parte 2 mecanica cuantica

8/16/2019 Parte 2 Mecanica Cuantica

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5.3 PARTICULA EN UNA CAJA.

LA PARTICULA CUANTICA BAJO CONDICIONES FRONTERA. CUANTIZACION DELA ENERGIA.

La partícula e ua ca!a " p"#" $e p"tec%al %&%t"' e( u pr")le*a +ue

c"(%(te $e ua ("la partícula +ue re)"ta $etr" $e ua ca!a %*,-%l $e lacual " pue$e e(capar' $"$e " p%er$e eer/ía al c"l%(%"ar c"tra (u(pare$e(.

De($e el put" $e -%(ta cl0(%c"' (% ua part%cula re)"ta el0(t%ca*ete 1ac%a

atr0( 1ac%a a$elate a l" lar/" $el e!e x etre $"( pare$e(

%*peetra)le( (epara$a( p"r ua $%(tac%a L ' c"*" (e *ue(tra e la

&/ura 2' (e *"$ela c"*" ua part%cula )a!" rap%$e# c"(tate. S% la rap%$e#

$e la part%cula e( u ' la *a/%tu$ $e (u cat%$a$ $e *"-%*%et" mu

per*aece c"(tate' al %/ual +ue (u eer/ía c%et%ca. La í(%ca cl0(%ca "%*p"e re(tr%cc%"e( a l"( -al"re( $e la cat%$a$ $e *"-%*%et" eer/ía$e ua part%cula. El platea*%et" *eca%c" cuat%c" para e(te pr")le*a e(*u $%erete re+u%ere +ue (e ecuetre la uc%, $e "$a apr"p%a$a +ue(ea c"(%(tete c" la( c"$%c%"e( $e e(ta (%tuac%,.

4a +ue la( pare$e( (" %*peetra)le(' " e%(te pr")a)%l%$a$ al/ua $e

1allar la part%cula uera $e la ca!a' $e *"$" +ue la uc%, $e "$a ψ ( x)

$e)e (er cer" para x<0 x> L . Para (er ua uc%, +ue tra)a!e )%e

*ate*0t%ca*ete' ψ ( x) $e)e (er c"t%ua e el e(pac%". N" $e)e

e%(t%r (alt"( $%(c"t%u"( e el -al"r $e la uc%, "$a $e cual+u%er put". 2

P"r l" tat"' (% ψ e( cer" uera $e la( pare$e(' ta*)%6 $e)e (er cer" e

la( pare$e(' e(t" e(' ψ (0 )=0 ψ ( L )=0. S"l" (e per*%te la( uc%"e(

$e "$a +ue (at%(a/a e(ta( c"$%c%"e( $e r"tera.

E la &/ura 2)' la repre(etac%, /ra&ca $el pr")le*a $e la part%cula e uaca!a ' *ue(tra la eer/ía p"tec%al $el (%(te*a part%cula7et"r" c"*" uauc%, $e la p"(%c%, $e la part%cula. 8%etra( la part%cula e(ta $etr" $e laca!a' la eer/ía p"tec%al $el (%(te*a " $epe$e $e la u)%cac%, $e la

part%cula e( p"(%)le e(c"/er (u -al"r %/ual a cer".

Fuera $e la ca!a' $e)e a(e/urar(e $e +ue la uc%, $e "$a (ea cer". Pue$e1acer e(t" al $e&%r la eer/ía p"tec%al $el (%(te*a c"*" %&%ta*ete/ra$e (% la part%cula e(tu-%era uera $e la ca!a. P"r l" tat"' la 9%ca "r*a$e +ue ua part%cula pue$a e(tar uera $e la ca!a e( (% el (%(te*a t%ee uacat%$a$ %&%ta $e eer/ía' l" +ue e( %*p"(%)le.

La uc%, $e "$a para ua part%cula e la ca!a (e epre(a c"*" uauc%, (e"%$al real:;

ψ ( x)= Asen

(

2πx

λ )



D"$e λ e( la l"/%tu$ $e "$a $e De Br"/l%e a("c%a$a c" la part%cula.

E(ta uc%, $e "$a $e)e (at%(acer la( c"$%c%"e( r"tera e la(

pare$e(. La c"$%c%, r"tera e ψ (0 )=0 a e(ta (at%(ec1a p"r+ue la

uc%, (e" e( cer" cua$"

x=0

. Para la c"$%c%, $e r"tera eψ ( L )=0 ' (e t%ee

ψ ( L )=0= Asen(2πL

λ )<ue ("l" pue$e (er -er$a$era (%

2 πL

λ =nπ λ=

2 L

n

D"$e n=1,2,3,… E c"(ecuec%a' ("l" (e per*%te c%erta( l"/%tu$e(

$e "$a para la part%cula. Ca$a ua $e la( l"/%tu$e( $e "$a per*%t%$a(

c"rre(p"$e a u e(ta$" cuat%c" para el (%(te*a' n e( el u*er"

cuat%c". Al %c"rp"rar la( ecuac%"e( (e pr"p"rc%"a

ψ ( x )=√ 2

Lsen( nπx L )

La &/ura ; a ) (" repre(eta$ac%"e( /ra&ca( $eψ

e ua uc%,

xy|ψ |2 e uc%, $e x para n=1,2 y 3 para la part%cula e ua ca!a.3

O)(er-e +ue au cua$" ψ pue$e (er p"(%t%-a " e/at%-a' |ψ |2 (%e*pre

e( p"(%t%-a. C"*" |ψ |2 repre(eta ua $e(%$a$ $e pr")a)%l%$a$' u -al"r

e/at%-" para |ψ |2 " te$ría (et%$".

Ua %(pecc%, *a( a "$" $e la &/ura 2) *ue(tra +ue |ψ |2 e( cer" e

la( r"tera(' l" cual (at%(ace la( c"$%c%"e( r"tera. A$e*a(' |ψ |2 e(

cer" e "tr"( put"(' $epe$%e$" $el -al"r $e n . Para n=2 ' |ψ |2=0

e x= L/2 = para n=3,|ψ |2=0 e x= L/3 e x=2 L

3 . El u*er" $e

put"( cer" au*eta e u" ca$a -e# +ue el u*er" cuat%c" au*ete eu".

P"r+ue la( l"/%tu$e( $e "$a $e la part%cula e(t0 re(tr%/%$a( p"r la

c"$%c%, λ=2 L/n ' la *a/%tu$ $e la cat%$a$ $e *"-%*%et" $e la



part%cula ta*)%6 e(ta re(tr%/%$a a -al"re( e(pecí&c"( +ue (e 1alla apart%r $e la epre(%, para la l"/%tu$ $e "$a $e De Br"/l%e.

p=h

λ=

h

2 L /n= nh

2 L

Se (e>ala la eer/ía p"tec%al $el (%(te*a %/ual a cer" cua$" la part%culae(ta $etr" $e la ca!a. P"r l" tat"' l"( -al"re( per*%t%$"( $e la eer/ía $el(%(te*a' +ue (%*ple*ete e( la eer/ía c%et%ca $e la part%cula' ("pr"p"rc%"a$"( p"r

En=1

2mu

2= p

2

2m=(nh/2 L)2

2m

En=( h2

8mL2 )n2

n=1,2,3,…

E(ta epre(%, *ue(tra +ue la eer/ía $e la part%cula e(ta cuat%#a$a. Laeer/ía *%%*a c"rre(p"$%ete al e(ta$" u$a*etal' +ue e( el e(ta$" $eeer/ía *%%*a para cual+u%er (%(te*a. Para la part%cula e ua c!a' el

e(ta$" u$a*etal c"rre(p"$e a n=1 ' para el cual En=h2/8m L

2

.

C"*" En=n2 E1 ' l"( e(ta$"( ec%ta$"( c"rre(p"$%ete( a n=2,3,4,…

t%ee eer/ía( $a$a( p"r 4 E1

,9 E1

,16 E1

, …

La &/ura 3 e( u $%a/ra*a $el %-el $e eer/ía +ue (e $e(cr%)e l"( -al"re($e eer/ía $e l"( e(ta$"( per*%t%$"(. P"r+ue la *e"r eer/ía $e uapart%cula e la ca!a e( $%erete $e cer"' (e/9 la *ec0%ca cu0t%ca' lapart%cula uca pue$e e(tar e rep"(". La *%%*a eer/ía +ue pue$e teer'

c"rre(p"$e a n=1 (e $e"*%a eer/ía $el e(ta$" u$a*etal. E(te

re(ulta$" c"tra$%ce el put" $e -%(ta cl0(%c"' e el +ue E=0 e( u

e(ta$" acepta)le' c"*" (" t"$"( l"( -al"re( p"(%t%-"( $e E .

E!e*pl" 2.

Part%cula( *%cr" *acr"(c,p%ca( e ca!a(.

A? U electr" e(ta c"&a$" etre $"( pare$e( %*peetra)le( c" ua

(eparac%, $e 0.200nm . Deter*%e l"( %-ele( $e eer/ía para l"( e(ta$"(

n=1,2 y 3.



I*a/%e +ue la part%cula e( u electr" la( pare$e( e(t0 *u !uta(.

Su(t%tu%*"( l"( $at"( e la ecuac%,

En=( h2

8mL2 )n2

n=1,2,3,…

Para n=1 :

E1=( h

2

8mL2 )(1)2= (6.63 x10

−34Js)2

8(9.11 x 10−31

kg)(2.00 x10−10

m)2

¿1.51 x 10−18

J =9.42 eV

C" la ecuac%, En=n2 E ' ecuetre la( eer/ía( $e l"( e(ta$"( n=2

n=3 :

E2=(2 )2 E

1=4 (9.42 eV )=37.7 eV

E3=(3)2 E1=9 (9.42 eV )=84.8 eV

B? Ecuetre la rap%$e# $el electr" e el e(ta$" n=1 .

Ut%l%#a$" la epre(%, cl0(%ca para eer/ía c%et%ca para la rap%$e# $e uapart%cula:

K =1

2 me u2

u=√2 En

me

Rec""#ca +ue la eer/ía c%et%ca $e la part%cula e( %/ual a la eer/ía $el

(%(te*a (u(t%tua En p"r K .

Su(t%tua l"( -al"re( ")te%$"( $el %c%(" A?:



u=√2 (1.51 x10

−18J )

9.11 x10−31

kg=1.82 x10

6m/s

C" ("l" t"car el electr" e la ca!a re(ulta e ua rap%$e# *%%*a $el

electr" %/ual a 0.6 $e la rap%$e# $e la lu#.

@al"re( e(pera$"( para la part%cula e ua ca!a.

EL OSCILADOR AR8ONICO SI8PLE.

C"(%$ere ua partícula +ue (e ("*ete a ua uer#a re(taura$"ra l%eal

F =−kx $"$e x ' e( la p"(%c%, $e la partícula re(pect" al e+u%l%)r%"

( x=0) k e( ua c"(tate. El *"-%*%et" cl0(%c" $e ua partícula

("*et%$a a e(ta uer#a e( ar*,%c" (%*ple. La eer/ía $el (%(te*a e(:

U =1

2kx

2=1

2mω

2 x

2

D"$e la recuec%a a/ular $e -%)rac%, e( ω=√ k /m . E la *ec0%ca

cl0(%ca' (% la partícula (e $e(pla#a $e($e (u p"(%c%, $e e+u%l%)r%" (e

(uelta' "(c%la etre l"( put"( x=− A x= A ' $"$e A e( la

a*pl%tu$ $el *"-%*%et". A$e*0(' (u eer/ía t"tal E

E= K +U =1

2kA

2=1

2mω

2 A

2

E el *"$el" cl0(%c"' cual+u%er -al"r $e E e( per*%t%$"' %clu(" E=0 '

+ue e( la eer/ía t"tal cua$" la partícula e(ta e rep"(" e x=0 .

A1"ra %-e(t%/are*"( c"*" e( trata$" el "(c%la$"r ar*,%c" (%*ple $e($e el

put" $e -%(ta cu0t%c". La ecuac%, $e Sc1r$%/er para e(te pr")le*a (e

")t%ee al (u(t%tu%r U =1

2mω

2 x

2

e la ecuac%, $e Sc1r$%/er %$epe$%ete $el t%e*p":

−ℏ2

2m

d2ψ

dx2 +

1

2mω

2 x

2ψ = Eψ

N" ")(tate' e( $e ut%l%$a$ pe(ar e ua ("luc%,. T"*e la (%/u%ete

uc%, $e "$a para el c0lcul":



ψ =Be

−c x2

Al (u(t%tu%r e(ta uc%, e la ecuac%, $e Sc1r$%/er %$epe$%ete $elt%e*p"' (e ecuetra +ue e( ua ("luc%, (at%(act"r%a a la ecuac%, $eSc1r$%/er (%e*pre cua$"

c=mv

2ℏ E=1

2ωℏ

Re(ulta +ue la ("luc%, +ue (e 1a calcula$" c"rre(p"$e al e(ta$"

u$a*etal $el (%(te*a' +ue t%ee ua eer/ía1

2ωℏ . C"*" c=mω /2ℏ

(e $e$uce +ue la uc%, $e "$a para e(te e(ta$" e(:

ψ =Be

−(mω/2ℏ) x2

Fuc%, $e "$a para el e(ta$" u$a*etal $e u "(c%la$"r

ar*,%c" (%*ple.

D"$e B e( la c"(tate a (er $eter*%a$a $e($e la c"$%c%, $e

"r*al%#ac%,. E(te re(ulta$" e( ("l" ua ("luc%, a la ecuac%, $eSc1r$%/er %$epe$%ete $el t%e*p". La( ("luc%"e( re(tate( +ue$e(cr%)e l"( e(ta$"( ec%ta$"( (" *0( c"*pl%ca$a(' per" t"$a( la(

("luc%"e( %clue el act"r ep"ec%al e−cx2

.

L"( %-ele( $e eer/ía $e u "(c%la$"r ar*,%c" e(t0 cuat%#a$"(' c"*"

e( $e e(perar(e' p"r+ue la partícula "(c%lat"r%a e(t0 $e(t%a$a a e(tar cerca$e x=0 . La eer/ía para u 9*er" cu0t%c" ar)%trar%" n e(

En=(n+1

2) ωℏ n=0,1,2,… Eer/ía( cuat%#a$a( para u "(c%la$"r

ar*,%c" (%*ple.

El e(ta$" n=0 c"rre(p"$e al e(ta$" u$a*etal' cua eer/ía e(

E0=

1

2ωℏ ' el e(ta$" n=1 c"rre(p"$e al pr%*er e(ta$" ec%ta$"' cua

eer/ía e( E1=

3

2ωℏ ' a(í (uce(%-a*ete.

El $%a/ra*a $el %-el $e eer/ía para e(te (%(te*a (e %lu(tra a c"t%uac%,.O)(er-e +ue la( (eparac%"e( etre %-ele( a$acete( (" %/uale( e(t0$a$a( p"r

Δ E0= ωℏ

A$-%erta +ue l"( %-ele( $e eer/ía para el "(c%la$"r ar*,%c" $el $%a/ra*ae(t0 %/ual*ete e(pac%a$"(' tal c"*" l" pr"pu(" Plac para l"("(c%la$"re( e la( pare$e( $e la ca-%$a$ +ue (e ut%l%#, e el *"$el" $e



ra$%ac%, $e cuerp" e/r". La ecuac%, $e Plac para l"( %-ele( $eeer/ía $e l"( "(c%la$"re( (e $%(t%/ue $e la ecuac%, $e eer/ía(

cuat%#a$a( para u "(c%la$"r ar*,%c" (%*ple' ("l" e el ter*%"1

2

a$%c%"a$" a n . E(te ter*%" a$%c%"al " aecta la eer/ía +ue (e e*%te

e ua tra(%c%,' $a$a p"r la ecuac%, Δ E0= ωℏ . El 1ec1" $e +ue Plac

/eerara e(t"( c"cept"( (% la ecuac%, $e Sc1r$%/er e( te(t%*"%" $e(u /e%".

EFECTO TUNEL A TRA@ES DE UNA BARRERA DE ENERGIA POTENCIAL.

C"(%$ere la uc%, $e eer/ía p"tec%al +ue (e %lu(tra e la &/ura. E e(ta(%tuac%,' la eer/ía p"tec%al t%ee u -al"r c"(tate U e la re/%, $e

ac1" L e( cer" e t"$a( la( "tra( re/%"e(. Ua uc%, $e eer/ía

p"tec%al $e e(ta "r*a (e lla*a )arrera cua$ra$a' U (e $e"*%a

altura $e la )arrera. U e,*e" *u %tere(ate pecul%ar (e pre(etacua$" la partícula e *"-%*%et" ecuetra ua )arrera c"*" e(ta $e

altura ac1" &%t"(. Sup"/a +ue ua partícula $e eer/ía E>U %c%$e

e la )arrera $e($e la %#+u%er$a. Cl0(%ca*ete' la partícula (er%a ree!a$a

p"r la )arrera. S% la partícula $e)ería e%(t%r e la re/%, II' (u eer/íac%6t%ca (er%a e/at%-a' l" cual " e( per*%t%$" $e($e el put" $e -%(tacl0(%c". E c"(ecuec%a' la re/%, II' p"r tat" la re/%, III' ("cl0(%ca*ete pr"1%)%$"( para la partícula +ue %c%$e $e($e la %#+u%er$a. S%e*)ar/"' (e/9 la *ec0%ca cu0t%ca' t"$a( la( re/%"e( (" acce(%)le( a lapartícula' cual+u%era +ue (ea (u eer/ía. Au cua$" t"$a( la( re/%"e( ("acce(%)le(' la pr")a)%l%$a$ $e +ue la partícula (e ecuetre e ua re/%,cl0(%ca*ete pr"1%)%$a e( *u )a!a.? De acuer$" c" el pr%c%p%" $e%cert%$u*)re' la partícula pue$e e(t0 $etr" $e la )arrera *%etra( el%ter-al" $urate el cual e(t0 e ella (ea c"rt" c"(%(tete c" la ecuac%,$e Sc1r$%/er %$epe$%ete $el t%e*p". S% la )arrera e( relat%-a*ete

a/"(ta' e(te c"rt" %ter-al" pue$e per*%t%r +ue la partícula (e *ue-a atra-6( $e la )arrera.

Ilustración 1 Diagrama del nivel de energía para un oscilador armónico simple.



Platea$" e(ta (%tuac%, c" ua repre(etac%, *ate*0t%ca. La ecuac%,

$e Sc1r$%/er t%ee ("luc%"e( -al%$a( e la( re/%"e( I' II' III. La(("luc%"e( e la( re/%"e( I III (" (%u("%$ale(' c"*" e la ecuac%,Fuc%"e( $e "$a para ua partícula e ua ca!a?. E la re/%, II' la("luc%, e( ep"ec%al (e*e!ate a la ecuac%, ("luc%, /eeral

ψ = A ecx+B e

−cx

?. S% (e apl%ca la c"$%c%, $e r"tera $e +ue la( uc%"e(

$e "$a $e la( tre( re/%"e( (u( $er%-a$a( $e)e u%r(e 0c%l*ete e la(r"tera(' e( p"(%)le 1allar ua ("luc%, c"*pleta' c" la +ue repre(eta lacur-a $e la &/ura +ue (e pre(et,. P"r+ue la pr")a)%l%$a$ $e l"cal%#ar la

partícula pr"p"rc%"al a |ψ 2| ' la pr")a)%l%$a$ $e ec"trar la partícula

*0( all0 $e la )arrera e la re/%, III e( $%erete $e cer". E(te re(ulta$"e(t0 e c"*plet" $e(acuer$" c" la í(%ca cl0(%ca. El *"-%*%et" $e lapartícula e el la$" le!a" $e la )arrera (e $e"*%a eect" t9el "peetrac%, $e )arrera.

La pr")a)%l%$a$ $el eect" t9el (e pue$e $e(cr%)%r c" u c"e&c%ete $e

tra(*%(%, T u c"e&c%ete $e ree%, . El c"e&c%ete $e

tra(*%(%, repre(eta la pr")a)%l%$a$ $e +ue la partícula peetre al "tr" la$"$e la )arrera' el c"e&c%ete $e ree%, e( la pr")a)%l%$a$ $e +ue lapartícula (ea ree!a$a p"r la )arrera. Pue(t" +ue la partícula %c%$ete' e(

ree!a$a " tra(*%t%$a' (e re+u%ere +ue T + =1 . Ua epre(%,

apr"%*a$a para el c"e&c%ete $e tra(*%(%, +ue (e ")t%ee cua$"

Ilustración 2 Efecto túnel



T ≪1 ua )arrera *u ac1a " ua )arrera *u alta' e( $ec%r' U ≫ E ¿

e(

T ! e−2"L

D"$e " =√ 2m(U − E)ℏ

E(te *"$el" cu0t%c" $e peetrac%, $e )arrera e(pecí&ca*ete la

ecuac%, T ! e−2"L

*ue(tra +ue T pue$e (er $%erete $e cer". El

1ec1" $e +ue el e,*e" $e eect" t9el pue$a ")(er-ar(eeper%*etal*ete pr"p"rc%"a *0( c"&a#a e l"( pr%c%p%"( $e la í(%cacu0t%ca.

Ilustración 3 Parte de la energía puedeatravesar la barrera.

Ilustración 4 Imagen tomada con un microscopio deefecto túnel. uestra la super!cie del cobre" dondelos #tomos de cobre est#n contenidos dentro de unrecinto cu#ntico de 4$ #tomos de %ierro.

parte 2 mecanica cuantica

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