8 apr 2021 quantum density matrices . L13{1 Pure and Mixed States in Quantum Mechanics Review of the Basic Formalism and Pure States • Definition: A pure quantum state is a vector Ψ = j i in a Hilbert space H, a complex vector space with an inner product hφj i. This defines a norm in the space, k k := h j i1=2, and we will usually assume that all vectors are normalized, so that k k = 1. For a particle moving in a region R of space these vectors are commonly taken to be square-integrable functions (r) in H = L2(R; d3x), with inner product defined by Z hφj i := d3x φ∗(x) (x) : R • Choice of basis and interpretation: Any state can be written as a linear combination j i = P c jφ i of pα α α i k·x elements of a complete orthonormal set fjφαi j hφαjφα0 i = δαα0 g (for example φk(x) = e = V ), where iθα the coefficients cα = jcαj e are the probability amplitudes for the system to be found in the corresponding states. • Observables: A quantum observable is an operator A^ : H!H that is self-adjoint (if the corresponding ^ classical observable is real). The possible outcomes of a measurement of A are its eigenvalues λα, satisfying ^ ^ A φα = λαφα, where φα are its eigenvectors. The expectation value of A in a given state Ψ is Z ^ ^ 3 3 ∗ ^ P ∗ hAi = h jAj i = d q1::: d qN (q) A (q) = α,α0 cαcα0 Aαα0 : ^ • Time evolution: It is governed by the Hamiltonian operator H. If fφαg is a basis of eigenfunctions of the ^ Hamiltonian, with Hφα = Eα φα, the Schr¨odingerequation and the time evolution of a state are given by @ (x; t) i R ^ = − H^ (x; t) ; (x; t) = U^(t; t ) (x; t ) = e−i Hdt=h¯ (x; t ) = P c φ (x) e−iEαt=h¯ : @t ¯h 0 0 0 α α α • Density-matrix notation: Given any state Ψ, define the operator ρ = j i h j. If the vector Ψ is normalized this operator satisfies ρ2 = ρ, ρy = ρ (it is a projection operator), and tr ρ = 1, and we can rewrite expectation ^ ^ P values as hAi = tr ρA. Then, if Ψ = α cα Φα, in terms of a complete orthonormal set P ∗ ρ = αα0 ραα0 jφαihφα0 j ; 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 ρy = 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 trace, we choose as basis fφαg one whose elements ^ 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 ^ ^ P hAiρ = tr ρA = α ρααλα : • Mixed states from averaged-out phase information: In quantum statistical mechanics mixed states often 2 arise as follows. If the probabilities jaαj that a quantum system will be found in each of the jφαis 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 = jcαj . The new density matrix 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 entropy. 8 apr 2021 quantum density matrices . L13{2 Example: Mixed State for the Spin of an Electron 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 j i = p1 (j" i + eiθj# i) or ρ = j ih j = 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 = j ih j, by taking a time derivative of ρ(x; x0) = ∗(x) (x0), @ @ ∗(x) @ (x0) 1 ρ(x; x0) = (x0) + ∗(x) = [H;^ ρ ] : @t @t @t i¯h By linearity we can then extend the validity of the expression (1=i¯h)[H;^ ρ ] for the time derivative to all density matrices ρ. This is the Liouville-von Neumann equation, and the operator L^ = (i¯h)−1[H;^ · ] is sometimes ^ ^ y ^ R ^ called the Liouvillian. Alternatively, ρ(t) = U(t; t0) ρ(t0) U(t; t0) , with U(t; t0) = exp{−i Hdt=¯hg. • 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;^ ρ ] = 0. So, ρ must be a function of some set of 3N commuting operators including H^ , 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 2 (E − ∆=2;E + ∆=2) in an incoherent superposition of states, start with a coherent sum over such states, = P ja j eiθn φ , and write ρ as an n n n average j ih j over all phase angles θn. The assumption of equal a priori probabilities implies that all janj 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 = ja j ja j e = δ 0 for E 2 (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 motion, 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 H^ and N^, 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(H^ −µN^) ;Z := tr e−β(H^ −µN^) : Z 1 ~ ~ ~ • Example: A spin- 2 particle in a magnetic field B = B z^, with Hamiltonian H = −~µ · B = −µ ~σ · B, where the σi are the Pauli matrices 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)..