Instantons in quantum mechanics

W-double wellaassumeeven,-2degenerate ground states

Quantum

proximateEU(x)=U( - ).

weknowthat

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methods

Foll *
What:··atx =-a.

ground states of the same energy
W=i

theOne at x = a:

W "(a).

degeneracy is lifted by quantum

tunneling eﬀect Only one state Foll

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aoaClassicalButUp),

evenTop Fote)
=0

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

82

Take any JT()

Fx;

*En(i)

spaceitInner productChoose

X3F: =Obeying B C.

(((t)

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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.
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As60

dominant contributions,Sel].

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=

=

path-integral comeof[ Um]+minimum.

valuestheCTCT)c(t) + 3(t)

Se(s]

.

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smallest) + U-+=

J 1

SupposezwithCCE)'sfromto=-ESE(z)] O3*(1 0(t))&"Giuss)]
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family of configurations

space[CIS [I]seMan m-parameter(S!
,

S=Again-s(=>Pals e),,) ++=Ukil i),zero

are,ThenOX

N

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

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3n(n(S T),Orthonormal basisof Goals

.

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7 1.

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g2UK)

2=,N.

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0(t) correction=w=+.

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(T) = 0minimum(0 0)Harmonic Oscillator.

2=++w?

()·Zeroesat

WT = Ittin
n=o= 10 0).

as

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

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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=
8is(,·TIT) = laIa,T/2&As

interestedare

weSolutions [T(T)Stin.

T-r(T)+-,

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

T + IG

Conservation ofenergy:↓Uki) = const = o
TM=>.

i .
e

IUG

T=1↑+St)dx

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

RemarksCCT)·monotone

isSel

Near

2U() >0.
for-a < <a(d / Ulii] [d==+(Uki)

=>

·as

(( = Ia,=[dk2U()=:So(() ~ + w2(x = a)-Daleinstantaneouswith
as[ + Isjump“instanton”

e.

g.

U()

=a-(t)=atanh(ECT-[)

=IG

1-

1 +WIT-Tewit

for TET .

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aparameter of the solutionZeromode associated with T

C , (T)=The(0x 8),(T i)[dt) -=0(2):Thisitashas

It is themakesapoint

ground statesure+U" ics) < 0

Schrodinger operator
zeronothat I IisSo-

have

shift(t)-,Al other modes

Regard=,.

is(as.

CT) isAll OthertheThe E=modeomonotone).

state

has EXO.

only parameter.
End of RemarksThus,

the contributiontoZola -a),is-t-NetUs)] CHOTheintegrandisT-independent

But for finite (and large) T,>=TheJdT

integral diverges
T-.

.

Thiswould

Z + (a,/give-a)) instanton

TNde( Us)]

Note thatthe JCCT)'s.

= (a -as/instanton

onlyis,that the approximation is

For+= [U() does

solving

Satisfy the B C. for finite T·(1 0(7)).

+=and

*

anbetter,and the above

notformula for

But we expect

approximation.

and better

as

Tis

largerlarger.

(f=-a ,2xi=G,

anti-instantons

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

above>-but

Tantiinstanton(same value).

.

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approximate
Chain>

k>k·:~(Integration of parameters

kJat....

kzo

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kN

dindt ,.

1Th--n!

#Sinb-- < TzIT ] E,u=anti-instantonsk·>&Fluctuation determinant:>of instantons

:

~n So
action·>saddle points-(*.

(det ( - w2)] (H O(H)

TN

=++.

(detIdet+w)/det( V"(EC)))](1 0())++

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

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

=-=
The:

ground state
Eo=t (a)1stJ(H 0()+e:( -ke-So)(I 0(t)+=Eota)=(t)(H0()

even

excited state :

Ei=

Elas:

--Eo

(t=

=+-E+ke(a)So)(1 0()+*=(()) (( 0(t)2tke So (1 O(t)++oddk =det+w)/det( V"(()))]

I+canbe computed.

U"(EC)) "guinitiThe

Spectra of-+02 c-atlarge 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

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=

2twSoltin Kale

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=>

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above

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=

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calledvalid

theconfigurations.

a=

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standard

using

when

dilutethegas approximation .
contribution

the dominantcomes

instantons/antiinstantons

where.
well-separated

Relevant

terms
ndensity ofTheinthek+sum

theasKewithisSo/th-SoDt

dilute gasnS

instantons/antiinstantons=
longthosee-Solt-

Asare,

it isvanishingly small

approximation,

isvalid.

andfromarepotential

Periodic

infinitely many classical vacuaumonat,·+ (ifa jia) In instantonsz(ja jia)=-=+-

in antiinstantins·Jdin---dTC

dEF-- dE-,

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

- ? En ?. -zEi2 Th-(ninSo3(kTe Sot)ni O(n -n-eWi exp)TeI+in !!

im2π=>(n n j ji),C,ja(j = 2)

x=jf ji)+- Sot if-+KTeESo -10)--instantons

anti-instantons

2KTcosPeSo/t=>Continuum ofenergy eigenstates To
labelledby OIR/2.

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

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matcheswell-known result

withinQ M.
.

(j)Suppose Estates &(j) Jjez localized

&/1/
1ja
Sitatx) = ja

,

<x(j)

=

f(x-ja),,< /Hli]=[j jEx-=8j 5-=1)"fight bindingapproximation(j -j1 > 20Then,To: =[e-it(j)arejezEp-=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)"