Chng 8: C hc lng t

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CHNG 8

C HC LNG T

Cui th k 19 u th k 20 vt l hc thu c mt lot nhng thnh tu mi: s khm ph ra tia X, s ph thuc khi lng ca electrn vo vn tc chuyn ng, bc x nhit ca vt en tuyt i, hiu ng quang in, hiu ng Compton Nhng hin tng ny khng th gii thch c nu da vo nhng quan im ca Vt l c in, iu chng t c s ca Vt l c xy dng trc bt u lung lay v ngnh Vt l ang ng trc nhng thch thc mi. Ngi ta nhn thy khi i vo th gii ca nguyn t, phn t (kch thc 10-9 - 10-10 m, c gi l th gii vi m) cc quy lut ca Vt l c in khng cn ng na. y chnh l tin cho mt mn khoa hc mi ra i l mn C hc lng t.

C hc lng t l mn khoa hc nghin cu nhng tnh cht ca vt cht trong th gii vi m. C hc lng t gii quyt nhiu vn c lin quan n cc tnh cht vt l ca vt cht mc su sc hn, do cng c bn hn so vi vt l c in. Do ta c th ni c hc c in l trng hp gii hn ca c hc lng t khi ta chuyn t vic nghin cu vi m sang nghin cu v m. C hc lng t cung cp cho ta kin thc hiu cc hin tng xy ra trong nguyn t, ht nhn, vt rn...

8. 1. LNG TNH SNG HT CA VI HT

8. 1. 1. Lng tnh sng ht ca nh sng Nh chng trc chng ta thy nh sng va c tnh sng va c tnh ht: hin tng

giao thoa, nhiu x th hin tnh cht sng, cn hiu ng quang in, hiu ng Compton th hin tnh cht ht ca nh sng. Lng tnh sng ht ca nh sng c Einstein nu trong thuyt phtn: nh sng c cu to bi cc ht phtn, mi ht mang nng

lng v ng lng = hE =hp .

Ta thy cc i lng c trng cho tnh cht ht (E,p) v cc i lng

Hnh 8-1. S truyn sng phng nh sng

c trng cho tnh cht sng ( ) lin h trc tip vi nhau. Chng ta s thit lp hm sng cho ht phtn.Xt chm nh sng n sc, song song. Mt sng l cc mt phng vung gc vi phng truyn sng. Nu dao ng sng ti O l

,

t2cosA)t(x = (8-1)

Chng 8: C hc lng t

174

th biu thc dao ng sng ti mi im trn mt sng i qua im M cch mt sng i qua O mt on d l:

)d2-tcos(A

)d-t(2cosA)cd-t(2cosA)

cd-t(x

=

== (8-2)

trong c l vn tc nh sng trong chn khng, l bc sng nh sng trong chn khng: ==

ccT , vi T l chu k , l tn s ca sng nh sng. T hnh 8-1 ta c: n.rcosrd == (8-3)

n : vect php tuyn n v. Thay (8-3) vo (8-2) ta nhn c:

)n.rt(2cosA)cdt(x = (8-4)

l hm sng phng n sc. S dng k hiu cho hm sng v biu din n di dng hm phc ta c

=n.rti2expo (8-5)

Nu thay hE= , =

hp v = 2hh vo (7-5) ta c:

( ) = rpEtiexpo h (8-6) 8. 1. 2. Gi thuyt de Broglie (bri)

Trn c s lng tnh sng ht ca nh sng, de Broglie suy ra lng tnh sng ht cho electrn v cc vi ht khc.

Gi thuyt de Broglie:

Mt vi ht t do c nng lng, ng lng xc nh tng ng vi mt sng phng n sc. Nng lng ca vi ht lin h vi tn s dao ng ca sng tng ng thng qua h thc: hay . ng lng ca vi ht lin h vi bc sng ca sng tng ng

theo h thc:

= hE = hE

=hp hay kp h= .

k l vect sng, c phng, chiu l phng, chiu truyn sng, c ln = 2k . Sng de

Broglie l sng vt cht, sng ca cc vi ht.

8. 1. 3. Thc nghim xc nhn tnh cht sng ca cc ht vi m 1. Nhiu x ca electrn qua khe hp:

Cho chm electrn i qua mt khe hp. Trn mn hunh quang ta thu c hnh nh nhiu x ging nh hin tng nhiu x ca nh sng qua mt khe hp. Nu ta cho tng electrn ring bit i qua khe trong mt thi gian di s electrn i qua khe ln, ta vn

Chng 8: C hc lng t

175

thu c hnh nh nhiu x trn mn hunh quang. iu ny chng t mi ht electrn ring l u c tnh cht sng.

Hnh 8-2. Nhiu x ca electrn qua mt khe hp

2. Nhiu x ca electrn trn tinh th

Th nghim ca Davisson v Germer quan st c hin tng nhiu x ca electrn trn mt tinh th Ni (hnh 7-3). Khi cho mt chm electrn bn vo mt tinh th Ni, chm e- s tn x trn mt tinh th Ni di cc gc khc nhau. Trn mn hnh ta thu c cc vn nhiu x. Hin tng xy ra ging ht hin tng nhiu x ca tia X trn mt tinh th Ni. Tinh th Ni nh mt cch t nhiu x. Hin tng electrn nhiu x trn cch t chng t bn cht sng ca chng. Thay Ni bng cc tinh th khc, tt c cc th nghim u xc nhn chm electrn gy hin tng nhiu x trn tinh th. Cc vi ht khc nh ntrn, prtn cng gy hin tng nhiu x trn tinh th.

Cc kt qu th nghim trn u xc nhn tnh cht sng ca vi ht v do chng minh s ng n ca gi thuyt de Broglie.

Cui cng, ta phi nhn mnh v ni dung gii hn ca gi thit de Broglie. Bc sng de Broglie t l nghch vi khi lng ca ht:

mvh

ph ==

do i vi nhng ht thng thng m khi lng rt ln, thm ch l v cng ln so vi khi lng ca electrn chng hn th bc sng de Broglie tng ng c gi tr v cng b v khng cn ngha m t tnh cht sng na.

Hnh 8-3. Nhiu x ca electrn trn tinh th

Nh vy, khi nim lng tnh sng ht thc s ch th hin cc ht vi m m thi v sng de Broglie c bn cht c th lng t, n khng tng t vi sng thc trong vt l c in nh sng nc hay sng in t...

Chng 8: C hc lng t

176

8. 2. H THC BT NH HEISENBERG

Do c lng tnh sng ht nn qui lut vn ng ca vi ht trong th gii vi m khc vi qui lut vn ng ca ht trong th gii v m. Mt trong nhng im khc bit l h thc bt nh Heisenberg. tm h thc chng ta xt hin tng nhiu x ca chm vi ht qua mt khe hp c b rng b.

Sau khi qua khe ht s b nhiu x theo nhiu phng khc nhau, tu theo gc nhiu x , mt ht nhiu x trn mn s cc i hoc cc tiu. Xt ta ca ht theo phng x, nm trong mt phng khe v song song vi b rng khe. Ta x ca ht trong khe s c gi tr trong khong t 0 n b ( ). Ni cch khc, v tr ca ht trong khe c

bx0 Hnh 8-4. Nhiu x electrn qua khe hp

xc nh vi bt nh bx . Sau khi ht qua khe, ht b nhiu x, phng ng lng p thay i. Hnh chiu ca p theo phng x s c gi tr thay i trong khong

, ngha l sau khi i qua khe, ht c th ri vo cc i gia hoc cc i ph v c xc nh vi mt bt nh no . Xt trng hp ht ri vo cc i gia

, l gc ng vi cc tiu th nht:

sinpp0 xxp

1x sinpp 1 bsin 1= . Do ta c:

= .psinp.bp.x 1x Theo gi thuyt de Broglie =

hp . Thay vo biu thc trn ta nhn c h thc bt nh

Heisenberg: hp.x x

L lun tng t: (8-7) hp.y y hp.z z

H thc bt nh Heisenberg l mt trong nhng nh lut c bn ca c hc lng t. H thc ny chng t v tr v ng lng ca ht khng c xc nh chnh xc mt cch ng thi. V tr ca ht cng xc nh th ng lng ca ht cng bt nh v ngc li.

V d: Trong nguyn t e- chuyn ng trong phm vi 10-10 m. Do bt nh v vn tc l:

s/m10.710.10.9

10.625,6xm

hmp

v 61031

34

eex

x ===

Ta thy kh ln cho nn exv - khng c vn tc xc nh, ngha l e- khng chuyn ng theo mt qu o xc nh trong nguyn t. iu ny chng t rng trong th gii vi m khi nim qu o khng c ngha.

Chng 8: C hc lng t

177

Ta xt ht trong th gii v m khi lng ca ht m = 10-15 kg, bt nh v v tr

. Do bt nh v vn tc l m10x 8=s/m10.6,6

10.10

10.625,6x.m

hv 11815

34x

==

Nh vy i vi ht v m v x xv u nh, ngha l v tr v vn tc c th c xc nh chnh xc ng thi.

Theo c hc c in, nu bit c to v ng lng ca ht thi im ban u th ta c th xc nh c trng thi ca ht cc thi im sau. Nhng theo c hc lng t th to v ng lng ca vi ht khng th xc nh c ng thi, do ta ch c th on nhn kh nng vi ht mt trng thi nht nh. Ni cch khc vi ht ch c th mt trng thi vi mt xc sut no . Do qui lut vn ng ca vi ht tun theo qui lut thng k.

Ngoi h thc bt nh v v tr v ng lng, trong c hc lng t ngi ta cn tm c h thc bt nh gia nng lng v thi gian:

ht.E (8-8) ngha ca h thc bt nh gia nng lng v thi gian: nu nng lng ca h

mt trng thi no cng bt nh th thi gian h tn ti trng thi cng ngn v ngc li, nu nng lng ca h mt trng thi no cng xc nh th thi gian tn ti ca h trng thi cng di. Nh vy trng thi c nng lng bt nh l trng thi khng bn, cn trng thi c nng lng xc nh v thp nht l trng thi bn.

8. 3. HM SNG

8. 3. 1. Biu thc ca hm sng Do lng tnh sng ht ca vi ht ta khng th xc nh ng thi c ta v ng

lng ca vi ht. xc nh trng thi ca vi ht, ta phi dng mt khi nim mi l hm sng.

Theo gi thuyt de Broglie chuyn ng ca ht t do (tc l ht khng chu mt tc dng no ca ngoi lc) c m t bi hm sng tng t nh sng nh sng phng n sc

( ) ( )[ ]rktiexprpEtiexp oo = = h (8-9) Trong kp;E hh == v l bin c xc nh bi: o

*22o == (8-10) * l lin hp phc ca .

Nu ht vi m chuyn ng trong trng th, th hm sng ca n l mt hm phc

tp ca to r v thi gian t

)t,z,y,x()t,r( =

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178

8. 3. 2. ngha thng k ca hm sng Xt chm ht phtn truyn trong

khng gian. Xung quanh im M ly th tch bt k (hnh 8-5) V*Theo quan im sng: Cng sng ti M t l vi bnh phng bin dao ng sng ti M:

I ~ 2o

Hnh 8-5. Chm ht phtn truyn qua V

*Theo quan im ht: Cng sng ti M t l vi nng lng cc ht trong n v th tch bao quanh M, ngha l t l vi s ht trong n v th tch .T y ta thy rng s ht trong

n v th tch t l vi . S ht trong n v th tch cng nhiu th kh nng tm thy ht

trong cng ln. V vy c th ni bnh phng bin sng

2o

2 ti M c trng cho kh nng tm thy ht trong n v th tch bao quanh M . Do 2 l mt xc sut tm ht v xc sut tm thy ht trong ton khng gian l dV2

V . Khi tm ht trong ton khng gian,

chng ta chc chn tm thy ht. Do xc sut tm ht trong ton khng gian l 1:

1dV2V

= (8-11)

y chnh l iu kin chun ho ca hm sng. Tm li:

- m t trng thi ca vi ht ngi ta dng hm sng . - 2 biu din mt xc sut tm thy ht trng thi . - khng m t mt sng thc trong khng gian. Hm sng mang tnh cht thng k,

n lin quan n xc sut tm ht.

8. 3. 3. iu kin ca hm sng - Hm sng phi hu hn. iu ny c suy ra t iu kin chun ho, hm sng

phi hu hn th tch phn mi hu hn. - Hm sng phi n tr, v theo l thuyt xc sut: mi trng thi ch c mt gi tr xc

sut tm ht.

- Hm sng phi lin tc, v xc sut 2 khng th thay i nhy vt. - o hm bc nht ca hm sng phi lin tc.

Chng 8: C hc lng t

179

8. 4. PHNG TRNH SCHRODINGER

Hm sng de Broglie m t chuyn ng ca vi ht t do c nng lng v ng lng xc nh:

( ) = = Etiexp)r(rpEtiexp)t,r( o hh (8-12) trong

= rpiexp)r( o h (8-13)

l phn ph thuc vo ta ca hm sng. Ta c th biu din )r( trong h ta cc nh sau:

++= )zpypxp(iexp)r( zyxo h (8-14) Ly o hm , ta c: x/

)r(pix x

=

h

Ly o hm bc hai ca theo x:

)r(p)r(pix 2

2x2

x2

2

2

2==

hh

(8-15)

Ta cng thu c kt qu tng t cho cc bin y v z. Theo nh ngha ca ton t Laplace trong h to cc :

)r(zyx

)r( 22

2

2

2

2

+

+= (8-16)

ta c:

)r(p)r(ppp

)r( 22

2

2z

2y

2x =++=

hh (8-17)

Gi E l ng nng ca ht, ta vit c:

E m2p

2mv 22 == hay p2 =2mE

Thay p2 vo (8-17) v chuyn sang v tri ta thu c:

0)r(Em2)r( d2 =+ h (8-18) Phng trnh (8-18) c gi l phng trnh Schrodinger cho vi ht chuyn ng t do. M rng phng trnh cho vi ht khng t do, ngha l vi ht chuyn ng trong mt trng lc c th nng U khng ph thuc thi gian. Nng lng ca vi ht E = E + U. Thay E = E - U vo (8-18) ta c:

[ ] 0)r()r(UEm2)r( 2 =+ h (8-19)

Chng 8: C hc lng t

180

Bit dng c th ca U( r ), gii phng trnh Schrodinger ta tm c )r( v E, ngha l xc nh c trng thi v nng lng ca vi ht. Ta gii hn ch xt h l kn hay t trong trng ngoi khng bin thin theo thi gian. Nng lng ca h khi khng i v trng thi ca h c gi l trng thi dng. Phng trnh (8-19) c gi l phng trnh Schrodinger cho trng thi dng.

Cho n nay ta vn xt ht chuyn ng vi vn tc v

Chng 8: C hc lng t

181

kxcosBkxsinA)x( += (8-22) A, B l nhng hng s c xc nh t iu kin ca hm sng. Theo u bi th ht ch trong ging th, do xc sut tm ht ti vng ngoi ging th bng khng v hm sng trong cc vng cng bng 0. T iu kin lin tc ca hm sng ta suy ra:

Thay iu kin ny vo (8-22) ta c ,0)0( = 0)a( =0B)0sin(A)0( =+= B = 0

v 0)kasin(A)a( == B = 0 nn A phi khc 0 (v nu A = 0 th lun bng 0 v l mt nghim tm thng). Do ta c:

== nsin0kasin vi n = 1,2,... T rt ra:

ank = (8-23)

Nh vy ta c mt dy nghim hm sng c dng:

xa

nsinA)x(n= (8-24)

tha mn iu kin bin ca min. Hng s A c xc nh t iu kin chun ha (8-11) ca hm sng. V ht khng th ra khi ging nn xc sut tm thy ht trong ging l chc chn:

1dx)x(a

0

2 = Tnh gi tr tch phn:

12

aAdx)xan2cos1(

2Axdx

ansinA

2a

0

22

a

0

2 === Ta tm c:

a2A =

Nh vy hm sng c xc nh hon ton:

xa

nsina2)x(n

= (8-25) Nng lng ca ht trong ging th cng c tm thy khi ta thay biu thc (8-23) vo

22 mE2k

h= v nhn c

22

22n n

ma2E h= (8-26)

T cc kt qu trn ta rt ra mt s kt lun sau:

a. Mi trng thi ca ht ng vi mt hm sng )x(n

Chng 8: C hc lng t

182

b. Nng lng ca ht trong ging ph thuc vo s nguyn n, ngha l bin thin gin on. Ta ni rng nng lng b lng t ha.

Vi n = 1 ta c mc nng lng cc tiu 0ma2

E 222

1 = h ng vi hm sng xasina2

1= ,

m t trng thi chuyn ng c bn ca ht. Hm sng )x(1 khc khng ti mi im trong ging, ch c th bng 0 ti cc v tr bin (Hnh 8-7).

Khong cch gia hai mc nng lng k tip nhau ng vi cc s nguyn n v n+1 bng:

)1n2(ma2

EEE2

22n1nn +== + h (8-27)

nE cng ln khi a v m cng nh. iu c ngha l trong phm vi th gii vi m, s lng t ha cng th hin r rt. C th, nu xt ht electrn m = 9,1.10-31kg, a ~ 5.10-10m th E ~ 1eV, khong cch gia En+1 v En tng i ln, nng lng b lng t ha. Nhng nu xt mt ht c m ~10-26kg chuyn ng trong min a ~ 10cm th khong cch gia cc mc nng lng E~ 10-20eV kh nh. Trong trng hp ny c th coi nng lng ca ht bin thin lin tc.

c. Mt xc sut tm ht trong ging:

xa

nsina2)x( 22n

= (8-28)

Hnh 8-7. Ht trong ging th nng mt chiu, cao v hn

Mt xc sut cc i khi: 1xa

nsin =

. Do xc sut tm thy ht ln nht ti:

n2a)1m2(x += < a m = 0,1....

Chng 8: C hc lng t

183

V d: Khi n = 1, xc sut tm thy ht im 2ax = l ln nht. Khi n = 2 xc sut tm thy

ht im 4ax = v

4a3x = l ln nht...

Mt xc sut cc tiu khi: 0xa

nsin =

. Do xc sut tm thy ht nh nht ti

nmax = < a

Kt qu c biu din trn hnh 8-7.

8. 5. 2. Hiu ng ng ngm

Ta xt ht mang nng lng E, chuyn ng theo phng x t tri sang phi p vo hng ro th nng nh hnh 8-8. Theo quan im ca c hc c in, nu E < Uo ht khng th vt qua hng ro. Theo quan im ca c hc lng t ta s thy ht vn c kh nng xuyn qua hng ro th nng. Hin tng xuyn qua hng ro th nng nh vy c gi l hiu ng ng ngm.

Hnh 8-8. Hng ro th hnh ch nht

Chng ta s nghin cu trng hp hng ro th nng dng n gin nh hnh 8-8:

Chng 8: C hc lng t

184

)ax(ik3

)ax(ik33 11 eBeA)x(

+= (8-33) S hng th nht ca phng trnh (8-33) biu din sng xuyn qua hng ro v truyn t tri sang phi. S hng th hai biu din sng phn x t v cc v, nhng sng ny khng c, nn ta c th cho B3 = 0.

H s truyn qua hng ro D c nh ngha l t s gia s ht xuyn qua c hng ro v s ht i ti hng ro. V s ht li t l vi bnh phng ca bin sng. Bin sng ti hng ro l A1 v bin sng xuyn qua hng ro l A3, do ta c

21

23

A

AD = (8-34)

H s phn x R c nh ngha l t s gia s ht phn x v s ht i ti hng ro, do ta c:

21

21

A

BR = (8-35)

trong B1 l bin sng phn x trn mt hng ro. Do iu kin bo ton s ht, ta phi

c 212

12

3 ABA =+ , do : D + R = 1 (8-36)

tnh c h s D v R ta phi tnh c cc bin sng. Mun vy ta da vo iu kin lin tc ca hm sng v o hm ca n ti cc v tr bin (x = 0 v x = a). T cc iu kin bin:

)a()a()a()a()0()0()0()0(

32

32

21

21

====

(8-37)

ta rt ra cc h thc sau

2211 BABA +=+ (8-38) )BA(k)BA(ik 222111 = (8-39)

3ak

2ak

2 AeBeA 22 =+ (8-40) 31

ak2

ak22 Aik)eBeA(k 22 = (8-41)

T (8-40) v (8-41) ta c th biu th A2, B2 qua A3: ak

32 2eA2in1A = (8-42)

ak32 2eA2

in1B += (8-43) Trong :

EUE

kkn

02

1==

Chng 8: C hc lng t

185

V in1 = in1+ , nn ta suy ra 22 BA >> . Do , c th t B2=0. T (8-38) v (8-39) ta rt ra c A1 theo A2, sau s dng (8-42) ta tnh c:

ak31 2eAn2

ni2in1A

+

= (8-44) T y ta thu c h s truyn qua:

ak222

2

21

23 2e

)n1(n16

A

AD +== (8-45)

Nu ( )222

n1

n16

+ vo c 1 (U0 vo c 10E) th c th vit:

ak2 2eD hay ( )

EUmaD 022exp h (8-46)

T (8-46) ta nhn thy rng, ngay khi nng lng E ca ht nh hn th nng ca ro (E

Chng 8: C hc lng t

186

Hin tng phn r cng c gii thch tng t. Ht nhn nguyn t gm c cc ht prtn (p) v ntrn (n). Trong ht nhn cc ht p v n tng tc vi nhau bng lc ht nhn, cho nn c th xem nh chng nm trong ging th nng. Ht gm hai ht p v hai ht n, mc d nng lng ca ht nh hn cao ro th nhng do hiu ng ng ngm, ht p v n ca

Hnh 8-9. Hin tng phn r

ht vn c th bay ra khi ht nhn, hin tng ny gi l hin tng phn r (hnh 8-9).

8. 5. 3. Dao ng t iu ha lng t

Mt vi ht thc hin dao ng nh iu ha xung quanh v tr cn bng l mt v d v dao ng t iu ha lng t. Dao ng ca nguyn t trong phn t, dao ng ca cc in xung quanh nt mng tinh th... u l nhng v d v dao ng t iu ha. Dao ng t iu ha l mt hin tng rt quan trng ca vt l ni chung v c hc lng t ni ring.

Ta xt vi ht dao ng (mt chiu) trong trng th nng. Trong phn dao ng ta bit th nng ca dao ng iu ha mt chiu bng:

2xmkx

21U

222 == (8-47)

trong m l khi lng ca vi ht, l tn s gc ca dao ng. Phng trnh Schrodinger cho dao ng t iu ha c dng:

02

xmEm2dxd 22

22

2=

+

h (8-48)

C hc lng t gii phng trnh (8-48) v tm c biu thc nng lng ca dao ng t iu ha

+=21nEn h vi n = 0,1,2... (8-49)

Ta thy nng lng ca dao ng t ch ly nhng gi tr gin on, c ngha rng nng lng ca dao ng t b lng t ha. Nng lng thp nht ca dao ng t iu ha ng vi

n=0: 2

Eo= h

Nng lng ny c gi l nng lng khng. Nng lng khng lin quan n dao ng khng ca dao ng t, ngha l khi T = 0K, dao ng t vn dao ng. iu ny c thc nghim xc nhn trong th nghim tn x tia X. Tia X b tn x l do cc dao ng nguyn t trong mng tinh th gy ra. Theo c hc c in, khi nhit cng gim, bin dao ng ca cc nguyn t gim n khng, do s tn x ca nh sng phi bin mt. Nhng thc nghim chng t, khi nhit gim, cng tn x tin ti mt gi tr gii hn no . iu c ngha rng, ngay c khi T 0, s tn x nh sng vn xy ra v cc

Chng 8: C hc lng t

187

nguyn t trong mng tinh th vn dao ng, tng ng vi mt nng lng Eo no . Nh vy thc nghim xc nhn s ng n ca c hc lng t.

S tn ti ca nng lng khng cng ph hp vi h thc bt nh Heisenberg. Thc vy, nu mc nng lng thp nht ca dao ng t bng 0, nh th c ngha l ht ng yn v vn tc v ta ca vi ht c xc nh ng thi (u bng 0), iu ny mu thun vi h thc bt nh. S tn ti ca mc nng lng khng ca dao ng t iu ha l mt trong nhng biu hin c trng nht ca lng tnh sng-ht ca vi ht.

HNG DN HC CHNG 8

C HC LNG T

I. MC CH - YU CU

1. Nm c gi thuyt de Broglie v lng tnh sng - ht ca vi ht. T i n biu thc ca hm sng v phng trnh Schrodinger. 2. Hiu v vn dng c h thc bt nh Heisenberg. 3. Hiu v vn dng phng trnh Schrodinger gii mt s bi ton c hc lng t n gin nh ht trong ging th, hiu ng ng ngm, dao ng t iu ha lng t.

II. TM TT NI DUNG

1. Lng tnh sng ht ca vi ht Trn c s lng tnh sng ht ca nh sng, de Broglie m rng ra cho cc vi ht.

Theo gi thuyt ny, mi vi ht t do c nng lng xc nh, ng lng xc nh tng ng vi sng phng n sc. Lng tnh sng ht ca cc vi ht c biu din bng cc h thc: E = h v p = mv = h /. Ngoi ra, theo thuyt tng i Einstein, mi ht vt cht c khi lng m u mang nng lng bng E = mc2

trong 22

o

c/v1

mm

=

mo l khi lng ngh ca ht (khi v = 0).

2. Hm sng Hm sng ca vi ht t do c dng ca hm sng phng:

Chng 8: C hc lng t

188

( ) ( )[ ]rktiexprpEtiexp oo = = h trong = h/2 gi l hng s Planck rt gn v = /2k c gi l s sng. Hm sng khng nhng m t nhng tnh cht ca h ti mt thi im no , m n cn xc nh c ng thi ca h nhng thi im tip theo. Hm sng c ngha thng k.

2 l mt xc sut tm thy ht ti mt im no i vi mt trng thi lng t ang xt. Nh vy, hm sng khng m t mt sng thc, m m t sng xc sut. Do hm sng phi tha mn ba iu kin: hm sng phi lin tc, hu hn v n tr. iu kin chun

ha ca hm sng l 1dV2V

=

3. Nguyn l bt nh Heisenberg Nguyn l ny thu c t lng tnh sng ht ca vi ht, c biu din qua h thc

di y khi xt v tr x v ng lng p ca vi ht hp.x x

Nu x cng nh (v tr cng xc nh) th px cng ln (ng lng cng bt nh) v ngc li. Nh vy i vi vi ht, v tr v ng lng khng c xc nh chnh xc ng thi. Do , trong th gii vi m khi nim qu o khng c ngha. Nu ta bit c v tr x thi im t, th n thi im t + dt ta ch c th xc nh v tr ht vi mt xc sut no thi. i vi cc vi ht khi nim qu o c thay th bng khi nim xc sut tm thy ht ti mt v tr no trng thi lng t ang xt. Ngoi h thc gia v tr v ng lng, vi ht cn tun theo h thc bt nh cho nng lng

ht.E ngha ca h thc bt nh gia nng lng v thi gian: nu nng lng ca h mt trng thi no cng bt nh th thi gian h tn ti trng thi cng ngn v ngc li, nu nng lng ca h mt trng thi no cng xc nh th thi gian tn ti ca h trng thi cng di.

4. Phng trnh Schrodinger v ng dng

T biu thc ca hm sng, Schrodiger a ra phng trnh c bn ca c hc lng t mang tn ng cho vi ht.

i vi vi ht t do: 0)r(Em2)r( d2 =+ h

i vi vi ht trong trng th [ ] 0)r()r(UEm2)r( 2 =+ h Cn ch rng cc phng trnh Schrodinger thu c trn c s ca gi thuyt de Broglie, thuyt lng t ca Planck v thuyt phtn ca Einstein, do cng c coi l cc tin . H thc bt nh Heisenberg v phng trnh Schrodinger l nhng nguyn l c bn ca c hc lng t.

ng dng ca phng trnh Schrodinger:

Chng 8: C hc lng t

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- Phng trnh Schrodinger c p dng gii mt s bi ton n gin ca c hc lng t nh tm nng lng v hm sng ca vi ht khi lng m trong ging th nng, c b rng a v thnh cao v hn. Kt qu ta c nng lng ca vi ht trong ging th b lng t ha:

2222

n nma2

E h= Mi gi tr ca nng lng En tng ng vi mt trng thi lng t

xa

nsina2)x(n

= T y ta tm c xc sut tm thy ht ti cc im khc nhau trong ging ng vi mi trng thi lng t. - Vn dng phng trnh Schrodinger, ta xt chuyn ng ca vi ht qua hng ro th Uo. T pht hin hiu ng ng ngm. l hiu ng mt vi ht c nng lng E < Uo vn c xc sut vt qua c ro th Uo. y l hiu ng thun ty lng t, v trong c hc c in mt ht c nng lng E < Uo th khng th vt qua c hng ro th nng. - Mt ng dng na hay gp ca c hc lng t l dao ng t iu ha. l mt vi ht thc hin cc dao ng nh bc nht quanh v tr cn bng. Chuyn ng nhit ca mng tinh th cng c biu din di dng tp hp ca cc dao ng t iu ha tuyn tnh. Thay biu thc th nng U ca dao ng t iu ha vo phng trnh Schrodinger, ta tm c cc mc nng lng ca dao ng t:

+=21nEn h

Nu n = 0, ta tm c mc nng lng thp nht ca dao ng t 2

Eo= h . Eo c gi l

nng lng khng. Kt qu ny c thc nghim xc nhn. N ni ln rng cc nguyn t ca mng tinh th khng bao gi ng yn. Suy rng ra, s vn ng ca vt cht khng bao gi b tiu dit. l c s khoa hc ca trit hc duy vt bin chng.

III. CU HI L THUYT

1. Pht biu gi thuyt de Broglie v lng tnh sng ht ca vi ht. 2. Vit biu thc hm sng cho vi ht v nu ngha ca cc i lng c trong biu thc . 3. Vit phng trnh Schrodinger cho vi ht t do v vi ht chuyn ng trong trng lc th. Nu ngha cc i lng c trong phng trnh. 4. Hy nu bn cht v ngha thng k ca hm sng. Cc iu kin ca hm sng. 5. Pht biu v nu ngha ca h thc bt nh Heisenberg cho v tr v ng lng. 6. Pht biu v nu ngha ca h thc bt nh cho nng lng. 7. Phn tch ti sao trong c hc lng t khi nim qu o ca vi ht khng cn c ngha. Khi nim qu o ca vi ht c thay th bng khi nim g ? 8. Hy tm biu thc ca hm sng v nng lng ca vi ht trong ging th nng mt chiu, c chiu cao v cng.

Chng 8: C hc lng t

190

9. nh ngha dao ng t iu ha lng t. Vit phng trnh Schrodinger v biu thc nng lng ca dao ng t iu ha. T rt ra biu thc ca nng lng khng, nu ngha ca biu thc ny.

IV. BI TP

Th d 1: Electrn chuyn ng tng i tnh vi vn tc 2.108m/s. Tm: 1. Bc sng de Broglie ca electrn. 2. ng lng ca electrn. Bi gii 1. p dng c hc tng i tnh:

m10.72,2vmcv-1h

cv-1

mm;

vmh 12

0e

2

2

2

2e0 ====

2. ng lng ca electrn: s/m.kg10.44,2hp 22==

Th d 2: ng nng ca electrn trong nguyn t hir c gi tr vo c 10eV. Dng h thc bt nh hy nh gi kch thc nh nht ca nguyn t. Bi gii: Nm 1927 bng l thuyt C hc lng t Heisenberg tm ra h thc bt nh gia v tr v ng lng

2.hpx x = h (8.50)

Biu thc ny c khc cht t so vi biu thc (8.7) tm ra t hin tng nhiu x electrn, nhng ngha vt l ca h thc th khng thay i. Khi gii cc bi tp vt l phn c hc lng t chng ta s s dng biu thc (8.50). Gi s kch thc ca nguyn t bng , vy v tr ca electrn theo phng x xc nh

bi:

l

2x0 l , ngha l

2x l

T h thc bt nh: x

x php

hll 22

Mt khc m ppx eEm2p = , trong E l ng nng. Vy gi tr nh nht ca kch thc nguyn t: m

Em e10

min 10.24,122 == hl

Th d 3: Dng ht c nng lng E xc nh chuyn ng theo phng x t tri sang phi n gp mt hng ro th nng xc nh bi:

=

0)(00

00 xEUUx

U

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191

Xc nh h s phn x v h s truyn qua hng ro th i vi electrn . Bi gii: Gii phng trnh Schrodinger hai min I v II. Trong min I hm sng ( )x1 tho mn:

0Em2

dx

d12

e21

2=+

h

t 22e kE

m2 =h

, nghim ca phng trnh:

( ) ikxikx1 BeAex += S hng Aeikx m t sng truyn t tri sang phi (sng ti), s hng Be-ikx m t sng truyn t phi sang tri (sng phn x trong min I).

Trong min II, hm sng tho mn: ( )x2 ( ) 0UEm2dx

d202

e22

2=+

h

t ( ) 2102e kUEm2 =h , phng trnh c nghim tng qut: . Trong min II ch c sng truyn t tri sang phi, khng c sng phn x t v cng v nn D = 0.

Vy .

xikxik2 11 DeCe

+=

xik2 1Ce=

tm A, B, C ta vit iu kin lin tc ca hm sng v ca o hm cp 1 ca hm sng:

( ) ( ) ( ) ( )dx

0ddx

0d,00 2121==

Ta c: ( )1

1

11 kk

kkAB,

kk

BABACkBAk,CBA +

==+==+

H s phn x:

2

0

02

1

12

1

12

2

EU

11

EU

11

kk

1

kk

1

kkkk

A

BR

+

=

+

=

+==

H s truyn qua: ( )211

2

1

1

kk

kk4kkkk1R1D +=

+==

Bi tp t gii

1. Tm khi lng ca cc lng t sau: a. nh sng ( = 0,7m) b. Tia Rngen ( = 0,25 ) c. Tia Gamma ( = 0,0124 )

2. Tm nng lng, khi lng v ng lng ca phtn c bc sng = 0,016

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192

3. Electrn phi c vn tc bng bao nhiu ng nng ca n bng nng lng ca phtn c bc sng = 5200A0.

4. Tm vn tc ca electrn ng lng ca n bng ng lng ca phtn c bc sng = 5200A0.

5. Tm bc sng de Broglie ca a. Electron c vn tc 108 cm/s b. Mt qu cu c khi lng m = 1g v vn tc 1 cm/s

6. So snh t s gia cc bc sng de Broglie ca electron v qu cu khi lng 1g c cng vn tc.

7. Tm bc sng ca phtn c nng lng bng 1eV

8. Vn tc ca electron v prtn bng 106 m/s. Xc nh bc sng de Broglie ca chng. (mp=1,67.10-27 kg)

9. Bc x gm cc phtn c nng lng 6,4.10-19J. Tm tn s dao ng v bc sng trong chn khng ca bc sng .

10. Vn tc lan truyn ca tia tm c tn s = 7,5.1014 Hz trong nc bng v = 2,23.108 m/s. Tm bin thin tn s v bin thin bc sng ca tia khi chuyn t nc vo chn khng.

11. Tm s phtn c trong bc x xanh bc sng 520 nm trong chn khng. Cho bit nng lng ca chm bc x bng 10-3 J.

12. Tm ng lng ca electrn chuyn ng vi vn tc c8,0v = 13. Tm bc sng de Broglie ca:

1. Electrn c tng tc bi hiu in th 1V, 100V, 1000V. 2. Electrn ang chuyn ng tng i tnh vi vn tc 108m/s.

14. Tm s ph thuc gia bc sng de Broglie ca ht tng i tnh v hiu in th tng tc U. Khi lng v in tch ca ht l m v e.

15. Xc nh bc sng de Broglie ca electrn c ng nng

1. E = 100eV.

2. E= 3MeV

16. Mt ht mang in c gia tc bi hiu in th U = 200V, c bc sng de Broglie = 0,0202.10-8m v in tch v tr s bng in tch ca electrn. Tm khi lng ca ht .

17. Electrn c bc sng de Broglie = 6.10-10m. Tm vn tc chuyn ng ca electrn. 18. Electrn khng vn tc ban u c gia tc bi mt hiu in th U. Tnh U bit rng sau khi gia tc ht chuyn ng ng vi bc sng de Broglie 10-10m.

19. Ht chuyn ng trong mt t trng u theo mt qu o trn c bn knh r = 0,83 cm. Cm ng t B = 0,025T. Tm bc sng de Broglie ca ht .

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20. Ht electron c vn tc ban u bng khng c gia tc bi mt hiu in th U. Tm bc sng de Broglie ca ht sau khi c gia tc trong hai trng hp U = 51 V v U = 510 kV.

21. Electrn c ng nng E = 15eV, chuyn ng trong mt git kim loi kch thc d = 10-6m. Xc nh bt nh v vn tc (ra %) ca ht .

22. Ht vi m c bt nh v ng lng bng 1% ng lng ca n. Xc nh t s gia bc sng de Broglie v bt nh v to ca ht.

23. Ht vi m c bt nh v v tr cho bi 2/=x , vi l bc sng de Broglie ca ht. Tm bt nh v vn tc ca ht .

24. Dng h thc bt nh Heisenberg hy nh gi ng nng nh nht Emin ca electron chuyn ng trong min c kch thc l c 0,1 nm.

25. V tr ca mt qu cu khi lng 2g c xc nh vi bt nh bng 2m. Trong trng hp ny, bt nh v vn tc bng bao nhiu ? Ht c th tun theo c hc c in khng ?

26. c lng bt nh ca ng lng electron b giam trong trng th mt chiu

>= 00)( xeEx xxU

trong trng hp nng lng ca ht c gi tr cc tiu kh d. Cho cng in trng E = 3.107V/cm.

27. Mt vi ht c khi lng m chuyn ng trong trng th c dng 3xkU = . Da vo h thc bt nh Heisenberg c lng kch thc di ca min trong vi ht tn ti vi nng lng cc tiu kh d. 28. Da vo h thc bt nh cho nng lng c lng rng ca mc nng lng electron trong nguyn t hyr trng thi

a. C bn (n = 1) b. Kch thch vi thi gian sng t ~ 10-8s

29. Vit phng trnh Schroedinger cho ht chuyn ng di tc dng ca lc F=-kx.

30. Vit phng trnh Schroedinger cho electron chuyn ng trong trng Coulomb gy bi ht nhn ng yn mang in tch Ze.

31. Vit phng trnh Schrodinger i vi ht vi mn chuyn ng mt chiu trong trng

th 2

kxU2

=

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194

32. Tm hm sng v mc nng lng ca cc trng thi dng ca ht khi lng m nm trong ging th mt chiu c dng vung gc vi cc thnh cao v hn, b rng 2a (hnh v)

33. Ht electron nm trong ging th su v cng, c b rng l a. Tm hiu nh nht gia hai mc nng lng k st nhau ra n v eV trong hai trng hp a=10cm, a=10. C nhn xt g v kt qu thu c ?

34. Ht nm trng thi c bn (n = 1) trong ging th mt chiu b rng a, c cc thnh tuyt i khng thm (0 < x < a). Tm xc sut tn ti ca ht trong cc min: 0 < x < a/3 (min I) v a/3 < x < 2a/3 (min II).

35. Mt vi ht chuyn ng trong ging th nng mt chiu c b rng a v thnh cao v cng: