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Instantons in quantum mechanics

W-double wellaassumeeven,-2degenerate ground states

Quantum

proximateEU(x)=U( - ).

weknowthat

anotherno zeropoint& nestion:

methods

Foll *
What:··at

x =-a.

ground states of the same energy
W=itheOne at x = a:

W "(a).

degeneracy is lifted by quantum

tunneling eﬀect Only one state Foll

Twopotential

aoa

ClassicalButUp),

evenTop Fote)
=0

Es ,

has the lowest
energy.

istheenergy

usual

WKB

instantons<splitting ?

,2

ZE T(xiki)
zu-(ox: =,e== En(t)e-measure

82

Take any JT()

Fx;

*En(i)

spaceitInner productChoose

X3F: =Obeying B C.

(((t)

Write&xEn=.
T- A

asand,

lying spectrum can be studied by looking at thebehaviour
The+(T() xy x) T(z)3)

Omit below

The low( V()anin=(t) + 3(t)

=such

F:

be(3. 3u)

written.

3([T()

;

==x:

0.

3(T)'s.

=,Orthonormal basis

canNof(T() + , -T/.

Ja ,E(x ] [F.

uniquelyN=aas

31=[5nP(i)fixed number to be.
determinedS

Computation by saddle point approximation
As60

dominant contributions,Sel].

For((T)

SE(k]

=

=

path-integral comeof[ Um]+minimum.

valuestheCTCT)c(t) + 3(t)Se(s].

Itmustobey

theE-LequI w - &EOM for "upside down" potential,+3([- U" Gins]3(5)++013%(xf xi),=Se](N .
3=smallest) + U-+=J 1

SupposezwithCCE)'sfromto=-ESE(z)] O3*(1 0(t))&"Giuss)]
Ne+fluctuation determinant+4Suppose Si(k]minimizedisbya

family of configurations

space[CIS [I]seMan m-parameter(S!
,S=Again-s(=>Pals e),,) ++=Ukil i),zero

are,ThenOX

N

=:

Es

= 0modesoa= 1,-of+m

U"(C 2) ,,necessarily Orthonormal).

((T) = c[(s i) +[xn)SM)Usi))](notWrite. . .,3n(n(S T),Orthonormal basisof Goals

.

1 CF.
etka.xs).

nz + (xfixi)=> e)N. (det (*-determinant

for+non-zero

U "((smodes*,es))] (1 0(t)only+

5Example
7 1.

z+.

g2UK)

2=,N.

(det(w)j=:

detA.

*ExactNo.

0(t) correction

=

w=+.

ASin(( + ) diagonalizes A with eigenvalue (I)Un(T) &:= O.
(T) = 0minimum(0 0)Harmonic Oscillator.

2=++w?

()·Zeroesat

WT = Ittin
n=o= 10 0).

as

WT +((N wi)=--3->-IIsinh (8T)/T(2wT)

as= Elo -as Tmatch

T+ a:

toSide1,tOperator result+ 100)-

1 , 2 3,(with=(N=) vEnlistent) =wealsoholds,+W

The double well

Compute Zy(taita) & ETIIa·For c = :=Ea Ia)~For >

xi,

t==-a,=-YYza>a)a-a> xthe unique minimumaso (w"T+U "(a):=look for trajectories likewe,T=8

is(,·TIT) = laIa,

6

T/2&Asinterestedare

weSolutions [T(T)Stin.

T-r(T)+-,

look for99 i-+T+A+o-AT + IG

Conservation ofenergy:↓Uki) = const = oTM=>.

i .
eIUGT=1↑+St)dx

2U(x)integration constant[JCT)↑[1-= 0aa7

7RemarksCCT)

Sell·Near

2U() >0as.
for-a < <a[ (* (ii)==·monotone

is+(2U :(( = Ia(() ~ + w2(x = a)

,-cal

=>

Rizal

=witheas[ + Is1~e.

·g.

The

instantaneous~Yo“instanton”

>
-U()

=Zero

mode associated with T

C , (T)=(0x 8),=(T i)(T) ==-,[dt) -jump(t)Soatanh(T-)
shiftisg·All other modes

Regard8,(2):Thus,

itas

has-haveapoint

ground state

It is thethe contribution

U" ics) < 0

Schrodinger operator
zerono+to.

(as.

CT) isAll OtherZola -a)
,

The E

integrandis

T-independent

But for finite (and large) T= (a -all instanton

state·For(f=-a ,2xi

=.

is-t

integral diverges

TheJdT,

has EXO.

T.

-TNde(U)]=,>=modeomonotone).

-Net(U)
The=G,

anti-instantons

contribute= -a a) (anti-instanton,=>thesameasinstanton

above>-but

Tantiinstanton(same value)
.

.

9
There

otherare>

Chain>

k>k·::>(kIntegration of parameters(k&In. . .

dEzdT 1.

u
7
=>

=

1n!

N.

-((det - w4]+=TkTh--#Sinb-- < TzIT ] E,zoanti-instantonsk·>&Fluctuation determinant~>of instantons

:

~n So
action·n- 1

saddle points

approximate=(detN. (det+w)/det(+V"(E())))

I10(kTe ES) (1 0(t)-&=+z+ (ta[a),=2 znn : even

T J(H 0()=>E+(19 [a),+I

=-=
The:

ground stateEo=t (a)
1ste:( -ke-So)(I 0(t)
+=

E (a)

=( ) (1 0(t)+excited state :

Ei

=

Elas:

--Eo

(t==++ke- E(-a)=So)(1 0()++ *(() )2tke So (1 O(t)+(( + 0(t)J(H 0()+11

k =det+w)/det( V"(()))]

I+canbe computed.

U"(EC)) "guinitiThe

Spectra of-+

02 c-

at

large eigenvalues (x(2)-TheItratiocan*So A

be:=+of the determinants

computedw

im(x-aandisU"((T)

is?

ZwA

areclosewell-defined.

withIs-alexp)/s)(see the additional note ]·= SA· Cdr

Ei-Er

=

2twSoltin Kale

So 2U(x)Hot

12

Exercise

Derive thisExerciseUK)
=>

E-Es

above

The=

is

Pow

calledvalid

theconfigurations

.

a=

I is

WKB

standard

using

when

dilutethegas approximation .
contribution

the dominantcomes

instantons/antiinstantons

where.
well-separatedRelevantterms
ndensity ofTheinthek+sum

theas

Ke

withis

So/th-

SoDt

dilute gas

nS

instantons/antiinstantons

=long

those

e-Solt-Asare,

it isvanishingly small

approximation,

isvalid.

andfromare13

potential

Periodic

infinitely many classical vacua

um
on

at,

·+ (ifa jia) In instantonsz(ja jia)=-=+-

in antiinstantins·Jdin---dTC

dEF-- dE-,

T/2 ? TnL-- = [ 2 T/2-.

- ? En ?. -zEi2 Th-(ninSo

3(kTe Sot)ni O(n -n+ijf ji)+-e- Sot ifWi exp)TIn !!

im2π=>(n n j ji),C,ja(j = 2)

x=-+KTeESo -10)--instantons

anti-instantons-2KT cosPeSolt

14

=>

Continuum ofenergy eigenstates To
labelledby OIR/2.

Eof-2tK cost -Sot) (HO(t)() j(1 0(t)-Ep(ia)=+Fo·.

I-

T> 0O

TL

15

.....

matcheswell-known resultwithinQ M.
.

(j)Suppose Estates &(j) Jjez localized&/1/1jaSitatx) = ja,<x(j)=f(x-ja),,< /Hli]=[j jEx-=8j 5-=1)"fight binding
approximation(j -j1 > 20Then,

To

: =

[e-it(j)arejez

Ep

-=

ei-u

plain waveh=(Block's theorem (Up()--:

Eo-28 cost

=

Fo(k)

energy eigenstates

periodic Up(x + a):=

M/2a (Brioullin ZoneFolia)=

eiioMolial

j independent-.

10(k)"