An introductory lecture on Feynman path integrals

Adolfo del Campo∗

July 16, 2004

Dept. of Chemistry, University of North Carolina at Chapel Hill,USA;Dept. of Physical chemistry, Basque Country University, Spain

1 Background

1.1 Classical Mechanics

Let’s consider a classical system defined by a kinetic energy T (x) and apotential V (x). Then its Lagrangian is given

L = T (x)− V (x) (1)

Theorem (Hamilton’s principle) 1 The motion of the system from t1to t2 is such that the time integral S (called action or action integral) has astationary value for the actual path of the motion

S ≡∫ t2

t1

L (x, x, t) dt⇒ δS = 0 (2)

The corresponding equation of motion (the well-known Euler-Lagrangeeqs.) are

∂

∂t

L

∂xi− L

∂xi= 0 (3)

which are equivalent to Newtons laws:

Def. : pi =∂L

∂xi⇒ ∂tpi = Fi =

∂L

∂xi= −∂V

∂xi(4)

⇒ pi = −∂V∂xi

(3rdNewton′s law) �

∗E-mail: isdd@ole.com

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Using the Einstein’s summation convention, we can write the Hamilto-nian as the total Legendre transform of the Lagrangian:

H = qipi − L (5)

1.2 Quantum Mechanics

It is postulated that all the information that we can known about a givensystem is contained in a vector |ψ〉 belonging to an abstract Hilbert spaceH.

Recall the completeness relationst∫Rdx|x〉〈x| = 1H;

∫Rdp|p〉〈p| = 1H

then,

|ψ〉 =∫

Rdx|x〉〈x|ψ〉 =

∫Rdx|x〉ψ(x) (6)

where we have defined the wavefunction on the coordinate representation asψ(x) = 〈x|ψ〉.

Similarly, using the resolution of the identity in the momentum repre-sentation we get,

|ψ〉 =∫

Rdp|p〉〈p|ψ〉 =

∫Rdp|p〉ψ(p). (7)

Notice that

〈x|p|p〉 = p〈x|p〉 = i~∂

∂x〈x|p〉 ⇒ 〈x|p〉 =

1√2π~

eipx (8)

where we have chosen the normalization 〈x|x′〉 = δ(x−x′) Eq.(8) is just theplane wave solution for the free particle Hamiltonian.

The ket |ψ〉 obeys the time dependent Schrodinger equation (TDSE)

i~∂t|ψ〉 = H|ψ〉 (9)

Let H be time independent, then the state at time t provided that the systemwas at ψ(t0) at t = t0 is given by

ψ(t) = U(t, t0)|ψ(t0〉 = e−iHt/~|ψ(t0〉 (10)

(Thm.: Iff H is hermitian, then U is unitary).

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2 On the Path integral

2.1 Introduction

In this section a derivation a la Swanson of the path integral formula is givenassuming quantum mechanics [2]. In 1948, in his ”Space-time Approach tonon-relativistic Quantum mechanics”, Feynman gave a derivation just inthe opposite way; using the classical Hamilton’s principle he introduces thisthird formulation of QM. The basic idea is that for the motion of a particlebetween two points in space, all possible connecting trajectories q(t) areconsidered, and a probability amplitude Φ[q(t)] is assigned to each of them.

Next, it is postulated [1] that:

1. If an ideal measurement is performed to determine whether a particlehas a path lying in a region of space-time, then the probability thatthe result will be affirmative is the absolute square of a sum o complexcontributions, one from each path in the region.

2. All paths contribute equally in magnitude, but the phase of their con-tribution is the classical action (in units of ~); i.e., the time integralof the Lagrangina taken along the path.

Φ[q(t)] ∼ eiS[q(t)]

~ (11)

2.2 Derivation of the Path Integral formula

Suppose that the wavefunction of a given system at (ta, qa) is known and wewish to find it at (tb, qb), considering an integral equation of the form

ψ(qb, tb) =∫dqaZ(tb, qb; ta, qa)ψ(qa, ta). (12)

Using the completeness relation we can also write

〈qb, tb|ψ〉 =∫dqa〈qb, tb|qa, ta〉〈qa, ta|ψ〉. (13)

This is called the composition property. In probability theory (thereforedealing with true probabilities instead of probabilitiy amplitudes) the anal-ogous expresion is known as the Chapman-Kolmorogov equation and in dif-fusion theory is called Smoluchowski equation (in the context of Wiener in-tegrals).

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Therefore, the kernel Z is

Z = Z(tb, qb; ta, qa) = 〈qb, tb|qa, ta〉 = 〈qb|U(tb − ta)|qa〉. (14)

We have then identify the Z with the probability amplitude using the Heisen-berg picture, for a system originally prepared at time ta and position eigen-value qa to be found at a later time at (tb, qb). This particular inner productis called transition amplitude.

Let’s now study the partition ε = tb−tan . Inserting again the resolution

of the identity,Z(tb, qb; ta, qa)

=∫

R

n−1∏i=1

dqi〈qb, tb|qn−1, tn−1〉〈qn−1, tn−1|qn−2, tn−2〉 · · · 〈q1, t1|qa, ta〉. (15)

Take,

〈qj+1, tj+1|qj , tj〉 = 〈qj+1|−iεH(P,Q)~|qj〉 =∫

Rdpj〈qj+1|pj〉〈pj | U(ε)|qj〉

≈∫

R dpU(pj , qj ; ε)〈qj+1|pj〉〈pj |qj〉;

but since〈qj+1|pj〉〈pj |qj〉 =

12π~

eipj(qj+1−qj)/~ (16)

using

limε→0qj+1 − qj

ε=dqjdt

= qj

we get

〈qj+1, tj+1|qj , tj〉 =∫

R

dpj

2π~e−

iε~ [pjqj−H(pj ,qj)] =

∫R

dpj

2π~e−

iε~ L(pj ,qj), (17)

where L(pj , qj) = pjqj −H(pj , qj) is the Lagrangian density.Therefore,

Z =∫ ∞

−∞

dp0

2π~· · · dpn

2π~dq1 · · · dqn−1e

− i~

∑n−1j=1 εL(pj ,qj). (18)

Note that for wanishing ε the Riemann sum reduces to the integral

limε→0

∑εL −→

∫dtL = S (19)

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Introducing the notation

DpDq ≡ limε→0dp0

2π~· · · dpn

2π~dq1 · · · dqn−1 (20)

we can utterly write the Feynman path integral formula

Z(tb, qb; ta, qa) =∫ qb

qa

DqDpei~

∫ tbta

L(q,p)dt = 〈qb, tb|qa, ta〉. (21)

We see that the integrand assigns to each trajectory q(t) a different phase(complex number). For this reason path integrals are also called functionalintegrals. Besides, integrating over momenta, and calling∫ qb

qa

D [q(t)] ≡ limn→∞(m

2πi~ε)(n−1)/2

∫dq1 · · · dqn−1

we can rewrite Eq.(21) as

〈qb, tb|qa, ta〉 =∫ qb

qa

D [q(t)]eiS[q(t)]

~ (22)

It is legitimate to ask ourselves how this formulation reconcilliates in thecorresponding limit with the classical world. For negligible ~, the integrandin eq. (22) becomes a highly oscillatory function for small variations of thepath. Therefore, only those paths close to the classical one will contribute,since the phases will add up contructively according with the variationaltheorem δS = 0. The contributions of the rest of the possible paths willcancell each other due to destructive interference [3, 5, 6].

2.3 Example: the free particle case

By definition for a free particle V=0, so the Lagrangian and Hamiltonianare:

L(q, q) =12mq2; H(p, q) =

p2

2m(23)

The ”propagator” Z is easily computed as follows

Z(tb, qb; ta, qa) = 〈qb|U(tb − qa)|qa〉 =∫dp

−ip2(tb−ta)

2m~ 〈qb|p〉〈p|qa〉 (24)

=∫

dp

2π~

−ip2(tb−ta)

2m~ − ip(qb−qa)

~=

√m

2πi~(tb − ta)e

+im(qb−qa)2

2~(tb−ta) (25)

This equation shows the spreading of a gaussian wavepacket with time.

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3 Path Integrals & Statistical Mechanics

The partition function in quantum statistical mechanics for a {N,V, T}-canonical ensemble is

Qβ =∑

i

〈i|e−βH |i〉 = Tre−βH = Trρ (26)

where we have introduced the thermal density operator ρ ≡ e−βH

Given Qβ we know ”everything” about the system. Namely, we cancalculate the ensemble average values

〈E〉β = − ∂

∂βlnQβ =

Tr(ρH)Trρ

(27)

Fβ = − 1β

lnQβ (28)

S = β2∂ lnFβ

∂β(29)

P = −∂ lnFβ

∂V(30)

etc.Now, consider Z(qb, qq; tb − ta) with qb = qa and β = i(tb−ta)

~ (Wick’srotation). Then, the quantity∫

dqaZ|qb=qait~ =β

=∫dqaZaa =

∫dqa〈qa|e−βH |qa〉

=∑

i

〈i|e−βH |i〉 = Tre−βH (31)

where we have use the fact that the trace is basis invariant.

∴ Qβ =∫dqa

∮qb=qa

DpDqe−∫ β0 dτ [H−ipq](32)

since τ = it; ∂q∂τ = −i ∂q

∂τ .The Wick rotated path integral is as well behaved as the Statistical mechan-ics version of the same problem. Only those paths starting and finishing atqa will contribute to the integral, in other words, trajectories have to beperiodic with period T = β.

A further consequencem is the well-known Kubo-Martin-Schwinger (KMS)condition: Wick rotated observables are periodic with T = β.

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3.1 The Feynman-Kac formula

As a simple application of path integral techniques, the Feynman-Kac for-mula turns out to be an excellent way of computing the ground states ofenergy leves [4].

E0 = −limτ→∞1τ

log Tre−τH (33)

Here we just outline a simple proof. We know that for a physical sys-tem, in the limit of vanishing absolute temperatures, β →∞, the partitionfunction can be approximated as Qβ ≈ eE0 .

Besides, ∫dqK(q, t; q, 0) =

∑n

e−itEn/~ (34)

Using the Wick’s rotation τ = it/~,

limτ→∞eτE0

∫dqK(q,−i~τ ; q, 0) = limτ→∞e

τE0Qτ = 11

∴ E0 = −limτ→∞1τ

log∫dqaZa = −limτ→∞

1τ

log Tre−τH �

References

[1] R.P.Feynman: Space-Time Approach to Non-Relativistic Quantum Me-chanics; Rev.Mod.Phys. 20 (1948) 367.

[2] M. Swanson: Path Integrals and Quantum Processes (Academic Press,Inc., 1992).

[3] H.Kleinert: Path Integrals in Quantum Mechanics, Statistics and Poly-mer Physics (WorldScientific, Singapore, 1990).

[4] L.S.Schulman: Techniques and Applications of Path Integration (JohnWiley & Sons, New York, 1981).

[5] R.P. Feynman, Statistical Mechanics: A Set of Lectures, Benjamin,1972.

1If the ground state is degenerated the RHS of this eq. would be g0 instead of 1.

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