Quantum Two 1. 2 Evolution of Many Particle Systems 3.

Quantum Two

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Evolution of Many Particle Systems

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The evolution of a many particle quantum system is, as the basic postulates assert, governed through the Schrödinger equation

where represents the Hamiltonian operator describing the total energy of the many particle system.

For an isolated system of particles, the Hamiltonian can often be written in the non-relativistic limit in the form

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The evolution of a many particle quantum system is, as the basic postulates assert, governed through the Schrödinger equation

where represents the Hamiltonian operator describing the total energy of the many particle system.

For an isolated system of particles, the Hamiltonian can often be written in the non-relativistic limit in the form

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As for conservative single particle systems, evolution is then most easily described in terms of the eigenstates of , i.e., the solutions to the energy eigenvalue equation

Note that the Hamiltonian

is independent of any possible spin variables of the particles involved. It only depends on spatial variables.

This allows for a conceptual simplification of the problem: we can essentially factor out (and ignore!) the spin degrees of freedom.

To see this write the total many-particle state space as . . . 6

Since direct products commute, we can combine all the spatial parts and all the spin parts to write this in the form

where

describes all the spatial variables, and

describes all the spin variables. 10


where



Thus, we can solve the energy eigenvalue problem in , spanned by the direct product position states , without worrying about spin at all.

After we are done, we can form direct products of the complete set of spatial energy eigenstates, with a complete set of spin states, for , thus obtaining a full set of solutions to the energy eigenvalue problem spanning the total direct product space.

Implementing this idea, we project the eigenvalue equation onto the many particle position representation, obtaining a partial differential equation

for the many-particle energy eigenfunctions

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Problem:

when is greater than two this equation (except in very special situations) is analytically intractable (i.e., non-separable).

This analytical intractability often arises from the important physical situation in which the potential energy of the system arises, at least in part, from two-body inter-particle interactions of the form

An obvious example is the Coulomb interaction.19

Problem:




Solutions to problems of this sort are fundamental to the study of atomic and molecular physics when is relatively small (, typically)

and to the study of other forms of matter (i.e., condensed phases, liquids, solids, etc.) when is very large ().

Under these circumstances one often seeks approximate solutions developed using, e.g., the techniques of perturbation theory, which attempts to develop perturbation expansions around some already previously solved problem.

An important, and in principle exactly soluble special case about which one often expands, in these situations, is that of a collection of non-interacting particles, in which each particle moves separately in its own potential , i.e.,

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Indeed, using single particle potentials of this type it is often possible, to accurately treat using perturbation methods, more complicated two-body interactions, like the Coulomb interaction described above.

Imagine for example, that all but one of the particles in the system were fixed in some well-defined set of single particle spatial states.

This remaining particle could then be treated as moving in the potential field generated by all the others. This can be iteratively repeated for each particle.

In this way, suitable potentials can thus be generated for each particle which, in some average sense, self-consistently take into account the states that the remaining particles actually end up in.

This is precisely the goal of so-called self-consistent theories, such as the Central Potential Approximation, the Hartree-Fock approximation, and the Thomas-Fermi model. 27

Once a set of single particle potentials of this type has been found, the actual Hamiltonian

can then be rewritten (exactly) in the form

where

it is to be hoped, now represents a small correction to the total final energy. 32



where


In principle, the exact solution can then be expanded about the eigenstates of the non-interacting Hamiltonian

in which

is a single particle operator that acts only on the part of any direct product state associated with the single particle space .

But suppose that in each single particle space we solve the problem

for the part of the Hamiltonian associated with that particle. 36

In principle, the exact solution can then be expanded about the eigenstates of the the non-interacting Hamiltonian

in which





in which





in which

is a single particle operator that acts only on the part of any direct product state associated with the single particle space . How do we solve this?




in which

is a single particle operator that acts only on the part of any direct product state associated with the single particle space . How do we solve this?

Suppose that in each single particle space we solve the eigenvalue problem


The states then form an ONB of single particle states for ,

and in the many particle direct product space we can produce a direct product basis from these states in the usual way.

These states are eigenstates of the single particle operators .

But, we claim that these direct product states are then automatically eigenstates of the proposed (non-interacting) zeroth order Hamiltonian

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To show this, we just compute

where the total energy eigenvalue

is just the sum of the non-interacting single particle energies (as it is classically). 45

The corresponding wave function associated with such a state is then a product

of the associated single particle eigenfunctions of the operators , the same result that one would find by using the process of separation of variables.

For electronic systems, the (anti-symmetrized) direct product state that is closest to the actual ground state (which because of interactions is an entangled superposition of such states) is called the Hartree, or Hartree-Fock approximation.

The difference between the energy of the Hartree-Fock solution and the true ground state energy is referred to as the correlation energy, and in many interacting systems it provides only a small correction to the total energy.

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Thus, as we have seen although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom.

In our discussion of many-particle systems, we have ignored the fact that in many systems of interest the particles are elements of classes of identical or indistinguishable particles.

This empirical fact puts additional constraints on the symmetry properties of the state vectors of many particle systems.

We will explore the consequences of these constraints at a later point in the semester.

In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have alluded to above. 56





In the next segment we begin a discussion of approximation methods for solving eigenvalue problems, of the sort we have already alluded to above. 60

Quantum Two 1. 2 Evolution of Many Particle Systems 3.

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