Quantum statistical ensembles The basics of quantum mechanics • Quantum mechanics is a more fundamental description of nature than classical mechanics. In general, a quantum microstate can be specied by a wavefunction Ψ(Ω), which is a complex function of the classical microstate Ω (with a caveat that pairs of so called canonically conjugate variables, such as position and momentum, cannot both be included in Ω). Quantum mechanics adds another layer of uncertainty to the needed statistical treatment of enormously complicated Ω. Measurement outcomes of any classical variable in Ω (e.g. the position of a certain particle) are generally random even in microscopic (non-statistical) quantum systems. The quantity jΨ(Ω)j2, then, expresses the probability (per unit phase space volume) that the classical state Ω would be observed in a measurement. One can view a quantum state as a simultaneous realization or superposition of many classical states Ω in a single system, where the wavefunction Ψ(Ω) gives the weight of any classical state in the superposition. Since the wavefunction is complex, a quantum superposition can exhibit phenomena such as (de Broglie) waves and interference. • For a system of N identical particles, the microstate is given by the positions of all particles Ω = (r1; r2; ··· ; rN ) up to a permutation. The wavefunction is then: Ψ(Ω) ≡ Ψ(r1; r2; ··· ; rN ) Alternatively, the wavefunction can be expressed as a function of momenta of all particles by taking a Fourier transform. Since identical particles are fundamentally indistinguishable, the probability density must be symmetric under permutations P: 2 2 jΨ(r1; r2; ··· ; rN )j = jΨ(rP(1); rP(2); ··· ; rP(N))j (8P) This implies (in three dimensions): Ψ(r1; r2; ··· ; rN ) = σP Ψ(rP(1); rP(2); ··· ; rP(N))(8P) where σP = ±1. For a class of particles called bosons, σP = 1. Multiple bosons can occupy the same single-particle state, or position ri = rj. For a class of particles called fermions, σP = 1 if the permutation is even (an even number of pair exchanges) and σP = −1 if the permutation is odd (an odd number of pair exchanges). Two fermions cannot be in the same single-particle state due to Pauli exclusion principle: Ψ(r1; r2) = −Ψ(r2; r1) ) Ψ(r; r) = −Ψ(r; r) = 0 • Quantum measurement: An operator is a mathematical object that operates on a wavefunction and generally transforms it to a dierent function of the microstate. Examples of simple operators are multiplication by a constant, taking a derivative, etc. In quantum mechanics, an operator A^ is associated with every measurable physical quantity a. The possible measurement outcomes a and the corresponding quantum states are generally obtained by solving an eigen-equation: A^ a(Ω) = a a(Ω) This is generally a dierential equation. One has to determine such values a (called eigenvalues of the operator A^) and the corresponding wavefunctions a (called eigenfunctions of the operator A^) that the action of A^ on a amounts to multiplying the same a by the constant a. Then, the wavefunctions a(Ω) are the quantum microstates in which the measurement of the quantity A^ would yield the value a with absolute certainty. In any other quantum state Ψ(Ω), the measurement of the same quantity A^ yields one of its eigenvalues randomly. The probability that a particular eigenvalue a will be measured in the quantum state Ψ(Ω) is given by: 2 ∗ P (a) = dΩ (Ω)Ψ(Ω) ˆ a 43 provided that all wavefunctions are normalized: 2 2 ∗ ∗ dΩ (Ω) a(Ω) = dΩΨ (Ω)Ψ(Ω) = 1 ˆ a ˆ • A quantum state is stationary if its probability density jΨ(Ω)j2 does not change in time. Since energy is always conserved, stationary states are discovered by solving the eigen-equation for energy: H^ E(Ω) = E E(Ω) This is known as stationary Schrodinger equation; H^ is the Hamiltonian operator associated with energy. For a system of N interacting particles, the Hamiltonian is analogous to the one in classical mechanics: N X p^2 1 X H^ = i + U(^r ) + V (^r − ^r ) 2m i 2 i j i=1 i6=j th but ^ri and p^i are position and momentum operators respectively for the i particle. These operators, in turn, can be represented as: @ ^ri = ri ; p^i = −i~ @ri th i.e. the position operator simply multiplies the wavefunction by the position vector ri of the i particle, and the momentum operator takes the gradient of the wavefunction with respect to ri and multiplies it by −i~, where ~ = h=2π is a variant of the Planck's constant h (sometimes called Dirac constant). Since the Hamiltonian is an operator associated with energy, Schrodinger equation identies the possible energy measurement outcomes (eigenvalues) E and the corresponding eigenfunctions E(Ω) that carry a well-dened energy E. • A stationary quantum state described by an energy eigenfunction E(Ω) is generally a superposition of many classical microstates Ω, and thus hard to identify by some typical classical congurations Ω. It is always more practical to label and enumerate the stationary quantum states by quantum numbers. In the case of so called bound states (where particles cannot escape to innity: electrons inside an atom, gas inside a box), quantum numbers always take integer values. Also, there is always one quantum number per degree of freedom. So, for example, a gas of N particles trapped in a box has quantum numbers M = (m1; m2 ::: mN ), where mi = (mix; miy; miz) and mix; miy; miz are integers. It is understood that permutations among the quantum numbers of individual identical particles do not correspond to dierent quan- tum states (in other words, M is dened only up to a permutation of particles). The type and meaning of quantum numbers is obtained from the solution of Schrodinger equa- tion, and depends on the given problem's context. If there are conserved quantities like angular momentum, then some of the quantum numbers label the corresponding conserved eigenstates. For a particle trapped in a box of dimensions (Lx;Ly;Lz), stationary quantum states are standing waves (r) / sin(kxx) sin(kyy) sin(kzz), where the wavevectors ki, i 2 fx; y; zg are quantized by the requirement that an integer number of wavelength halves t exactly in mi 2 N λi = 2π=ki the box: 1 . The integers are quantum numbers. 2 miλi = Li mi Diatomic and other molecules that can rotate have two additional quantum numbers that specify a rotational state: an orbital quantum number l 2 f0; 1; 2 ::: g and a magnetic quantum number m 2 {−l; −l + 1 : : : l − 1; lg. Elementary particles also have spin, an internal rotation-like quantum number ms 2 {−s; −s + 1 : : : s − 1; sg, where s is determined by the type of particle. For all fermions, s is a half-integer ( 1 for electrons, protons, neutrons, etc.). For all bosons, is an integer ( for hydrogen, s = 2 s s = 0 s = 1 for photons, etc.). Vibrational modes of linear oscillators are enumerated by an integer quantum number m 2 f0; 1; 2 ::: g. 44 • Stationary states can be degenerate, i.e. there could be multiple stationary quantum states (with dierent combinations of quantum numbers M) that have the same energy E. Therefore, quantum numbers M rather than energy E provide an unambiguous labeling scheme for stationary quantum states. One normally labels the Hamiltonian eigenfunctions E(Ω) by quantum numbers, M(Ω). • Any wavefunction Ψ(Ω) can be expressed as a superposition of energy eigenfunctions N(Ω): X Ψ(Ω) = CM M(Ω) M The coecients CM are complex constants that can be calculated from: C = dΩ ∗ (Ω)Ψ(Ω) M ˆ M if the wavefunctions are normalized. As a consequence, we can give up using the typical classical microstate specication Ω (e.g. the positions of particles) and switch to quantum numbers M as a complete enumeration scheme for quantum microstates. Equilibrium quantum statistical ensembles • An equilibrium quantum statistical ensemble is specied by a classical probability distribution p(M) of quantum stationary microstates labeled by quantum numbers M. By construction, p(M) depends on the microstate energy E(M), which is required in equilibrium. p(M) is the classical probability that an ensemble member would be detected in the quantum microstate M upon measurement. If indeed a system is found in the quantum state N, then measurements of various observables can still yield random outcomes due to the nature of quantum mechanics, but a measurement of energy yields the conserved value E(M) of the found microstate with certainty. • Using the same arguments as in classical physics, one denes: 1 p(M) = δ ··· in microcanonical ensemble Ω(E) E(M);E 1 p(M) = e−βE(M) ··· in canonical ensemble Z 1 −β E(M)−µN p(M) = e ··· in grand canonical ensemble Z where: X Ω(E) = δE(M);E ··· the number of quantum states with energy E M X −βE(M) Z = e ··· canonical partition function at temperature T = 1=kBβ M 1 X X −β E(M)−µN Z = e ··· grand canonical partition function at chemical potential µ N=1 M • The most convenient ensemble for dealing with identical particles is grand canonical. Instead of xing the total number of particles N, one species the chemical potential µ. Then, the quantum microstate of identical particles can be completely described by a set of occupation numbers : the M nm 2 N number of particles in every possible single-particle state (identied by its quantum numbers m). The occupation numbers for bosons can be any non-negative integers. For fermions, however, the occupation numbers can take only two values, zero or one (Pauli exclusion prohibits occupying a single-particle state with more than one fermion).