QuantumPhysics.dvi

Translation operators in both bases satisfy

e+iaP/ ̄hQe−iaP/ ̄h=Q+a e−iaP/ ̄h|q〉=|q+a〉

e−ibQ/ ̄hPe+ibQ/ ̄h=P+b e+ibQ/ ̄h|p〉=|p+b〉 (14.10)

and their mutual overlap is

〈q|p〉 = e+iqp/ ̄h

〈p|q〉 = e−iqp/ ̄h (14.11)

This summarizes all that will be needed to derive the functional integral representation.

We now use the position basis to express the time-evolution of the wave functionφ(q,t) =
〈q|φ(t)〉associated with a general state|φ(t)〉. It is given by

φ(qb,tb) =〈qb|U(tb−ta)|φ(ta)〉=

∫

dqa〈qb|U(tb−ta)|qa〉φ(qa,ta) (14.12)

Hence, the evolution of the wave function is given by the matrix elements of the evolution operator
in a position space basis. It is these matrix elements for which we shall derive a functional integral
formulation. We can split this evolution into the consecutive evolutions of two shorter time intervals,
tc−ta= (tc−tb) + (tb−ta), withtc> tb> ta. Inserting a complete set of position eigenstates in
the product on the right hand side of the formulaU(tc−ta) =U(tc−tb)U(tb−ta), we obtain,

〈qc|U(tc−ta)|qa〉=

∫

dqb〈qc|U(tc−tb)|qb〉〈qb|U(tb−ta)|qa〉 (14.13)

Clearly this process can be repeated.

14.3 The evolution operator for a free massive particle

Before launching a full attack on this problem, let us calculate the matrix elements of the evolution
operator for a a free massless particle with Hamiltonian,

H=

1

2 m

P^2 (14.14)

SinceHis diagonal in the momentum basis, we calculate as follows,

〈qb|U(tb−ta)|qa〉 =

∫ dp

b

2 π ̄h

∫ dp

a

2 π ̄h

〈qb|pb〉〈pb|U(tb−ta)|pa〉〈pa|qa〉

=

∫ dp

b

2 π ̄h

∫ dp

a

2 π ̄h

ei(pbqb−paqa)/ ̄h〈pb|U(tb−ta)|pa〉 (14.15)

Using the fact thatU(tb−ta) is diagonal in the momentum basis, we find,

〈pb|U(tb−ta)|pa〉= 2π ̄hδ(pb−pa)e−i(tb−ta)p

(^2) a/(2m ̄h)
(14.16)