix elements ∗ ραα0 = cαcα0 over 0 < θα < 2π. The off-diagonal entries in ραα0 will average to zero, while the diagonal 2 2 entries will give ραα = |cα| . The new ρ in general no longer satisfies ρ = ρ.

• Additional comments: (1) For a system consisting of two subsystems A and B, H = HA ⊗ HB, and a mixed state for A can be obtained from any state ρA,B by tracing over subsystem B, ρA = trB ρA,B. (2) As a measure of the mixedness of a quantum state ρ one can use its n = 2 R´enyi . 8 apr 2021 quantum density matrices . L13–2

Example: Mixed State for the of an h¯ h¯ • Density matrix: Suppose that an electron has a 50% probability of Sz being + 2 , and 50% of being − 2 . A pure state, its corresponding density matrix, and a mixed state which give these values are, respectively,  1 e−iθ   1 a  |ψi = √1 (|↑ i + eiθ|↓ i) or ρ = |ψihψ| = 1 , ρ = 1 . 2 pure 2 eiθ 1 mixed 2 a∗ 1

• Mean value and fluctuation of spin: Check that the mean value of Sz vanishes in both states and calculate 2 the mean value of Sx in both states; why is the result reasonable? Calculate their variances (∆Sz) and 2 (∆Sx) , and compare the results for the pure state and the mixed state; comment on the results. State Evolution • Evolution equation: Working in the Schr¨odingerpicture, we start by obtaining the time evolution of a ρ corresponding to a pure state, ρ = |ψihψ|, by taking a time derivative of ρ(x, x0) = ψ∗(x) ψ(x0), ∂ ∂ψ∗(x) ∂ψ(x0) 1 ρ(x, x0) = ψ(x0) + ψ∗(x) = [H,ˆ ρ ] . ∂t ∂t ∂t i¯h By linearity we can then extend the validity of the expression (1/i¯h)[H,ˆ ρ ] for the time derivative to all density matrices ρ. This is the Liouville-von Neumann equation, and the operator Lˆ = (i¯h)−1[H,ˆ · ] is sometimes ˆ ˆ † ˆ R ˆ called the Liouvillian. Alternatively, ρ(t) = U(t, t0) ρ(t0) U(t, t0) , with U(t, t0) = exp{−i Hdt/¯h}. • Quantum equilibrium density matrix: An equilibrium density matrix is one which is time-independent or, given the form of the evolution equation for ρ, one satisfying [H,ˆ ρ ] = 0. So, ρ must be a function of some set of 3N commuting operators including Hˆ , and be diagonal in the corresponding basis of eigenstates.

The Quantum Microcanonical Density Matrix

• Form of the density matrix: To describe a system with energy En ∈ (E − ∆/2,E + ∆/2) in an incoherent superposition of states, start with a coherent sum over such states, ψ = P |a | eiθn φ , and write ρ as an n n n average |ψihψ| over all phase angles θn. The assumption of equal a priori probabilities implies that all |an| for states in this energy range are equal; if we also assume a priori uniformly random phases, we get 0 1 0 i(θα−θα) ρ 0 = |a | |a | e = δ 0 for E ∈ (E − ∆/2,E + ∆/2) , and 0 otherwise , αα α α Γ(N,V,E; ∆) αα α where Γ(N,V,E; ∆) is the number of states in this energy range.

The Quantum (Grand) Canonical Density Matrix • Form of the density matrix: If we can partition the system into two parts, each of which is similar to the whole system so that its ρ is the same function of the constants of , with little interaction between (1) (2) (1) (2) them, then as in the classical case H ≈ H1 + H2 and ραα = ρα1α1 ρα2α2 , so ln ραα = ln ρα1α1 + ln ρα2α2 . ~ This means that ln ραα must be an additive constant of the motion, or neglecting a possible overall ~p or L and in a basis of eigenstates of Hˆ and Nˆ,

c−βEα+βµNα ραα0 = e δαα0 . Then, in an arbitrary basis and calling ec =: Z−1 (K := H −µN is sometimes called the grand Hamiltonian), 1 ρ = e−β(Hˆ −µNˆ) ,Z := tr e−β(Hˆ −µNˆ) . Z 1 ~ ~ ~ • Example: A spin- 2 particle in a magnetic field B = B zˆ, with Hamiltonian H = −~µ · B = −µ ~σ · B, where the σi are the  0 1   0 i   1 0  σ = , σ = , σ = . x 1 0 y −i 0 z 0 −1 −1 βµB −βµB Then ρ = [2 cosh(βµB)] diag(e , e ) and one gets, for example, hµzi = µ tanh(βµB). Reading • Textbooks: Kennett does not discuss this material; Pathria & Beale, Chapter 5. • Other references: Part of this material is covered in Plischke & Bergersen (Sec 2.4), Halley (first half of Ch 2) and Schwabl (Sec 1.3-1.4). An extended treatment is in J A Gyamfi, arXiv:2003.11472 (2020).