QuantumPhysics.dvi

5 Some Basic Examples of Quantum Systems

It is will be very useful for later studies to examine some of the mostfundamental quantum

systems beyond the two-state models. They will include some finite systems with more than

2 states, time-dependent 2-state systems like NMR, the 1-dimensional harmonic oscillator,

and the angular momentum algebra.

5.1 Propagation in a finite 1-dimensional lattice

One of the most fundamentalmodelsconsists of states propagating on a 1-dimensional lattice

consisting ofN sites. We label the lattice sites by an integern= 1, 2 ,···,N. The Hilbert

space is spanned by the basis states|n〉. A state|n〉may be thought of as representing the

quantum system where the “particle” is at sitenwith probability 1, and probability 0 to

be on any of theN other sites. Such states are naturally orthogonal, and may be chosen

orthonormal,〈m|n〉=δm,nform,n= 1, 2 ,···,N.

A simple example of such a system is provided by an electron propagating on a lattice

of atoms, ions or molecules. In reality, there will not just be one electron, but many. Also,

the electron will be free to move in more than one dimension, and will have electromagnetic

and spin interactions as well. In this model, all these extra effects willbe omitted in favor

of just to 1-dimensional location of the electron. An example withN= 6 is provided by the

Benzene molecule, where 3 electrons approximately freely move over a 6-atom ring. In this

case the lattice is naturally periodic. More generally, the model can describe propagation of

electrons along long chains of atoms, ions or molecules withN≫1. If only bulk properties

are of interest, we are free to impose convenient boundary conditions on this lattice. We

choose these to be periodic, which allows for the simplest treatment. Therefore, it is often

convenient to identify|N+ 1〉=| 1 〉.

The dynamics of the quantum system is governed by the Schr ̈odinger equation, in terms

of a HamiltonianHfor the system. We want to use physical arguments to try and retain

only the most important dynamical information inH, and omit all else. To do this, we study

the Schr ̈odinger equation. Any state|ψ(t)〉may be decomposed onto the basis{|n〉}n,

|ψ(t)〉=

∑N

n=1

cn(t)|n〉 cn(t) =〈n|ψ(t)〉 (5.1)

The Schr ̈odinger equation

i ̄h

∂

∂t

|ψ(t)〉=H|ψ(t)〉 (5.2)

determines the time-evolution of the probability amplitudescn(t), as a function of the matrix

elements of the Hamiltonian.To show this, take the inner product of the above Schr ̈odinger