Lecture XIII
37
Lecture XIII: Double Well Potential: Tunneling and Instantons
How can phenomena of QM tunneling be described by Feynman path integral?
No semi-classical expansion!
! E.g QM transition probability of particle in double well: G(a, −a; t) ≡ #a|e−iĤt/!| − a$
V(x)
Potential
x
Inverted
potential
! Feynman Path Integral:
" ! t #
$%
i
" m 2
q̇ − V (q)
G(a, −a; t) =
Dq exp
dt
! 0
2
q(0)=−a
!
q(t)=a
Stationary phase analysis: classical e.o.m. mq̈ = −∂q V
%→ only singular (high energy) solutions Switch to alternative formulation...
! Imaginary (Euclidean) time Path Integral: Wick rotation t = −iτ
N.B. (relative) sign change! “V → −V ”
"
!
! q(τ )=a
$%
1 τ " #m 2
q̇ + V (q)
Dq exp −
dτ
G(a, −a; τ ) =
! 0
2
q(0)=−a
Saddle-point analysis: classical e.o.m. mq̈ = +V " (q) in inverted potential!
solutions depend on b.c.
(1) G(a, a; τ ) ! qcl (τ ) = a
(2) G(−a, −a; τ ) ! qcl (τ ) = −a
(3) G(a, −a; τ ) ! qcl : rolls from −a to a
Combined with small fluctuations, (1) and (2) recover propagator for single well
x
a
-V(x)
-a
a
x
o
Inverted
potential
-a
1/t
(3) accounts for QM tunneling and is known as an “instanton” (or “kink”)
Lecture Notes
October 2005
Lecture XIII
38
! Instanton: classically forbidden trajectory connecting two degenerate minima
— i.e. topological, and therefore particle-like
τ →∞
For τ large, q˙cl ' 0 (evident), i.e. “first integral” mq˙cl 2 /2 − V (qcl ) = $ → 0
precise value of $ fixed by b.c.& (i.e. τ )
Saddle-point action
(cf. WKB dqp(q))
! τ
!
!
!
$
#
τ
a
a
" m 2
"
2
dτ
dτ mq̇cl =
dqcl mq̇cl =
dqcl (2mV (qcl ))1/2
q̇cl + V (qcl ) '
Sinst. =
2
0
0
−a
−a
τ →∞
Structure of instanton: For q ' a, V (q) = 12 mω 2 (q − a)2 + · · ·, i.e. q̇cl ' ω(qcl − a)
τ →∞
qcl (τ ) = a − e−τ ω , i.e. temporal extension set by ω −1 ( τ
Imples existence of approximate saddle-point solutions
involving many instantons (and anti-instantons): instanton gas
x
a
o4
o5
o3
o2
o1
o
-a
! Accounting for fluctuations around n-instanton configuration
'
G(a, ±a; τ ) '
Kn
!
τ
dτ1
0
n even / odd
!
τ1
0
dτ2 · · ·
!
τn−1
0
An,cl. An,qu.
(
)*
+
dτn An (τ1 , . . . , τn ),
constant K set by normalisation
An,cl. = e−nSinst. /! — ‘classical’ contribution
An,qu. — quantum fluctuations (imported from single well): Gs.w. (0, 0; t) ∼
An,qu. ∼
n
,
i
-
G(a, ±a; τ ) '
=
'
1
sin(−iω(τi+1 − τi ))
'
Using ex =
Lecture Notes
n=0
i
e−ω(τi+1 −τi )/2 ∼ e−ωτ /2
n −nSinst. /! −ωτ /2
K e
e
e−ωτ /2
xn /n!,
(
!
0
n even / odd
n even / odd
0∞
∼
n
,
√ 1
sin ωt
/n
1 .
τ Ke−Sinst. /!
n!
.
τ
dτ1
!
τ n /n!
τ1
0
)*
dτ2 · · ·
!
τn−1
+
dτn
0
N.B. non-perturbative in !!
/
/!
G(a, a; τ ) ' Ce−ωτ /2 cosh τ Ke−Sinst.
/
.
G(a, −a; τ ) ' Ce−ωτ /2 sinh τ Ke−Sinst. /!
October 2005
Lecture XIII
39
Consistency check: main contribution from
0
nX n /n!
= X = τ Ke−Sinst. /!
n̄ = #n$ ≡ 0n n
X
/n!
n
no. per unit time, n̄/τ exponentially small, and indep. of τ , i.e. dilute gas
^
A
S
Exact States
Oscillator States
ht
A
S
x
! Physical interpretation: For infinite barrier — two independent oscillators,
coupling splits degeneracy — symmetric/antisymmetric
G(a, ±a; τ ) ' #a|S$e−#S τ /!#S| ± a$ + #a|A$e−#Aτ /!#A| ± a$
|#a|S$|2 = #a|S$#S| − a$ =
C
,
2
|#a|A$|2 = −#a|A$#A| − a$ =
Setting: $A/S = !ω/2 ± ∆$/2
/
C . −(!ω−∆#)τ /2!
e
G(a, ±a; τ ) '
± e−(!ω+∆#)τ /2! = Ce−ωτ /2
2
1
C
2
cosh(∆$τ /!)
.
sinh(∆$τ /!)
! Remarks:
(i) Legitimacy? How do (neglected) terms O(!2 ) compare to ∆$?
In fact, such corrections are bigger but act equally on |S$ and |A$
i.e. ∆$ = !Ke−Sinst. /! is dominant contribution to splitting
V(q)
ïV(q)
q
q
m
q
qm
o
(ii) Unstable States and Bounces: survival probability: G(0, 0; t)? No even/odd effect:
"
%
3
2
Γ
−ωτ /2
−Sinst /! τ =it
−iωt/2
exp τ Ke
= Ce
exp − t
G(0, 0; τ ) = Ce
2
Decay rate: Γ ∼ |K|e−Sinst /! (i.e. K imaginary) N.B. factor of 2
Lecture Notes
October 2005