n

dr 1 · · ·

G(r f , tf ; r 0 , t0 ) = lim C(„ ) dr n’1 exp iSk / , (3.20)

„ ’0

k=1

where

tk

L(r, r, t) dt,

™

Sk = (3.21)

tk’1

over the linear path from r i’1 to r i . Note that a normalizing constant C(„ )

is incorporated which takes care of the normalizing condition for G, assur-

ing that the wave function remains normalized in time. This normalizing

constant will depend on „ : the more intermediate points we take, the larger

the number of possible paths becomes.

Equations (3.20) and (3.21) properly de¬ne the right-hand side (3.19).

For small time intervals Sk can be approximated by

Sk ≈ „ L(r, r, t),

™ (3.22)

™

with r, r and t evaluated somewhere in “ and with best precision precisely

halfway “ the interval (tk’1 , tk ). The velocity then is

r k ’ r k’1

™

r= . (3.23)

„

3.3 Path integral quantum mechanics 47

3.3.2 Equivalence with the Schr¨dinger equation

o

Thus far the path integral formulation for the Green™s function has been

simply stated as an alternative postulate of quantum mechanics. We must

still prove that this postulate leads to the Schr¨dinger equation for the wave

o

function. Therefore, we must prove that the time derivative of Ψ, as given by

the path integral evolution over an in¬nitesimal time interval, equals ’i/

times the Hamiltonian operator acting on the initial wave function. This is

indeed the case, as is shown in the following proof for the case of a single

particle in cartesian coordinate space in a possibly time-dependent external

¬eld. The extension to many particles is straightforward, as long as the

symmetry properties of the total wave function are ignored, i.e., exchange

is neglected.

Proof Consider the wave evolution over a small time step „ , from time t to

time t + „ :

Ψ(r, t + „ ) = dr 0 G(r, t + „ ; r 0 , t)Ψ(r 0 , t). (3.24)

Now, for convenience, change to the integration variable δ = r 0 ’ r and

consider the linear path from r 0 to r. The one-particle Lagrangian is given

by

mδ 2

L= ’ V (r, t). (3.25)

2„ 2

The action over this path is approximated by

mδ 2

S≈ ’ V (r, t)„, (3.26)

2„

which leads to the following evolution of Ψ:

imδ 2 i

Ψ(r, t + „ ) ≈ C(„ ) exp ’ V (r, t)„

dδ exp Ψ(r + δ, t). (3.27)

2„

We now expand both sides to ¬rst order in „ , for which we need to expand

Ψ(r + δ, t) to second order in δ. The exponent with the potential energy

can be replaced by its ¬rst-order term. We obtain

‚Ψ i

≈ C(„ ) 1 ’ V „

Ψ+„

‚t

imδ 2 imδ 2

1 ‚2Ψ

—Ψ δx dδ + · · · (y, z) ,

2

exp dδ + exp (3.28)

2 ‚x2

2„ 2„

where Ψ, its derivatives and V are to be taken at (r, t). The ¬rst-order

term in δ and the second-order terms containing mixed products as δx δy

48 From quantum to classical mechanics: when and how

cancel because they occur in odd functions in the integration. The ¬rst

integral evaluates3 to (ih„ /m)3/2 , which must be equal to the reciprocal of

the normalization constant C, as the zeroth-order term in Ψ must leave Ψ

unchanged. The second integral evaluates to (i „ /m)(ih„ /m)3/2 and thus

the right-hand side of (3.28) becomes

i 1 i„

1’ V„ Ψ + ∇2 Ψ ,

2 m

and the term proportional to „ yields

2

‚Ψ i

=’ ’ ∇2 Ψ + V Ψ , (3.29)

‚t 2m

which is the Schr¨dinger equation.

o

3.3.3 The classical limit

From (3.20) we see that di¬erent paths will in general contribute widely

di¬erent phases, when the total actions di¬er by a quantity much larger than

. So most path contributions will tend to cancel by destructive interference,

except for those paths that are near to the path of minimum action Smin .

Paths with S ’ Smin ≈ or smaller add up with roughly the same phase.

In the classical approximation, where actions are large compared to , only

paths very close to the path of minimum action survive the interference

with other paths. So in the classical limit particles will follow the path of

minimum action. This justi¬es the postulate of classical mechanics, that the

path of minimum action prescribes the equations of motion (see Chapter 15).

Perturbations from classical behavior can be derived by including paths close

to, but not coinciding with, the classical trajectory.

3.3.4 Evaluation of the path integral

When the Lagrangian can be simply written as

L(r, r, t) = 1 mr 2 ’ V (r, t),

™ ™ (3.30)

2

the action Sk over the short time interval (tk’1 , tk ) can be approximated by

m(r k ’ r k’1 )2

’ V (r k , tk )„,

Sk = (3.31)

2„

3 This is valid for one particle in three dimensions; for N particles the 3 in the exponent is

replaced by 3N . The use of Planck™s constant h = 2π is not an error!

3.3 Path integral quantum mechanics 49

and the kernel becomes

dr 1 · · ·

G(r f , tf ; r 0 , t0 ) = lim C(„ ) dr n’1

„ ’0

n

m(r k ’ r k’1 )2

i

’ V (r k , tk )„

exp . (3.32)

2„

k=1

The normalization constant C(„ ) can be determined by considering the

normalization condition for G. A requirement for every kernel is that it

conserves the integrated probability density during time evolution from t1

to t2 :