Advanced Quantum Mechanics

We once did the whole classical mechanics to quantum mechanics thing here, and since there are courses on the advanced lessons, I might as well toss this in here too, and save myself the effort of thinking of another topic for this series.

The first immediate change is that more often we're dealing with many-body systems now, rather than the probability distribution of a single quantum particle. This will not immediately change much of the math, but it will give considerations like the operating space being the Hilbert space a more nuanced meaning. Other ideas, such as degeneracy of eigenvalues are elevated to more meaning as well. Still, the probability operators of the eigenvalues won't change in themselves, mainly due of a quirk in the bra-ket operators commuting with summation, and the sum over all states of some eigenvalue returning 1. We also retain the standard Ansatz of the Schroedinger equation for many of the basic problems, which gives rise to the Schroedinger picture with time-dependent wave functions, and Heisenberg picture which equips its operators with the time-evolution operator from both sides. Minorly, the Heisenberg equation emerges as a parallel Ansatz to the Schroedinger equation, though it's often more relevant in field theories, where we would like to avoid thinking too long about particle-particle interactions.

N-particle systems extend the Hilbert space by one orthonormal basis for each particle, where each entry of the basis ends up the product of N wave functions, each itself member of its particle's basis. This extends exactly in the same way to the rest of bra-ket calculus. This effectively increases the computational effort by an exponent of N, which means that for almost every system, the pure state contains too much information to handle effectively. We could associate some probabilities, which wouldn't change anything about the random property of the individual wave function, and decrease the computational effort. For this, we take once again the density operator, which has a trace of 1, is positive definite, and self-adjoint. It's the same density matrix perhaps familiar from intro to QM, or mathematical modelling courses.

Since the N-particle system hopefully retains the particle number. This qualifies it as a "Grand Canonical Ensemble". I always forget formulas from statistical physics, so I'll list them with relatively little explanation. The probabilities are normalized to

The understanding of the quantum N-particle system as a grand-canonical ensemble won't help the computation problem, but it's theoretically accurate.

Since we'll be dealing with ensembles of identical particles a lot of the time, the states need to be (anti-)symmetrized, so that we keep the Pauli exclusion principle happy. This is accomplished through the "second quantization" and the (anti-)symmetrization operator

Where P is the particle permutation operator. With the expanded description the state of an N-particle system comes a difficulty: The states in a tensor product don't commute. This means that a correct application of the particle exchange operator can't change the measurement result. If an exchange operator exists for a system, it doesn't adhere to P|Ψ⟩ = ±|Ψ⟩, then Ψ is actually an unphysical product state. For bosons, the rule is P|Ψ⟩ = +|Ψ⟩, and we call the Hilbert space symmetric, for fermions, the rule is P|Ψ⟩ = -|Ψ⟩ and the Hilbert space is antisymmetric.

States can also be described using the occupation-number representation, which lists all possible states as entries, and notes how many particles occupy it. It's also orthonormal and complete. When written out as a product state, it requires a normalization constant (for ⟨n|n⟩ = 1), which is generally

Though of course, fermions can only have n-entries of 0 or 1. The Fock-space is the direct orthogonal sum of all N = 0, 1, ... states, and the set of all occupation-number states form an orthonormal basis for it. From there, we also define our creation/annihilation operators:

For bosons, ε = 1, for fermions, ε = -1. The same operator can be applied to the product state, which adds the state from the left. Note, that the order is important still.

Observables usually consists of one and two-particle components:

Read the second term as the scattering of two particles from orbitals c, d into the orbitals a, b. Whereas the product of creation and annihilation operator once gave the particle number, this is now the occupation number operator, and the particle-number operator is the sum of all occupation number operators. The Hubbard-Model introduces one such operator

Which describes state transitions of particles within an N-particle system. It consists of a state-hopping part, and a particle-particle interaction part, which can be roughly translated to a Coulomb-interaction. The state occupation does not take into account the particle spin, so technically two particles can always occupy a state. This can be described "on foot" using the occupation operator for both, or the abbreviated double-occupation operator

For spin-1/2 fermions, the spin operator is still split into the spatial dimensions, though they of course need to be indexed for the particles.

In terms of unitary transformation of bases (of the same Hilbert-space), the creation operator is expected to transform in the same way as the one-particle basis.

Operators hence remain form invariant under unitary transformation. When changing to a continuous orthonormal basis (e.g. containing eigenstates of the position operator), all the summations transition into integrals, and the field operators require new representations. This will take the (hopefully familiar) form of quantum-field-theory.

For a first introduction of thermodynamics into quantum N-particle systems, a standard, and general method is that of the "Static Mean-Field Theory". As there are ideal gases (see the ideal gas law), there is a notion of an ideal quantum gas. It's a generic "free" system

where H and thus t are Hermitian. Unitary transformations apply in the same way, with a diagonal matrix that is transformed from t. This is the form that H takes in the second quantization. The grand-canonical Hamiltonian is H - μN, and the partition function is a product of the sum of all the individual partition functions.

The Bose and Fermi statistics remain the same. Checking over to the ultra-cold quantum gases in the Physics line is going to go into more detail on this. Static mean-field theory is derived from a generalized variational principle, which has static responses of the many-body system to a static and weak perturbation. Exploring these perturbations means checking the behaviour of observables in response to small changes of the Hamiltonian. Thermal expectation values are centered in this discussion. For observables, there will always be an option for decomposition of the Hamiltonian that isolates it up to a scaling parameter.

Using the Trotter decomposition and taking the limit to infinity creates a modified Heisenberg representation:

which doesn't include any imaginaries, so it will never transform unitarily. This can be remedied by the time-transformation used in the Wick rotation t = -iτ.

This is also called the "dissipation-fluctuation theorem".

For operators commuting with the Hamiltonian, the grand-canonical potential is a concave function of λ. From the density operator comes the grand-canonical potential as a functional, yielding some scalar.

The functional can be minimized for the grand-canonical density operator. For general Mean-Field theory, some approximations must be made, though the resulting Hartree-Fock approximation takes a familiar form. The approximations are the variational principle, then an invariance in λ, and after choosing a reference system characterized by λ, the correlation functions is assumed to be simple. Then the variated matrix t' can be described using the Hartree-Fock self-energy.

For an optimal one-particle Hamiltonian, fix t' to through Ω's invariance.

Said self-energy is computed through the use of Wick's theorem. It's hermitian and frequency/time independent. The first term is considered the "Hartree potential" and the second the "Exchange potential". Computing it for a continuous 1-particle ONB with U being the Coulomb interaction for spinless particles will give it a more familiar, but more cumbersome form.

From this, we can see that the exchange potential need not have a clear classical interpretation. Such expressions can be simplified using appropriate forms of quantization. For computations in the Hartree-Fock approximations, usually the familiar expression is set to equal the an unperturbated quantity, plus the expectation value of the appropriate amount of exchange terms.