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Lecture XII 35

Lecture XII: Applications of the Feynman Path Integral

Digression: Free particle propagator (exercise) cf. diffusion

Gfree(qF, qI; t) qF|eip2t/2m|qI(t) =

m2it

1/2exp

i

m(qF qI)2

2t

(t)

Difficult to derive from PI(!), but useful for normalization

Quantum Particle in a Single (Symmetric) Well: V(q) = V(q)

V

q

e.g. QM amplitude

G(0, 0; t) 0|eiHt/|0 (t) =

q(t)=q(0)=0

Dqexp

i

t0

dt

mq2

2 V(q)

Evaluate PI by stationary phase approximation: general recipe

(i) Parameterise path as q(t) = qcl(t) + r(t) and expand action in r(t)

S[q+ r] =

t0

dtm

2

qcl2 + 2 qclr + r

2 (qcl + r)

2

V(qcl) + rV(qcl) +

r2

2V

(qcl) + V(qcl + r)

= S[qcl] + t

0

dtr(t)

Sq(t)

=0

[mqcl V

(qcl)] +1

2 t

0

dtr(t)

2Sq(t)q(t)

m2t V

(qcl)r(t) +

(ii) Classical trajectory: mqcl = V(qcl)

Many solutions choose non-singular solution qcl = 0 (why?)i.e. S[qcl] = 0 and V

(qcl) = m2 constant

G(0, 0; t)

r(0)=r(t)=0

Dr exp

i

t0

dtr(t)m

2

2t

2

r(t)

N.B. if V was quadratic, expression trivially exact

More generally, qcl(t) non-trivial non-vanishing S[qcl] see PS3

Fluctuation contribution? example of a...

Gaussian functional integration: mathematical interlude

Lecture Notes October 2005

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Lecture XII 36

One variable Gaussian integral: (

dv eav

2/2)2 = 2

0r d r ear

2/2 = 2a

dv ea2v2 =

2

a, Re a > 0

More than one variable:dv e

12v

TAv = (2)N/2det A1/2

where A is +ve definite real symmetric N N matrix

Proof: A diagonalised by orthogonal transformation: A = OTDO

Change of variables: w = Ov (Jacobian det(O) = 1)

N decoupledGaussian integrations: vTAv = vTOTOAOTOv = wTDw =N

i diw2i

Finally,N

i=1 di = det D = det A

Infinite number of variables; interpret {vi} v(t) as continuous field and

Aij A(t, t) = t|A|t as operator kernel

Dv(t) exp

1

2

dt

dt v(t)A(t, t)v(t)

(det A)1/2

(iii) Applied to quantum well, A(t, t

) = i

m(t t

)(2

t 2

) and formally

G(0, 0; t) Jdet

2t 21/2

where J absorbs various constant prefactors (im, , etc.)

What does det mean? Effectively, we have expanded trajectories r(t)

in eigenbasis of A subject to b.c. r(t) = r(0) = 02t

2

rn(t) = nrn(t), cf. PIB

i.e. Fourier series expn: rn(t) = sin(nt

t), n = 1, 2, . . . , n = (

nt

)2 2

det

2t 21/2

=n=1

1/2n =n=1

nt

2 2

1/2 For V = 0, G = Gfree known use to eliminate constant prefactor J

G(0, 0; t) =G(0, 0; t)

Gfree(0, 0; t)Gfree(0, 0; t) =

n=1

1

t

n

21/2 m2it

1/2(t)

Finally, applying identity

n=1[1 (xn

)2]1 = xsinx

G(0, 0; t) m2isin(t)(t)(exact for harmonic oscillator)

Lecture Notes October 2005