Upload
sla-movics
View
251
Download
31
Embed Size (px)
Citation preview
BEYET€TE S
K\qrgi\(vs ll+iLqi \.teqdr+i{ts
- vesrccskc \-s{cl l I o. L\qssaikut- t^"[o,\* t^o"le l l I
po*Ye\.t saeF LHEcH A NikA)
a
kou!\vrvs* lvrecun *ixn)
.+
i : , 1 , 1 , 3 r . . . , t !
, r h
L q tr), Fi rr) J -,
f l <i ttot,i."ositds
tili\<ai tnEla(tetsrilcaes {i\i\.oi \aQ\^h,!isd3 )
^! tF, r)
Pl- : F( r , \ \ r E- t r ' t1 , B t r ' t1 t " '
e\{.-i qeY\csaf{sek -+ \^!\\ d\&\etr(s {3
sik\^-\o.u^:
^l:r i \r)= lurn(f-!)=
t (F,t) = A il. tli-"rl)
I
rr '; (nx - So) = A s.'. (\<x-t-t)
-> ult
t (Frt\ - d\to'^di
EF_(!t = a\\o,hdb
\<x - (Nt I o.l\qqoti)X
k - x - u J = v
- . \( \ot \sse\essca J
Y ( F , t ) = A . s r a ^ \ k r - ! ! t )
\ \ R r - w ! ,
t (c , r . )= A e-
I : 'l-l
tF , r )
FIZIKA III - Kvantuntnechanika - 0R0SZ L. Fizika Tanszek
ko ) t - . . . = ( ! o + ( ! i ' ( r ( - k r \ + "
- (.l t( t f ,,,]. i(. t ]otk =
) t I r tx-. . ] t )k.J e dL
I j (x-*" '+J.k. l
rl
-*"' r') k II
J
I
Jtx-.!it)) { , ^ (x -w; r ) Ak
( x - w 5 ' ! ) a 1 1
| { (xL-
j (!"-* r)
, j (x-- ;+)
( (k"x - (! , ,+)= co e ' LLV
k
Hv\\dr^cgs\^.o, q'+not
l l !x,r\ = I (U() eJ
i<"(k\- Ak) [k"+Ak)
f d * l / , .(! (r4) -- !. Lk") '
L6.l;"."-
v.+Ak
I i lt<x- u"tt (xr\= c" le"
Jko-Ak
i (("l k"- (-"
'lLQ-
.\;^ (x^\,.,J{) akt(x,t\ = <" b. (raox-\"J. lr)
( x - i ! q ' ! ) . A k2. Al<
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
'-
bt-taf_
z \ zf r ia+&3r -e
q!4
td
l,l1t\lalg
xfr4 .1,:! ii?:1"Eg A
zul
a .e
5
.4
U
z!?444
,
FIZIKA III - Kvantuflnechanika - 0R0SZ L. Fizika Tanszdk
,{. t(v AN-\ \rHtlEcvt ANt KA
B".,er."t[ (T.5v\a\^e{i,iu.elcirtis)
A {e!.ete- tesf s\r5i\ttd'so,
6=oo
a -?"
L-J r ,Io"r.[ lt",l::Y"1"*"t
I
* ^t ') 'L-t
A v' \rT
<i \i !ut\ i \*o L,( u^i <1vl o \< )
( e")
ILT
( lo,erei1u- leors \
?LANcl( L,r9$o)
9L l\:t
e\uileti u^aq5.vri+c!
( t " ) = -------> li- I
: sot asdqq,: (r)
? F----'iI I I | | I I I I ' .1,'',1,'/n,/r,'/ytl,r-i > -€ o € { ( L t E I ' L
\ev$^i<"s, eq\q\^\i{f
( tu ) = ( te )e. lg . t d tn \A . r . -
eE - ' \
_ 1 \ ,: G t 6 t S . \ o l . s ,
: l ( ! )
ir.--(:
FTZIKA III - Kvantunnechanika - BRoSZ L. Fizika Tanszdk
A { i t"5 e\el t tvsr+ss Je\cusdj
{a.r (eteL}'\,-o'{d3l^e\e: \^!\\,i\q)
{i'"^ - J -
\ \
\ e S :
- ltr l '-
o l.i\i1pti
E r!)
\i \a" ! e\U€p\e
t l EJ,T
c\{avqi4i^ r. {vuell i
I (* {"tf i \te\\i\nr.) N E M 1 ! |
EiNSrE i N t{e o s)
" {o\",^i
-
*,E,. = hl: -6 =hF- -i- I
g. Liral i i ;
f 6,^
---- v aslecs t e
iqc\ E \w '__-^.->, i i . \ i k
- , i -
E = ! . r l
csH?ToN ( {q 13 )
av" -,,<A
--------^d€- ---r - -
\ e c
( ^ ' * . )(.{ - .^4 $)
A\ <].s\a skel(ql!\(-
Lvrl{Evr-+s ID ( lqo:] \ {3| l ) I lo * ',-
I , -;- -.,i Pvoro{l r e ' [ ^+{te"}i r\ - l {-i*{-
E----::.-' +
--+-'--<-- ..i^ri-
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
@ A\o\$L e\aiss-riis
JHiavoqd,r- e{sw- L BALrlElr. r 8!s)
sei.-r.epe$
+-\o
B. l a , r r : " =R [+-i){ ^ E 3 , $ , 5 ' G RYOBEtsg-B.iTT
- , ^ / . r . l \'*,^=^\;.-^.Jt " ^ = .1 , r ,1 . \ . - .
Bot \ \ o\s.-uraetrie (1913)
?!st{ ! \dtv\ \oL:
? ' I E . , E . r E , r . . - . L o t p . i \ . 1 < i L , s t c c , i a u 6 . k t v s
E ^ - g _?a : \,lh,( E* --.i Ew'
? - 5 : L , - = r . l a r ^ = 4 r 1 r f r . . .
*\\c,prt E* v,.,*
.'^' ' .7;..'""/ :e j - - ' .Ir\l(Ei,,. . \
\ /.L "u.a*n-rE. : - : i
T
\^e\^^ \L-etiit! d-t\-us\!l-
E=o
l.6ti t!
E rtgdbetx - Ri|l
FRANCK- hqhjr! L{ 1.t Li)'t-
E"-+ en = e) + e._
EL- E \-\ = EiL-' Ei
r,!!a\qdtlo.!" GLLii\i!.[
i-@ =@1-a @ --i e <-
FIZIKA III - Kvantunrnechani.ka - 0R0SZ L. Fizika Tanszdk
tsoHv\ - {alc- L!(q\;\<i\i {el[ileltsoHu-- sonnsu-FELD (19.,t 6 )
d\ta\ci\os i tc;! o{ 1
Bosn- :?LANc t ( Iv(x). *
I<-
E u = h ' t ' t o
\^ 1 , t \1\3r. . -
{A tc\\es e.e-1ia\i' 1g'it^1
H(p,r)=S.*==
L L = P v - = v \ ! r
\ ^ - . 1 r 1 1 3 r , . .
(= hou^i\{sr- t S: ".-*)
t dr\"*di)
F - .2.(E
P r _ + . -
$ea' = n.l .-e.\ lr .€ = zrr 'e [,-Z*""-*",*-/ :-
6pa* = aT.+\1.8. J- = u. .LJ. ' rtir At-rluAuos\tAs
A\\q\i\^oa L6ovc^iqd\.il j
Inn,*",r,3, ,*+] ? ( r .=r, r , r , . . - { )
/ \ \Ld\\-{ad\. . tsotrvr. \'i\n- cj\s\a({odc(\ \cr"-
P
II
( t =.)$pr" = p.zvlr = L. L'ir = \,\.l-!
$ p,- oq,-
FIZIKA III - Kvantunrnechanika - 0R0SZ L. Fizika tansz€k
De tsRo cLiE t,,t3 z j)
E\e\etv ov'^ cqv.ere s\av\l d,u-
w,7
{ou^<9psut
ErP
- - .+ ' [ i r " . , { o{o t^-
E,F
HUyA|4
v,7
--+ e\eL{rot t
"3)HF
21t r = t \.1 ( d\di""nov^)
I1=\
P
v!.t1 ("3)
P
Dlviss o N - CEtr.},lEt\ L^: r.+)
-..+ T\^aw^s!v. - t*."^ (tt .t *)
k'-7srv / B t q q q - r e { t s x i 6
HEisE NtseT. G (132s)
al{ ut,- b,tt .- .
oJLr (!L! aL! .--
<, j. urr_ all . - . :9J - -+xPxE: r - ) = )
r__-_____v+_r,i/L Ats!.rv\Ar.T
tnqtti x 4iqlohalilolds( or sqb rc)
I
scHq.do iNeE[ ( {frs26) +
/--v-v.V_v_y1_-) V\"\\i\Aeg)r.\\e. -. (XXI.:]L1Dc Bvlglie L,"\\,i,v^
FIZIKA III - Kvantummechanika - OR0SZ L. Fizika Tansz6k
,t. ,{. Hv\l ilr,,rr^ec-\,to,qiLo.-
A,l.^. A scuqosiscel\ e31cr^\e! i! o. \v\ri\ '\{ 'n 1l\' i\^t} .
t^"srqd t6.^e3\Do!^\
S i l.u " \\ i!.^I t [ t l ,-
1-?t Il i l l
\o le Svogl ie)
Arto"ai
IP,E | -
t:ro
J, (r.r - *d)
I lfl + vo = E (b^"'\"'*it') vo =
t'*I s! I vo = trur ("tis1pc'\i6s vq\d'cro)
o. vo\ \^u\\a\^^, o,fLsr h'\\aw'<q\c\^\eL!^eL is L"\\ \'u'te- \ "
I= (k", \ . \ ,11) Ir= (x \ \ r l )
| ^ l . ta ,+ l
g = L*.x + kt . \ t + k l '+ J
?t
d
_.1:_ +
. IF
. ?x.
a T :
1"t t , t \= A "
I.
l4 ?t"- jar-J '- t"
/ r 1 r \ . r L \- - ? \ q + t l + ( t ,
v__.-.Y+
4
e v l
^r.' ^v(.4F.+-+)
J
, 2" Y" z'ir" \" , : . -
a+-j
. 2z 'Y. 2't" \"ay"2*7
k'
) 1 ,
?
z
l j
(
_ u'-z\r\
- - q -zh,\
?r,l^v l-'t *u.
+ V.,*.
FIZIKA III - Kvantuffnechanika - 0R0SZ L, Fizika Tansz€k
Uj v^.t",\^^.tir.o,i sti-^bi\urqot vete\'."\c te :
?".{. 2a Y.zx l A :a
**)'*" = a,t"
Nave- 1 LAPLACE- opc!-dto !-
(v-UverLtti uto'sl\'iroL )
+s = [*..+
b.;J
' o
I f i) Ar-trr'AtositAsv \_1J
155 tcLab
- 11- /\ ^l
iv\a!{ , \arg1
+ Vs.^ ! . = -
e15 0j v-otcer^cti
v(F,r)]1 = ff '
-* of. + vtr,t\ ' t =-a" l ,) -
Vo '--r V (Frt)
*" -t * tF.t)
Id t {n99; scHYro D iNGeR<qt a\\ ct
sri,-bi\v'-ot
= s1,'r*+"* (ff)
HAHiL'toN tgenitr ' t2.
\ e l css " \ t \ e
f . tl - _ : - n . +L1*
Y
r.^a{qr.qa{i \co.,r r.^^[,,et"{c-L IoFevaci i L) so,.oao,td.
isl ,
H^t -- -ldS{n3tA sct^r;6rir^qq\^ <t){\ta,tope,.ot!\^ 5r\\^^\i\!!qo+ V\c.51\^dlvc\
-/--..r.---}-\--.\^K v qq+ !\^\^{ <r^ q \', L q ( V psqt"\.i\"*"..)
-.-_r.--z,___r-
?orcid\i!, dille$r{^11,i1 r1.1r*\e! -+ L!l6u1{3cs di{{ e\'cqtidt <9'1ru\rt (ct')
Qrerrcrr-As)
V(F\t) ---+ Vtt) k"urqtvc{iv t.qdstcr- E = <i\\oqdi
t tt,+) --+ ^{,CF).eLr) tsr"p-."tu*ri qrcLbc^ irta-\i)t
Ll ^t -J
-E" t -+ UH+=t-
HJY
(r)
\9.jg
(r)
4
Y.e
FIZIKA lII - Kvantunrnechanika - 0R0SZ L. Fizika Tansz6k
I t",r)
tsirou'1it.it :- 1 L l L r + o J - q a )
{ ^ t ' + q ) = Z e ' } ( t l . ' 6 l - \ ^ 4 )
- i L { x _ [ h _ , ] a \- . i o r ' - * - '
Y l x - I . - r ' l ^ \/. \ i.^^-: J
;v]A \^sv*i \dl i l . t \ i l - \ | \ ' A,t i ! -*)
(q\{) - - \AlEz - jLtx-h*-x+^' (1 1r-* '1 \ -1 '1rrr^ '1)
A
(t\ua>(* t tr12
(^\.-! \^"\"*" 1.""{*i)
L lv. osii,\^^-)H^{ = 1"1
Li\c\i\\ \-" q!\ "r d s L&
.l/
fr4=AE"I
Zh
1(x-."^1 =
-d- + Vtx\ - \e [r-qd) + u: tx-hd\& t "
=- V* tr)
A Z E f -S * +wtr-^^1111*- ' - ; +L l . * d x - I
A L e v k ( ) / ) I l t - h c l
fA- -f (r-\ ')
,1Luc
A Z e" \h tx ) I t r - ! ' . )
€r ( " t . lHt ) t A ("t \^y ,)
4
" {L('\-i) o\+ lA\-zz c ("x t'-* o lv"t'.r\r t* -""1)
et c sLd\.i,v s\o"rd{(\c\ (u-r^)-\i\
t,aq\\.vn$\i ttU\lq$,.l.
FIZIKA I1I - Szif6rdtestf izika - 0R0SZ L. Fizika Tanszek
l vd\ a\v.o r d, !or. i
?e'. s\d\o.<r.\
^l te,l) =
--> si!\.'\\6'\^^ | at Bt.3li"-)
Uoio ! 6\o.\\ l , iss"\
-.-t \^! \\ <i \^^ ( so \-^ o, 3(\olao\i\d,tr )
A . ed -
_b-sZn4
-l
/ (.ris\pe"aios va\d..i6)
v_..__-uu- /z\L \ ' )
IIrI
)ll
{y J-,,=no^ta..,^'"* J
-343.ro tt I r , = ,o'*
- t s l -{o \,\ J
't l(
2w\
r-) r
l+.L
$\r\r.CS A
Sldloc"ro\ u!19i v<! \ecst(e-
\r.- = d: = r'''" .J.,k
(er rn.AN
: lD- - u- (1)
,Snrer-vetuetb)i--!
t idipi\qq-{\'o\
{ tT. i d iPi \ \ov.. thax
r - 5 (cxJ-
o\ist\r{.v\ii is vq\
\^v\ \dvq<\a,vnql s+dtsor! i \ t
sr ),i lr^..L..sr t^ ei\""srisucrLe
)
rec E+ lr'oss1 I
(et e\rLtnL^ ,."<r* sto[\t )
t
aXo
?..ax" l"?o:
E\ Lc\ !eh^'l\^L ort ot I eq5ih\'a-\ tets+e-{5:ut\ \iul)
iic-lffiicTiaT-Gqi * t u,\\d\^ \-'is\i-\^: I'
v-r.*ir5t* -
"o5.1i Lnre1" - lew levles*e{l \^v\\d\)e--
l€,^si9qt ir le ) q*it q t'.{'isci\L \a\ri\
e\e\.troql*.ir^\ ac\e\-\.ilvk!.
( lr,'r l euvr.rAt t Hir.rcret I )
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
MAx tso!.N t {116 ) + (:**""' t'r^')
to,ao.r'r{a\^{i tiuq '. cso.\a (qis\ e\q\{t"qt tqiv'"'L
( e\{qi \ t l \es l"a*\ i \ ' i : )
?(F\t) = l t (€, t) \
\'v .'r.ti1' a^,--f-'l-v\
/icg\da\asi \
ttl;)vc\dsle\" <seti\
\t tF)J
, lI"lt?,t) I
.&= l^tc"l \ '={tr l n' tv=1
t\ao\$,p\adL^u!vs,i\t
\r,lscvrcfi q\\r\^i\- I i\
UX
c\eL+o!^ n{otrds N
E-----t-'-n<' cagae\ct'-lh* bl
xtN
\ e\.qu\
\i.virLil: \.crqpi<tdsi r)ou{aL
" \ste\.r-e*e*.i r,n sss*Ul.( eg5sr"t*e <sqq <qt'-t\c" <\tL{*o\' \ o \ q V c u o \ \ ! " t c u l )
FIZIKA II l - Kvantummechanika - 0R0sZ L' Fizika Tanszdk
,1,,1 . 2.. A \",\a'""^ {'t'lgvir^r v'o{e'-'li\'^i t"\qid"t"sd lqr
fiv=.^v l'c ^ z f. .r -r,"r.,tlU(.t)=n(..*) -Jt1t-ait=Jlrl av={ *l ' ' t
\.^d
i f = -9
!0,^ro\-{t^e,^o-L,uJ*tT*.,i*---'-"*.,,.0,f .,ti*..,.r.e
( $oa"'. \^^i\.re.\eL{e\^iLdi c:rtr'6r-6L'
p{r \< 'a*\u*. l ; t31et- )
EY
Vtx) t = EY + RE6vLA|rir AS- -:- ry" 1Z\&
t"I
,/* *-, -f t'- n
--'' -.1- E" - "+.e"r ^r, I g
- - ,
' * . ^ . \
-\
\
* ' . t*Jr. . \aeqo\d. is o\e qeu veg' \<i ! ' is isb
;iiii;i*; ".,^ t"r"r'-.ru.ti I
-> !qj,i!.vsL .qr.*r.! .. I*i ..lfiil$i",":
/.uv\ {= t lrt"r-e\1ia -EII vu)>s
^j,s - 1 ) o --+ f >o
@ * ' L<O ( - ^ t<ol Le ,
"{ ti\{"t V($ <E
.\. - ( > o €- .k..o" i
, , Y =
Lto <- ^t>o
vt^\=E a.=o J i*q,",.,;i *;i$l'n*"'") ^r = o ^,i= o J
tG)
o. (!o\ul \'aula\s\d\a6. w d$s:Ei\-
E r^c$o\h '-+ ti\! <r"'-\Po*t .-* ['"* - fupl
' ! iU d,\\op"t
I
t .z
{ (s".,"i$'
[r'
I \-t
FIZIKA III - Kventunnechanika - 0R0SZ L Fizika Tanszek
K6\6R d\\o,po{sL <s.\i\^
v ({)l(of\\L ) \^\q1 I
E3
E L
(n,J S\b
pb\^hE \^ e \ \d td tq \d lq
L;rf i"lt- ---+ qq*ri\itr"sq S c\tG cr.irr5eu
v(3\ \^\u^ebi[!s
i.! stb
r g5eule\ !^ a1o\ddsdvc\r^^l,arrq"et,lc\ \
f
i rde\(es! tq Iv (x)
A " l isrecsl"q ?>s \ a\\s1i\^'Jsqll c\r^^cq!^\d\\".\S a\ho- q taYvi\\ ' \i s i " u . \
o "L i , ^ . t i L " t e l ^1 " "1 ia !< "
IA t\r^r1p"ul src$\;\el hc\n Js
I
?>o
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika fanszek
1. , t . i .
4.,1.1. , { .
v (,})
ESgsrerii il"rdL l.bt6{t <i\\qps\o\-rcr
x(o tqcl.\a -\^\h (\ '.
" t"=+* ' tLE6v!A-\ l i l is
- - - )
X>L to.u{" u, i . \ \qk i
"t": + o(1 l ----
U€qlLAEi-rAs --)
o S x : L \ q v \ e q i . \ b q \ i
^i\ =_l-1 * --)
r- vo Ht=EI*- I ;^y *v(r\Y= E^P
I+ ltsGvr-hyri1'li< J
,[ tH t!q.{i L* )
*"=* ? [vL<r-e]1!-Y----J
2 ' " ^ r . - \ . 1+ -_;_ \ ! . -E) =+o(
-+ E =-LL
ls\er"eid\ v5\jj ( \ ,du-\"s td',1\4\dx)
o {X _6 (X
^ { ( x ) = A s e l B o e
L,"^ "+ (!) = o
, (Y - lX
\ ( ,q = Aoe +B.e
dr^ ^t(,4 =\)
l - ba (x ( +ia) tor\"wi5bc<
f+\t".^"r.- "\i++Rv (* ( t.i\\. -\i
A"=B
Boe = A - ! , t i ^LL +BcorL \ -
Bo Q- = LA c,s5 LL - \4 B 4^i- LL
[ret.a.t]
--5 o( -.> E
stdqitL-{i
fr"r..r^tl
- J ( x
{(A = Aai,^\.x + B c,e' l .x ([86!Lih' is\
nEeaLoAS d te\.s
t{Ee\,!Awi r is i *k), ,
x:a fc"*b a\ t
tx -- L Fo-t\q(
1L - <
t\elia3q!a! I 'i .riqqrxiboq \utu'rrl
\i; $tl c\\ artr' L
GYA\N[!A\I AL\rA!t\]lis t " [!a*\"$^
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
<q$*erue\ lA{"ra\d\^4! i )
( v x<o)
{o.touaultoL-
^*(t= o {-x.o, .,< >ul
- i ; ^t =.^t (o:r :11
Y =-'-= +\.---1/J
- r L
{"*t"r= 6 r;^o* + u.,-uxL (r^..{cu^eti Ldi \a{.5ot.{dr)
:o
ILL .. - l !q.{eh.riL*) 1' l r i l ioa
hlk1 t1 1ir r_ I Vzr^^ t:t '
I t^= r,.,r, ' . . ;n+; \lx j
l-r","- e : o
f o:x:L)L V(r)=o
fr"t - et
d .=
?o\t.^cta\ isbo+
v t{)
.<--.
, ' \ =O14
- -+ E.
"{ I,x) =
A.{L
Lf'
II
L
Yo.) \.\
o
L.L =O
11- T-.t
Kr I yL
l^tta^=l -t
1.4.3. t .
oI----'t---"1
IL
\e3"\ovitdsi Iqt\ i{c\ I
Yto)= ^? tL) =
"t(o)=p+b,lt (t-\ = Asi4
L
JI
H a{e.q eti \- di\ a j
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
A \ra,,,.rt \r'^<.go\dds {v!c\ra<r{se, I
E v d c r q c s r-eqvitsqdl,,r i
A" v^(Stdt. i \ds i v c\;srih'J s( 1 -
sit i : t5 * k\cssr. iL"s wqc\o*.1.q!leFt\t
?u" ax = A. a,tx 0, . - -A. ! t = A:p- t,L{_ .
. [^ -
i ' " -
t
i ltv o.*\ v,^*c c\... iL o. -t ltlarsri\<vs\a(eko.L.iLa.
( tt" tu.sp ",^ +e*. r o. e\,)A+ R.^e\niq :
E E w = E o \ -
A E,.
E K
2-!^+ 1
'' M dta,os+L:piLvs<\^aeg\a\i€vo (^., n)
" A E
(,"$- -=:-
A L"c^*1.,;11.s6a
A r^.sqto\.itdsi latiniu.\C
,; I \ \ lL
\aabtsi \..!\o5sd9
"HoLr"srGuiLvs
\..a\\q{'$ ;*it\r. "o \o\5u^iva5 ci\\cgc. : l -
L
- t t i i t . - t -E:h: - ----:= vL
"i, -- ll! aL- !4 x' u . l L L
LL
t
| I I IKA III - K vantummechan i k a - 0R0SZ L. Fizika Tanszdk
Y(F) =o h* F { L3
AY =EY hq l^ eL '
r+r, f +t* ' )
$*= r'r,s...)
(ur - r,r,r .. .)
(vr*=1.r3,.-
t
^'
Hy -- '="f
s\.rp.va\as . t (F) = Xcr. Yrp'11+)
/4. *+ * s\* = -\.'Y\axL ) , rL ?+,
xoY I + xY\t t xYL\ = -'L-a'xYI \ *I
u* Y\ l r ,L_l__ + _:- + __=- = -lzXYL.
\ - ' f - . - . J \ - 1 v
(,\ (9 G){,.u{
- t"t - L?t - Lot = -l"t -- e =.*: = 1'rr. tu*
I o^-T.l u"=I
L ;=*
_ *L\{
PEN.EYIFELT€.TELE L -)
( t ' tot=o ^.1.r .at)
p.\e{<id\ d![n+Hcivo'.*dit ^<.,^riiS
] .
Ad.i d.tt t.lat :
rr v1 I lcvou{"u,r s\d.^oL
*rraZ\qLL
,r=-{t',x,r1 = }} ,r
+h: + t1+ .l
, I 'qtr \^ r , " \ -x l . "^-($v)'"-Hl=)[ x ,h1 , rn* = { ,1 i r . . .
t
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
D e set^{vart Irl{. j ' , l l) d,\\6\aelok
f n,.,v,:,!rr 1 <q "f t*l( n] + r.r' + 'r*')
Es5 54ii\ a\,-t€L\"ea \;\L sojdt\;9s"i,",j
D<1eue-r i{ [ <i \ \cp"toL- i^ o
{ t , , l t E :3E.
la,'n i'
\^v0",l'
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz€k
,1,,{.3.3. Fl ov ".or,",i \<v s [i\e,ivts os\<i\\d\at-
l(\css+iler,s ps+h"a(\^o,\^iLa\ah
ov^-------#'*ts'==><-_
]i_- -kx. xtq= xs au^. (urd+ d)
vk) * = lI-*,\(x)= 5{r= L
,L a\btd\^osi!"'c
-*@vtx)
F. - -
es tdsr.i\.is
;; Yr.
J
JI
V l{) =
! eud!\,rh w i\d\!t (i. V t1)
!+xV(l) -: V(x;
uro*id\l * + LIr'vl x.*...- f L* lo . Ldr.J"= @
- ! - - , -=P = !"^*L
-4-') - - -> x
/ ' V" \ l )
-* y" + **,r '^y =E{ -- ^1ts = ? I scl;\".{.1-I {"t.a-!
" { " + L* (e -. t.- \
q\ + lz ,^ E _r \ + \\.-..!J
-= L
L * ' ) q
*') t
( i el6\ds te*ttrisi"..l )
\ V! [1)
| ' . .
l(rqt + r,w^ecLo.qi\.q,
FIZIKA III - Kvantummechanika - 0R0SZ L, Fizika Tanszdk
ttt o ( 1- r'x') "1 ' o.!r
2
isrue,.{edett d\\o\d.^os \i^( lga!. Jo \ <i s o \^
€b(olqe \ {c\"o.t\o\t"4{i i
! tr) V {x) = | w.*a '<\
\ t4 1 vt- r ' lEr-
e\s6 "t u) --
Y [x) "t (-x)-^+ (-,()
* (vi'^')
(ptv^ttor )
l*w1'=ltc*rl'
(ea* r 1
:v s tl"
L"q le '^ ( p" \ !J . \1- tas r)E,, --r .r4
^1"" o (,1" - x'*')t
Te\trit I
(-:. y + +yr x'
9J=-zTX-T"^{." = -ttt"-z$xt.)
= (_rf | \{rxl) 1"
+1" - "ttr') ^t" = o
:o
(t.- .,f) + (kS'- x') xL\--------v-J l---v,J
+o:o
tlier! )
( " * )
tv *)
lX '= ! ( + ^{ ' = A"e: ; X -
(r-+r*.riL-i Gq\!\"{d! )
-+
FIZIKA III - Kvantumnechanika - 0R052 L. Fizlka Tansz6k
1\<g-\o,vitasi {{tl i\"\
Ur.. nt"L<y =x - - 5 ! 6
As u^{5\atA"! \d\ .- 1
o "t" txr = a'- 4vL
L "
| , . \j l"t"t")l
z\4 / )-
& =t
Ie \ i \ I
1"=*d=
er^r'gio.) r^ e3 v,o,\ i r"qiy<. 1
= {u.-
EO = :
q\,J
A . ,
Y" = g. { ;^-
:
"t* (x1 =
\0. i ' b, i \ t oki\oL
(c^.r) t "
( c. + c. *1) ^Y,
( .^x+..xt) t"
1t-*t*t'{",\-<d. .to\".1 p"\i!"o\^.
ipqRno\ )
(e i *s )
Lp6,b^uq*)
( p.i""s vq9i
{ ( , 1/t.
t d\ua{ldk )
lt* t")
SeV,,t.Ii'^qer.
ti' * (^1^-*'*') 1,. "y"" r (1"-a'r.)1. =o
'.{".
rl1c". \"C'^{,'tXt)*) " tt"."t)t
_,qy "to
t h d { q q 6 [ i \ r o \ :
Xt"tx1 = 7t
FIZIKA IlI - Kvantummechani.ka - 0R052 L. Fizika Tansz6k
- 'La t
}t* - 2,(x X* + (^?.-'i.) X* = O
'1t,^ =Z c" xLk
' L-!l f "= t c , . l< .x
T*- t -L
X{- = :. cL' t4' (r'-{)' x
qo" r L'l"-ot'x') "t" + (r^-r")
tHEh.nitE- ori$- {\("r
ti.^ttt)+.
--Y
t -1L [L- { ) cL. x - I za"L.c,.x t [r.-1.) c"xo =p
,"r]L+,t) CL+L -
. 4 )
x\
fi \
r.
c{' a \
r){
tr
Ni,
. L
L+
t
;
I
.,v
_ (
T.
5-
zc !
L
Lcax +. . -
k -+ \<+2
t"r [. cu * ci(tr
=p?-,(k - ("t*-4,t!)
(L+1-) (\.+1)
!/9*+^...,,-.#
L s . - ! 1 r - r + 1 - G 1 \ 8 1 - - - .
\ 1 - \ t t \ l ) l s r \ 1 r ' r - r
tI pe.e.^'te\lil<\ qr. bl,\ \\
rL =o
lt,^ t*t
lt. tr'l
.t.PoLitror'\
v<Lvtriis lorr^^vl ort
(Henn\te r'.rl
\(c\l Lqq k$cq
.FIZIKA III - Kvantummechanika - OR0SZ L. Fizika Tansz€k
FIZIKA III - Kvantuflmechanika
Y"u) qLLo. \es+ psl i t^o*, lao' :
O C r + t = O t C . + * = O r
2aw- ('te- 1.)c!.
l'^ +r-) (u +r )g_,/-
=9 Uq\, \^s11 \<11<.-
2"rw-(1*-1")=t)
( 1,. r^^a5 is""tvc
\ - =
tEr. =
r ' l a r - \E = 3: n. . = +- (1. f 2o(vL) = E" +-V \ Zra L{
10 + Ll .w
v^.-1\^-!d \, "1\'St 'r( e- = Eo f Ntr ' ' -\a^ t
I t ,_L
: 1^-
: q -
4 t - __> _t^u\oPaqt\
e ( { ra idE=o
u..\rc. lL",----------r------a X
<\^r.v\ia o.raloLbd< I
( \ = o \ '1 ,1
t". !,)
. 1 . . . )
I -, r.
t= ls.t"\ " -"
Tansz6k- 0R0SZ L. Fizika
, ,1.1.\ .
1,,1. \ .4.
N e v-. ko lii t\ \\\opolo\r. \ d\di o.
\(x)
-r----_+----'x
A v{!\ec,t a v{1es. r,.$sri,^"-slfc,\, - *t i l : | | iql L\o'ss\iL!s \ae<uo{iLc'
s\q"i*t -UJ\:$ {o{ +s \4quloq \- ELi\ <! i J '
l\r ci\\oe"\ {;g"ar^1 {i\'L'i tovtd\q*:f )
k : l l ^ i ( \ -dx = iJ
+€*t r ) - - :^ t t l =
|
la{s* c,\\cl:o{ ese\n
I t*r'.t" = 'o ( t),^.* L-"l i tL . i \ . \ ,o\ ese{i '^: , -
i,1 \sqclor- : !dts\\,u[\41i "'":j$j\(1-1t-l)
a:gv:':rets: (ax '-- *x ) I
i U+a*1
a )(+AX&
i tx+ax\ - t L<) =l ' o
{ (r+ ax) - i tx)- - 4 -
- ft lt"rt- S \tt"
*l "t(akti \" i t . isi ej l{nlclI
",,. i'\e5 qle*reI"
&i
A vo\isri'^$sc <i"-w. sl,vlsC
gtx'x. .)=Jl t laax > o
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
t\iv".t o. t (r.t) -\ \&e3 u'{I.it"* agrtq\e+Sqt^^;ai,jae e31tr\el , €1i"1 ebb$\ \.4\\
ar ia'i\'i\ {;513
_-.--> da = - f1 \"tt'
j=,i:! (l) ,1 -
Eat L4\\
: L - a :Y "1 _Y ,Y
!cti,\iu)scAi d"oqsivusca.dr ! iq i i i t .^ i I
-ftr"ir'=-&*'.^t0^r =-i"i I(fi.iy =. i t- J
-a l+ t '?+ ' - '
i ( / ^ \ *= - -!- I "ttlrY -
-hL '+ '
i l- I +.4 "\'
\ at-. dJ(a
i +.' [ ^-t,+ ^v" .i i . ^ (
*^ Ir HY l
" )
* **"-]
i f r \ 4 . n 1q +J= -41 - : - a , ^ l - +5 / 9 'Y l
+ l 2 e ^ ' - t q - - l.J
= . t.fi (dc-tc')=4 ["*dx \ -
=4[5(r'Y'-crx L
.u) \
' )
s- \ \r)
J - r I \ , r l \.:- l* t '- tY )c*t -
\.or^trv.c\iw !tud\1eve\ e\e\cq
Y tx. t ) = t (x\ q '
Aha\d\^$\i \ds i
- r$9i:*i)2 t
- ---)ax)- ---_-
{? t
l" I, Ixvl}n)
Ax- '
t= q1t*v{-tvd)
V = SVq"\ =e:qt'
H"f --- + ^l-x
FIZIKA III - Kvantummechanika - 0R0SZ L Fizika Tanszdk
T<\do,'. Std\ada\ ..us\gi vasl<csLq :
!+ E.,F
iL,(x Y( t . t )= A.e"
Y(*)
t tr) '
{t"r =4{ (r) =
.r L*
.* - iL-
, iL." t
- iL. v*
I| '" & l,i'J = *'^*
lrf +iu\n\' 1I
= tr. lAl -- u,
=$\l'
\r\'T
o\c\ i l ic\ I
{ = * t"Yt'
. i va*sl t Ir i5 q L\oss{L"slrtt\\. i \d\.
^.1.) . L. A!!.,o'\^a ar otcqci I\ A\ow (tcLr.rilt sr,ir.',\;s)( eenow 1""*"r^)
{"i <- | --> ,A rasl<<j\.d ;,fj\,ldso\^q\v.G rr-i r"L : c "1 et= {Yq\ i} \ - i (s\ iS)
^t'tr\, E L
dsE
oI
@) 0 .
d.n
q. l^tn l 'u. \Y,\'
T
r ' 4 .I ^tj (r2) II^t.k\ I
. " \ l q r - , , .1 1 . - - \ s _ v u ) '
g--\/------J
. : , ( L
^l^ tx.)
t, (r")9r (r,)
9r (r-) \
t. -: "lL o ( x
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
r . l+,t ' .r l ' = I "t" ( '")
lt= e 124 (r'-1')
< L' l \ ( " . ) I I | (T . r . t i - . ( t 6 iq q , " { { l )
-24 (x1-x^)T: e
T.\sr:\cqer .t-LL
*|.ar
- z IJse tv,-E) . (/.-r.)
Vo\e\<ia\i . ; . t :
G Anow L"i\e\i\.! ' .
Tr, T* = 1I .r-I
-zlz; Jr_T_ -
E
FIZIKA III - Kvantummechanika - 0R052 L' Flzika Tanszdk
?e\d .o . :" - ' * (
l{ idel - c\-iss\ii \{i"-er*isrril )
E\cL{touoL L'\ipis. {i!-cLbi,\ e\eLt'rcsr\'!'\"s\^-ld'!d"A
)
L (I^"\-"q.- e\.\-d's\-"s !i.)
$" tr_
I = l-,T*
tx)-(v"-Q.; = Q"=e --+ Q. = g-1"
- 3 - ^ a
1F l - : (x"-r)- I -I . - I J I
- c.x
s:
I . - ' . o -
3_
o- dll ' -::-
f) - l
6 0r
3J
z l I ^ 'L5EY"
(La\"\ito bssr4rgges )I
+awrEvr-NiIDHETK b"*"\q
t i t
{i,-
tstr_>xT^
I ' I I IKA ]II - Kvantu'mechani ka - 0R052 L. Fizika Tanszdk
1..r.T,3. Atlro\a"\ds lo{,e\c.t,i\ i\s r,. st^rii\vii\ qqir-
N\ {9\ sra g\e Le s pofet < t , i \ ga\ : A
(r{o.({.ud,"i.! d\\.r"\)
Ht=Et
l__l vo
l l
J B E
t! i
a\^^^t +v (,.) l = E ^Ji.
. ! L w \ i ^ . _ \ - r
il^.\ (x) = A1 e-' +
d,r-Y3 t,() = Alq +
.Lx*'(4 = Are -
- lL ,Bneo
B! "'l (r =o)
*e"-e
na3"lari\" '\ r t (r) J.\s
t] (x^)^tl tx4^t. tr.)
tl tx.;
lf--j
(x . i^ x , - r ,<- is ! )
(x)al tx"1 -
tL (x.) =
^{,r (x.) =
J.. lr.f(*) lA' \ '
. T . _Glt rorw (e<v"1
( Fe\ada9
^,L^' (e-v.\ u* (e >"")
T
l TtEi )={
t
+E (v"-E)
FIZIKA III - Kvantumrnechanika - 0R0SZ L. Fizika Tanszek
Ncg5laeg\{ rcs p0 reL\cr d\ !s \q\ :
- - - - - + l \ c , = . . _ .
- x* l I
i l=' l---
-T-- -vo
] rat;u d\\o,osroL
Xet.,^,ir^s o\c1i,t e{{r-L\.,s
,__r Vo
I t - - * - l - E r I E r , E r r - - -
L {
J t*f ar
' l - - - - - l^TI
- r - - l - e 1 |
t\"ee 1i dsliq\.|-)
>> (\"N;r)4c
l^r u)f(o'r')
I{ o rd .q , i \ " q , . qq d
t is lecs l "c ' i ! .ssse.C' ri.r.et5 \orli'lLodjL
@x
Itor l'E d E i
FIZIKA III - Kvantuirimechanika - 0R0SZ L. Fizika Tansz6k
4.2. A kvo'^t"r^^r q.c\no.^iL o. qxio\^^a{'it<vr {e\ipl\ise-
Fiti\.qi $v"au1a!-
I.l,
,1.2.^,
Iv (F)
-+^lt-
A kvo.*tr",^ r^ e"\o\iLa \^ atelq^{iLo"i <*L;a.i
t
ArtruAtrositAs .l
Fq =tg 1 -->
----> * ler+) -
saawi{. i ! . ,
$etr\{Ero ) tq\6.\ vd\irsl e l r , t a35 \ t gy t v r .
F-*ei/
srdrqql!-{stFv^auq\isigek
L.q\ll\ .\ c d{{i*i.ii)i.,q} E ovevd{sr kdl.r
ctiqa!qiLci vd,\+o}i'
\" a"hqt. {t ri\.
',7)q
p (e",cr,n.) \ .r. ' ,F Lx,x, +) -J
P*,fr, i* i "i ,9,i j
FA l<\o.ssriL's ua<e t^ aq i \<l b o,r
pr p.l rpl +pl*r= ?G. = -----z*:--
V : V(r) = v(x,g,r)p = F (r ,1,1, F, ,p.r , t . ) =T(r '
A Lvqk\v\auecuc\qiL<ihc\
o\-rprclS opev.\ovsl .-
+ --- i =T(i,i,i?e1
|
Hqqn\r ! ( op, 6 = #(6:
,, I
1Pr1F11P+)
*6i *fi|)+vrl,|,i;
.^. :
FIZIKA lIT - (vantummechanika - 0R0SZ L. Fizika Tanszdk
rll
fi= * (i,"0,, *6il'= -* [* - *b..
Sc\^"iat,-qe":
PrPr J2\ \2z' )
v F,$,i)y 1x,5,r) ,
^ h6P":T5i' J -
. t -_h sre lz,^ _ q ?
$z
=z' ---- Au lr-Atosit'tr nr \
= + (i '{,i, 0,, ,6r,6.)
HarEHAt i r (A i oss te+oeL lL6 :
$;11.6\\dv or o1e",i\""o\. :
1L t 1u a"4i*,;,rr. \n15vir.1"L
llsg\r I
{=AF ( i"r .a,orosli\"e<ieis {ro\sr{oikAc.d :
u"r,-"a{) --{
K=- K)
A (..f. '..F.) = .^ Ap,!o,*p\e- ga6oL
+ ( , AG.- t -
Our.r,i+ou o\ocbro. I
(A+B ) .9 - Aq + Bq^ / ^ \
(A'B)Y=n[s,f)\--1-
f,
(o1or.d\s*oL gfi$eAt)
(sped\o,or. ::::e3}S_el
JI lac\u^r+ e\rt*eir^ct-
r d\t".ld!i\i{riEIYI-J^r = '{
{v\ai o{o{.ra.1oi (6aL a+
(drP) = L'*P\Q{ *{i\^-
c\to\\,EL t^e$,qittoL)
(SLATAYL s1nr1q$
J o(tY) P(F)&1(x, l} ' \)
]( y , o , t ) J
J.rr = {.rlF> (j.\6\"\)
l€tlatlales . L.o'ai\.i{o.r.<".a$c-
P'-"* *
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
f . l
"rN = I ld tF)l d,tt = (al"r) = ,\
, (d\11) - o
A\ 3E:N-ej!L opr.o.{or : A ---+ A
J d tt <ct) ( lon r r'1
(gtA t'a1( io t6 tor l )
(v p,tr e:r)(prAr) = qA*plt)ti!}\e\.ee\ l.\i*^ (
f | \r ,!tj Ss!{*t!t- "s.{it-)
Jn+ ittsJ:tS,-I o1,e-,i{or , A* = A
"-o\ (plir>= (Aplt) =Yds\lq\ese\ t"il',
". (p,[ 4nS]ij,^,lqf erelc\-)
IA+ 6tr lolvN gALT oge.dlosoLrq \ o,^.-\ L"+.
,{. { i \ . . \ i E95 6\4di.{aar* oper, i{or s. i i \ev\i1q va\iS
tsir.\^jl{d\ I
eqqe*\ij' $^cd I
A-=A L
Bia\L:i !d\ '.
("( \ A,(> = A (d\d>( la1"r ; - d(d l . ( )
Aj dl ---)
U*A : A d + q \ ! 4 \ i S
Ax - I
(Aa; ). . *
qrlrr^(E ,, r,,^e rt (a; lAai)= A.(r ; ld i )(Aa;) d; . ) = ,45 (* i \a i )
(r i - r j ) (ai la;)r-----v_- q/-
-oL{-[] \^o1! \ {"i\ {/^
R) 6\6\i!utd$so.\a\4i15! ir^)ei
oper;\t. \d;\-o!.hii\i sc\o\{"t{ i[ei\e r {o"\rsor\o1,r!\dUlqL a1tl\adsro
FIZIKA I1I - Kvanturnmechanika - 0R0SZ L. Fizika Tanszdk
ou. I \ t r 'J \ i1AL-[ \ , l lqgwe\ \ r (ud*{
fd^,a., <r, - .- , n., .- . . l| 4 t"o i=i
(ai1a;;=si=lo u^ i . r j
It\.cdYi!aL! ltjgc+\e\€\-*
A bL (1.5e"a*5){.r. rr5 }S!!
t o - - 5 q : d :
A {ir D\he!+ti lA * b,i\ts {Ggg\,dqlcL !\o,hd.
A\LrLrdL \.ll:5;- ."' ,
{ q: = l6L$lex staq\l
= HiLSEET r.i.
vYeK
A )t "55
K i. . Go <{,i)) d i t .
.riueqdi\l E\,\.ri.re\ri ltr(!) vi er Ar er
[6 \. e\ l.€t \{d r\ cL ,,
(qt.t> = (f.,", (a i l d j )
6iJ
Aa qi sLa\dv to u^v o,^e*raL \i^e5\dtdtoad\'\ :, c o
(ar\{) = (<r \Z a1a,) - Z oj (3 i l " t j ) = " i +, i i . {
lio,a,\ =ii "1",\ i a . / i j i
! o -
>- q:{, -r oi -- (4i\\Q>
t b . l r i J s i - \ r , rY . /
A X (.\r1s.a*1)\t. \a\ro' "\a'*"r*" (A,t'e'"' ) oi'e"'i\oa
sd1i,\$1.r.,i*tr r.q.^dsere \r\\'.
Aui: n;di - - )
B B; = B; [3; ---)I - t -
.T =c .Y - - - -
R, t4' l< ' ' " 1
1:5s.1-< \ | - r -\
B.i\i! \"o\s>tsv \-.icl iL _--_+ g d\viys\
Ao:rur:q"portci6 orve" i Y= c,9.+ c.f"-
t l ' t
Iekt5a9.r cl,ltc\no+oL
<4 i\'? )
FIZIKA III - Kvanturnmechanika - 0R0SZ L. Fizika Tansz€k
, t .2-.2.(6<vevetdciaL)
i=x'I - r ! -l 'x- J 2x
P* I r'r) '<t i (6- r) -+ [ fi,i - ii,l \ =9
LF,.' i l id6\'s
ALl . \< i taq r I - - -
fList= r't-e 'i \| \L\aq"!eto' )
L6,,il,r .\ | 3- x,o\?x
Y+x?qn,
LPrrxl
LPr ' ) I
LPr, l J
ij
!I
J1a
LP",61l = Iopeq;.toeoL $SSUi!
LPl , Pr I =u. ^ a " \ - &LPt rPr l - r "
I X . q I - p
r , i i1-er i i 1 =o'
. ^ ^ 1 - , 1LP") I -r
; stb
FIZIKA IIl - Kvantummechanika - 0R0sz L' Flzika Tanszek
A {e\csevi\e\i !6t'i"1eL L-(ieveYddd;L)(|^€w\ dela.^e\-d|{ esq{b{\) Last (!o\dta'
ts < s cl ti\ $ g5" ir'1 e\r
14,t1 = o Aa,=t4'
0$af = r;04;i ltar) = ri.(6"ri)
q- h/J
c ' 4 i c ' d i (\t"tt, \*55 le3'1r*)
B, F,0a;=cd; <-e BF.=I
AlL.\.i\o.,- teldt
i oLqo!' soit d{4'';95 !'q5
Ar ri,\\c\o\ or-.. sU \o' d.,^r.A
oo' L0- ,$ I =o _--., LAU),ts
A 4ri4
0 p,ttl
r'?,1 -o
= Ar4iu) ' f
= si Ptt l) J
A.{, =AiSi I6q,=u,.{. J
{.(1,}) = ,(Lt/) Pitt)
tcr."*.: so\d{.[rqg"i.:cL
? ( i _ ^ o \ t .lD , w. tx \ l=o - - ->
t
-fi r'
lA, { tA;l
A l = A : r { :. t ,
lA o ( r = A i Aq i .^ , I
A-d t = A d
i^ I r . ,A d i = n ! ^ L
:L!
Z o -A*d i = Z q " ' A . '$tAl =z o{ i
4 tl) ai= 4 {.Ai')' o(i:,*-^2q .,#
n Fi = B; fai.
FIZIKA III - Kvantunmechanika - 0R0SZ L' Fizika Tanszdk
^.2..1. A kvq,^\vr^ r-rc\,.ori\roi h^{vas.\\-i\rt a\oPio'i.
"t lF *) .haiLqi it,t lt \' dr
\1-{!\{- a
(C't - lLVf)\ . ' - q)
- rurid a lqr
H€ n-s:s --) j e\-a\is . '' neu.b ni qeu-"
b{veu{e\is )
" trEN DSaEf("
pq: eq5 e\eLt o\o.uet\iL di \\o$.tbd( vat^
gonr.r l"t\t&
\ ."r* = f"l - ;
Il . (E)r=E
I ara:r 1o
<.)^{ = J_bo
I Ar-r,r$
' f r r.\ " { \ -dx - - l / ' i t J , (
,)
-- Ha=Et --+
( . . * ^: I " { { "1I
o { ^.J
E\ t " td{J
!
t*rit "rx = (*\i\"t>
|ls Si -t AS
(A).? = (q\A\e>
!AL ls r i !LsdG
FIZIKA III - Kvantumrechanika - 0R0SZ L. Fizika Tansz6k
iI.
A kv cr..l v rq u^ cc\or^iLq, oxti\^.{i (= lo osr tv\ ri t v r^"si )
393S1:,L. oxri*o, . , ni\^deq tr\rtsi lcs (etrt{^*\. i l \rpo{!sr. ^ 1t u$\{ '{ Lt.
eg5 <\twi{ vau"rdii,\ -
lii
1T Mi\^.re\^ di\^q\aiLni v,i\\o\i,l^sl rqj o\^odiv(5,;\l+ op{vi\oYtv{q. lat i ; \ "r i . Ar . \ .p v, i \ {sr iULs\ ( ("e\ l {S ( \ \ l ' "11!c l {ohv!!-
t^ru(aLr. . . ) vru" lc\ l . or,arv; . lovo\ \ r i Lr\ \ , Vo11 e\ i l i ls i \^o.r \ r . \^. h{ i \ { . \ \ {$5- ic l{ (setev{ l . i t i i \ .* t .
Bdvqr\5 diudhi\.qi u d\Isa\ Lb\ ttuor.eR opqv,i{ov.sqiiudiLci .. diuoui\ai vi\{"rS H€Y\hErb i.!iLil\6.did\
) q\( \
f ,g;= +; t i - : l+n,r . ,+. . . . " , r r . -^, I
c*xii ruo : E9\ cli(a,\qiLqi vd,\{oai r-ivise\s* d r-ivd!vA!.HAra Eur€tLE- | l".\si.\alaE te\eL{h-\ A\q\a!tbd\tiit\i\t \*i*i\ ese\tr- l o+ o,\dr\i sLa\<iv !\5v{($o\\^d{d"! r.\*{i \u-eq i
tf, lF y7
l';Sfi l\^{\4 st..ci!!^iri,.,s) d\\dy}\\o\.dt
l ; t: i seHu6oiNGEvr e9,1ru\r! \^.{dv"\".Ar iat\t\or. idi\ l\
HY= -+"tt-
ld \" ,po{oL j - - 1a,p,y, . . . ,x , . . . I = 1L
t d t ! ^oh i l ( o . i va l t r r i L ! + i A , t s \ . . . , H \ . . .
lp r . r \=+, - - .A
F=F(Px, t r . . . )
l lt. F yi
lil 1r\. r.tr
v. H1|
FIZIKA TII - Kvantunrmechanika - 0R052 L. Flzika Tansz€k
H€Yrds c\
1^0,"n, " sl}lT
hv qq.{.v r-q e c\a qqiLc\o.,r
AILAIVIBAU \
;
Aa, =Ai 4 i (l l t. Arri\^^)
A i
urives (r".era^ sdjd+ d,\\-po{h.'q )
t-.,/*t(r)r
c'{=
(A)."
= <\lAtr)= Ai 4 i t i= r , r ,3 , . - - ) q
l<.,a,, a.,-.- ."r,,...]\ar
E ot <t ---- <\\,?) =>{
\o,.lt = 4
\i "iai)= <qtAtv) = (i "."4. \A=?i"'", ("'n?ao oo Ai { i
!o- L= Z Z .t e1 A; (dr\a;)
L j -:'-
. t -lq i l . A i
drj
,Letr-'re lnetes <-
- c\{\ai \Mdvis
Z lqt l -
zi.=t
- l
\o i l t .A l
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
faEh.ds a N d-lo t\c\-r v^evisJ
lA , , A . , A . , ' . . , A i , . - -( [ \ . A x i A k n )
\N
L lq . \ 'A i
rN il^ F= l " l \ ' = ? [r i )
u"iva! !\$U ----. e31 r-\c\Ai \,.^ivis
(\\. A,.ib\^^d\w1l q ; . \
_ l_I. L
(q id i ) \ ->, .z
T(j+.r;;= \(.elai>1
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
kevd iS , .
lV. A xi i \ed :
fdbq\ardso.-
<x>* = {tr i i .s) +> li lt a.* BoYrN
T-l ^lerel t
_-{_ 7 xvt
f--_l +. l
LTP I--a--a------a-,- 4- a
,A1 Ar Al Ai
o l i ia Uvd+ speLtvwq
(A) { = <q \A l f )A 3 \ar. 's u^qq\qlA\.o1dsq 1
.A*1 = n'-4.Yisrte!<se.^ Li\"" d r
f \ , x i L x \ = A r ' d i l x )
/ \
"" ' . .1 i . j vd\ lsrs
! : l l i ( r ) \" { = 4
Ylx\ =2L oi di t^)i=1
(A)^ =2 io i \ -n; -_
=Y 1(a. l .p) \ A:i - I
- ' t "
Q TAI
( - * . ' . < * - )
'{o\5{su5 spe\<{"vu
(x)r = (T\x\f)
i7 = xl 7
iaal:x)]1'1x.X (x,x.)= 1\ ) tx,x ' )
, ,", ."J \ o..."."\ to
no
3:{rr^,;11 ._ x , _ \ a
IY(4 =l q(x.) 1(x,x\) a*\
qx2^ = f 1-1rr \' x' "rr. =rJ
= i l(rt,i l lv,"r)l x\"r,.\-: ' ---
"'-- ---
7 (x, >a)
1.2..+. A Loor oti v' <ilo. is o1 irn^s.1au. kvcq\v t^^t-e e!. qt" { \. a,i
FTZIKA 111 - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
f\eg lcell o\"\-ui
x I LX)x ) :
x' ] (r , x ') -
(x-x ' ) . ) tx , i ) =
I l7t, .r ' ) lJ '
i saii{e.'-l..if
y' . ) (x,x ' )
x' . ), (x, x')o1
I&(=L J
(DiY\Ae)
( ' r (x,r ' ) lv( i ) )f *= t:r.i
(x \,(') { (A dxf_= j d ( r - r
)
.).1 txl "i1
- {(a)
rtx) = l<1tx,r ' t lvt*t)\ = l{ k)l-
- - - - - - - - -#x
?(*')ax' : \v r*'l1t ar.'9-a-
Av. \at - q va\ \s{ ! .J s ige\ \^o95
o r'iste<s t{ q X t\e\)\-oorc\i\dfo\i.\^o\\
o. u"i'dsi <v.e d. \^^a.\'t 4 (x'r x'+&')to.t"r^dt^fa esi.\c.
(eo w r.r )
,to!q\ i q. viste<s l-t (x\, x'+a"') ls-{sqd \bcrv. uq\^,
a ix,lr) J1x-r')
FIZIKA lI l - Kvantunmechanika - 0R052 L. Fizika Tansz6k
P,\i 2.?j 2x
? ri,2 l
= Pr. {
*P; Y
= r il LpI L''
-+ 9k\ = A e-
(r55.*rs3
L6'r0,tp.
s q;{{.r31!{\t
sao ; , f ds i " .6 .
s-\r i \ i" \e'-
es.\ i . . is ,)
r lt* I =o
I =u- t _^l
-w
sc^ii\\i91,.er"1ctr (
I r ! , x i q r?
!qqv\{al- a\-,pti\)
siL\" \ow. qasdJ.i{+! 33 ! e\-.1(
-+ Ae
( d{ ts[ocLiE)i upv\r"s speudtos
(= sdic i \d \ \d,eolq )
il,.8 =Prq =
P-E =
P*qP:Q
P*Q
Il:3e!1
-\e t i* ls;\- P' opevd{{r \" r
sq\.^{ \ ' .15 ! Q.\^Y c t
P*9^ =PrYr
0* {,= P.Y.
I t<r- i{s^ Li : ({. \{.) sL-\,ir \as*l_\nt \
)dr r ;r = ,r . o:r . -
FIZIKA lII - Kvantlrmmechanika - 0R0SZ L. Fizika Tansz€k
A-2. .5 . A h qtt i r otc{\ <rqsaqi }<\d( io
Nl
N
(B)
B,B,
5 g:1
B p, =
9+
Adi= 4 4
q \ ldi . l
AB
A- <o>ts-(B) AB
AA
Ats1
( . {
(cAA'
Ats
t92
\.p ) J AB
I AA I AR I - *4
D' ( rd -.r,-r)
h d{e\u q l iU a :
I g = q .1 ,1 + '
I ^ +
lG=1aA- jL "
(*'r \ix) =I te) : (q lG G
" . *
G'G
AB
AB
(
(
X \d\i,s \ad,\* )
( " " ."*-
cra! i ! i . i i , \c , , '^ i" t | )lq):o
Y)
= [? aA - j
= ? ' ^A- +
rr )(xar +j aB ) =
1n ( ln oe -AB A ) + aB---:
" Y:- .1rl AA , aB i = \ D- d
1(2) z4
(f \^A-t{> ( r0 I AB-l\p ) - q ( Lp \D lY )d '
(r>
(r)t
FIZIKA III - Kvantummechenika - 0R0SZ L. Fizika Tansz€k
I (g) = f of -?(D + ag' >o
5 (q"\poYabo\a - -+ Di :aLr iu iqc i \ r < O
( l) ' - * aAt a t3'
a,At a B'
AA.AB
:o
> | <D'_r
: i \'"> \
?e \ . l c . :
f l r iseq\rcv1 l i \e
i ' l, f 1A ,i1= 16'-B= r I
U.a!d\-\ d{\o.\sa\i ve\a cti'
tr =:
Ag*-ax : L
af r r "aP l :o
nI , I
--i\>..=-- , o"
d\a{ \ <se\<\ I AP* 'A { <: -L
E\r!.\ \ ,^ea<s .. tr i n is a\d\5\. i \ :
tE55t dr\! \r" eRs
\ t t ' rer r .?nst / ic i a tc ' , " \."u\s !q"qe'r.r-{ \'
I k\ass]il.."r sreu\d\et )q \Aevas \,\^t(\ \e qv ^\Loa.i5
r-eui w.\srw eqj \^^qusslt tFiLvs es\L;\
v.!!.u ic..^*!k"",L (BELL 'rs 6 8)
1 l.*lti,i lr)l- tA A . A ts :
t t 1(J
FTZIKA III - Kv€ntumrnechanikaOROSZL. Fizi l<a Tanszdk
4. 2. 6. A |<\ogsrilcvs \^ cc\o,\^il(o. {s q Lv a*{vr-q^ a c\^q\^tt \
4.2. 6. 4. E l-ll(E NFE ST tiLr-\
M;ris he!^ src dcioh(ivivs $:t jg: )
d\\o1oorl.ow
, 1
] t tc,r t I
N.,N
p
AAt{ ) t^.q {qv\o\qea1q
I i"r,i (po6*irr"{)
. l , ". . Jd t ' ' t r * t o l t
' ' Y l + )
(t 1a 1"12
f * ^l1 t AY J.r /
= (^P 1A V )
--?
+ (tlAt)
Ina ,;
I H+ =-r" |t - \ -1"I " * + : *I HY=++Y- A
A.r;diLj \^o9: :
+ =,4 H+* t
i ^ *" t = +g l rY
--1*".*.,
-9-tq) = +1d+
' t t ! ) L
=in
=tt,
(t l Ail t)
A 0t)(t
<r
HA^t \ -g ( t \
(HA-Au\^ f \' ' - /
. . . \I H r A I \ k a \ * d c l o h )
EHa€NFEST te. le(A \1i,At t )ol (r(A),rer =
FIZIKA 11I - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
A,1 .6.2-. A [\ossrile,:s rr^t.\,. ct^i \< o. rr^s\ g.i s al5.ste\ e.
(Na.r, lo\ l l . ar i iwa)
( . { d i q ) Ht=-: r ,)
^ad_i+"EhqENtesT - [ . i {e le.
#(r") = + (rltA,6"lr)
li,i,,lt = L+, 0,1"r *\-/-
p
f ^ ^ l
lv'P, lt = +I
(v.ot _ ? v.t)\ ar ax l
-v2e -+42x
-dx
d <r",;= (t\l-#)lt)
[ i t {eie l : r"e\ isdE v o." 1
V(,() 2V' ?X
I)
. . 2I t(r) l J.x
T
(x>f av lL-tl.,o
ft(p") r ov \\ 2x , / a (p"> 3 ' I lt\ "t{I
1-s c Htrc i b iND€F.
€5t{u\.{6i\
[v lut v trtrEcurr,itirI(ELL hosa!,,ilhi
*<t'"> I a"1l- t l<o
i (LAssl i \ , . !s HEc\{A\r i l (ATI E !r F1- a\ko\qaaq\
l1|la
2 . )
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
{s3.\our ki.r-r."1A s i.r. -l- v.aoqio"Ala{a:
I
"A- Pr--r-dt
(P") : p'.
-+ i-+lP'
hlossriLus+!i \aeqF\!t
F95 \<\ass1;Lu. lu^.r .""s+r."pi \<. : ) r- iv is v-.udi3, hdglu ,
qa35 sldr^^J
k v o - { w q q { < L \ d q : r ( o i e \ c \ ' ; u ^ i v i S s o r - . o l . i 5 o . ( N - { a - - } .
A k\as3liL's *"" i tnl,sr". e+e.^ e\e\^i r^^ivisaL ci\\aral' , e \qq " l i L^^eq o u^ i r \ dd "+ l .a \ t
(t ai*1 1 l * = - :Yt \ '
Px + l (x \Zw1
ET\Y!ENFES.\ Ii\e\ 1
j* +> =f, <^rltil,itt)
1r,ilt = L*,;, lY +
r \ / ^ a= - - I - l - 9 - x ^ t
z\a \ zxa
LV(r\\x. l ^{
9.,-^Ctl=-
- -* l*,- l*\ - l , Y + xt -x^ l
</> 1 (-*) . (t\c) ,{ i ( ^v l 0Y \i \ - ' 2 ^ ? /
t (r i6, t>
A l::!5l:\! l- \<r.sseiL"s
lrtL-dt
2 V2 t ] ," ,
. fi <c,) =f;{u
*Y' =="(srir.'. al=s+{,rrr.^)
* " ' = *(t")
FIZIKA ITI - Kvantummechanika - 0R0SZ L. Fizika Tanszek
,r . t . 6.3. es q:r ido l(;+6$i \^ *t dvuro{lousdqi
ve\,ici.1
Es9
EHfIENTEST
J { . -
AA .AE .
A>
('
- (
,l
A
_t_
^L
1t\ t r1,Hl t ) =
.*\t0,41176\a \ \o \sd3 i t e \o '
l {^I \Lr, \A I t>
o
cii r 1l
<E> = d.\\ c{d.d
-+ AA. AE :I
- l - : -? - t t
L
d
A>_{
A ve,.,r4 s1qe5q\\ lero\\i vo\\s+aqoraq+
I e\\{qli ( LqrqL{eti !,r{i\." s \d'S{q.\o.-)
\ 5{q l iLvs k. . \o i lJ
^ E = 6 cLs spcLt'..-
( A !-\dsdg .ti.-.wtl."s)
AE! = \,esa<, + o. s.pa\!{vlta !u-\k\3\eks?di\"
z
I(\AN'T U H E LEIT-TY\\) biN A$IUA
A F . r \ + : :?
D\NA$\V.!S t\sDELL
FIZIKA II l - Kvantummechanika - 0R052 L. Fizika lanszdk
, { ,3. evJ.ii\e+ is s- r^^d. \r. sr- c-\^{v\--
4 . \ . ^ . / \ P qo .pqvd \e t
l<\oss\ i lu s r^ ceuq*i l .a 1
L. .=. . -
L+ - xP: - ?P" \!nt;u ,i\\-\Di+ )
Xvo- tLb uc eL ,dq iLq i
(rr. r'r"** )
Il . e - , e \\ ^? ) t 2 , )
L{+11t
x = r . lna $.c)yr, = h '-l^;- t9 4M' Ya -_ V C$ \5
\ l +
20Q ( x , , 1 , t )
?0 ex-
?x 'Y
-?
= Q(x(r ) , t ( , f ) ) = Q ty)
aQ o* =2, ?9
[1. t,r; '- ') i.Q,,.A '1..rl&0=
LrQ * 6'o .'o t*tli:t"j
Q . Tt .---- rcx-\ati!,is , i".!,T::"i \_{3e;ru-{b€q5iv\iLU
(QlQ>= 1rf zn =r. "--' o= #r- l - , l - -r , .o
' : ' "
i f zrr - " )
u 'Qt.1*. . ,1=Qt ' t ) - - . r t ' '
( r . Axt i |A-)
a-! . ! - = * " = ot ^ ! t !11
FIZIKA lII - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
,{.3.X, A (t6,I t) pd\5o\&sl1d s rhd.qhcses {^^eleq\^{v\,\6.
k loss r iL ,s L^^€cqakiLq ' -
f ( o ,o r L \ )
Mr ( o ,o , M i )
IL_=1 - .n= r . . q . v
f1, = 1. v'rr
_ e v Y . = _L Zu^
l3tF
Mo = -fi; tr'n
|1 . = -S- L-
K!o { t ! qqnec \ ^oq iL q
Qr'. r"r" *-)
H. . f I " .M.
\ .1-o:M_Q
-:- i.6= v.6z h ! I
.A-+Yl'1+ = - : L _
( \ -L- 'o t l r t l3 r . - . )
= MD ( eoun .--n".r"*| "
. "^u;:;:-di f i \qoqck{uu^r(vq*r"kq
)
f r : 9 ,e+ to Ar- ri ,,..YQ = --l- e
lali
f4r = - f B.v^e
I L I lKA r1r KvanTumfoecnantKa - uKU5z L. rLz lKa tanszeK
1.3.3. fa61q;s c rl.tvd\is e.!I iv\q.\^-
,1. j . j .4. A pri\qopqrdir let h"cq\^atdvo\is\
K\<rssli lls r,.^c"\..o,*iLa I
F(v) - l -> V( r )=v(v)1-
I ps,
\ / rv \ ' l Q ' t - t )
\1t t o t1
L =rx i la' ' ct+
"L!--d+=T:F+FxF =ixF
\ - J 1 . . - - a - !
/ +*s-{su).u-{elq
p
l - . , . I' t - |
l { .q+etva{iv ve.^asrer\e\^ F=d\ I tlY iva\ r - = or \0\6ro - ->
- o\iv ^ v - x v = t 4 - \ . r x - ;
!'.'__\/+
!_ x <rr
l " - H a r H r
L= r rP= rxP !+
(,j9}l5- d\\..r.:),t
r xp r
d
^ 1 ^ r , , - aH . - r - l - v ( Y ) - Y Y l l ' a \ \ ( v l
2 q
u = F" + i - +v( r \ =F' ' z v - z \ - r -\_/+
= v",, (o( La = ,i.\l dq .Li \
L = r .po I-J
- -fi + l.'- + vrO =
(
H=$+v.ptr l
v"k(r) =*. +v(.)
!"rc.ciJ l )
FIZTKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
(eu t rd l i s . "S{a" t . * t^^. !sA vi \rc<ste \^. i \5, i !d\"d\< jet \c!*t i re
E > o
E = O
E < O
+qqt"s a9,1e- \ ' i t teh s{9eL i
t L '
E > Et""LA Lu;,
FIZIKA II l - Kvantummechanika - 0R0SZ L, Fizika Tanszdk
J(:::::::::**::X\dssliL!r la(el"dhi\<"
Hoq i l \oq \n99!e( f
k !dd"qqa(V.qq i \ .q
[,{ qhi\Lh n\aiv;tor
-S A",r- + v (x*,+)
Ii
- * Ar ,o , , t *v ( " )
, r ^L . - .?* P' +
z;;i L 1- vtk)
u=fi(n"'
1(u*.*r" ' )" ( p * . " )
H =.-, ni
+pi *pi ) +
IIII
V (x..tr,a )
+ V(r )
n -
-->H= J"a t,o,.1 = ar- *
*. o o,.t
o"=**("#)^i,{ = * ,,- #i"-**.) . .r x.
a^q!19^ 2{L
LI
= Ar|
o H opetd\ov Fn\d\-L"ovc^i \ ; . {dLL
+ r r " _ | \ , r F \- - t-.* ar -
,|-- rr 'r,n + t' t' 1
I a r - l , ^ 1 , \ / t r ): : - P " + - LZuvl_ ]
{.
$,' = -r.'a.-tlao *;l* fr'
lri,i.1=o II
lL,r-r l-o I""^-lLH,rrl=o J
-->
Ht
r.O ='+9 (.1.r.2)
tr,,\),{\:R(rrY (,},q) =e(") gtn) Qtq)
= y lrr,q)
FIZIKA lII - Kvantummschanika - 0R052 L. Fizika Tenszdk
A pet.r i \ . ! so\, i . farr;
L-Y = L.Y I- t
L.Q:L,O IY (\},.?)
\ M e g o l { d s o , ;
: orq.Q (e)
( i - r '^r +
-t 'Q aoo
A
4,r^49-
eL* lo0 =
f*'Q
LiQ,A
-d'^fo-$n =Lro
o"o.(*-S*";n."\vJ
-l-A * . '
( i d;\.s)
p"li\-o\- \-i, d \aay"e\
-lA 1 + \ - ?- -
( ,* .+\ ( LE GEN\nE\t ' - -
{ ( ( + 1 ) { - e . r , r , \ r . - . \ ' * " r
6 + l + 1 + l - n
u^etiiL t."{*\vq iad,\)
\u..i r t"". {Y t$, . { \ - - ?r ( t - , . } ) . q '
L ! - - t . ' { ( {+^ ) L= ( )1^ ,1 ,3 \ ' - -
L+ = t l i ^ ^ ( ( , { ^<=or t ^ ,+ -1 t - . _ tQ ,
GoH BF\l s€!EN(
A Pet"{ ; lq! \ r" t -pouewseL \anasUi csa€v.\ . iu i ;1. 1
M€nis ( \.. d{,i!{ \au o{1ci t iv.\dcii,L)
L,I
- - tI
_ _ iI
r lLqLtt l
l L,L ' r , - ' l
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
& p o\! opavdn \e+ kvqs{ i \ r i sg t^o l .
L =t r l t ( {+ r )
s ! c q l . i \ { e \ i ! c ;
( r - " = or r : r t . . * - 1 ;
Psr S= l+L =i F-i
r ^ " . = o t41 l l j l $
1
A p"\1o r^,or5 osb"l \^^oqueses \^^sr^ artv\.^ :
M= " \' ' L Z \ q
-
l - - - - - - - :__--e {
ML =lB \ {(s+{.
-IYr=fB\""1
t "o \ r \1 \3 r ._ -q 1 = o \ t ' \ \ t t \ ' ! { -
^.3.3.2. A suq,lv ivd\^\I, (vadid\is\ t^b\qds \ei '"asq.
- l= i 'Y I
)
rE€ r 1 ! ( v ) : v ( v ) ( ,1 . j . r .1 )
HY
lv "! r.,,.', .S) = ntrl . '( 1+, e)
l^
V , {! z\,*
,{
Y
L.t
= !r { t{+i)
vr {+ tF)
Pv+= E.R(*-u*r)]r'[ , .
Lz,^
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszek
L au"[, \" o95 r-h A .
L e (u,Y L a r \ * )
^ != _L- :L / " t JY l \ti1 -\& \ ct 1- ./
\_._/J
zr ?. + v' 'r-
P[r\ = v Rr"\ ----, T
vui T{ = E.R l "
+ v 4 Y r \ - - E r K
=fR+L
A,1 \ +
r . ArR
P'
zri + r.ri'
_ +.1Zk
- -1,z\q
l ?
V"1 t.) = -9- 1 v1";
Lz = Nr d(r+l) { = o r i r l r - . .
ce\t td\ \s "" .{"" \e-
\"^oa3i v€sae(\ \ae
t tF) ="t t , . i , , i ) = Rt"1 . P) ' \1.o.>1. "r* 'q
L = n ! t ( t+ l \\ ^ . ( = o t { l l t - . . I (
\ (r) --,- E rn(Y)
vc)= ]-+{r to
q l-e)I e\.1{-,^ .1 .t"*s--)
Y (rr; -- a r^.,-,!
v(,) . {a
(rau\.\i Lo".*.^it-,so1+{i\\d\!F
)
p o { { ( ( d \ ! ! l e \
o9o"* )FIZIKA III - Kvantummechanika _ 0R0SZ L. Fizika Tansz6k
''-i^
". Pot
"t d$.
2d{
f r 1'a.
"{r t" t
Lr =+i t(t+.r)[ = o , 1 , 1 , j , \ , , - -
I*> lP l d r . l o \ ,p
-J
(.),.?)
Yf lra;o arr
{ i+i\ .or 1e\e.^tase
"r "1"*^; {ev{sqet I
dv = v!,,i-, ') ctLldg dr
)^r t a" = \"t\4"r" lY l'.r.^^ * -ruat
t r )
1"t I "t{
ir
Jrz
r ti? t ' d" - l
vclAsriui)s igg.\ t . \d, \ \ - \A o vis\qcsl .{ qt 1 5r3avt ' dr ! ' t \ \o 'g -
sdgJ gij\4b\^elbqq ---3 ,f,. cR\{vvu{;\ r l,ivo\stt3v.. t,",-
\^i.t 'o1.! '^ s+e"U i".- ( o hia*"qi*. ..\sr*)
( <o,rcro , ." te-cr " i )
t = { ( q L t c { h l e , - - + o q )
^f (",
"^. , lLt r . . t \ l;".- \.*"-- \*,- T/l
+'-------.\.-- -----
I
lPl'or"
^f
,.{) -- R(.) Y. (,},.{)
I G d k h i . q e . e v \ \ e , w ( ). ! - - 4
? = E P t l e s o \ d d s q ip o l i \ s \ a h s d s \ e v v e \
IIII\g
r u , , t \ " ) \ ! A G ! e c a . € )
- - r E ^t r " . = - e - - '
e . . = 1 , 1 , l ) , - . . . 1 $ L . d v ! . q 1 r . o c
L = or.1,113, ir \ . - " ( \^-1)
E
fu
L\at
I r"=.rr""a";..;J
$
FIZIkA III - Kvantunmechanika - 0R0SZ L. Fizika Tansz6k
lelies \aegoto{ds \^idvoqi\s\evu isq ese\€.^ :
!3 ["a*{.k!\a\q
hETLEL v
hAGNESES -
tsalra \a. cr{.q i\
{3.6 ev ( a [ ta"o16.. d\as io,^ i \d, ! i i ,s !q.veidic.)
^ ( krotv"q{\ d{ou. r(,a\t\)
.. S(r\ui b\N CE\'I. *odr\ i!ssa.\a\oLl.\dt"' lA '\"q\
Bourl s(HtI6siN G €\4'
V"4, t"l = -L
<.t tii^Ij < r.t
E . - | 0 - o, . l , r , . rh-r)
_ \ r :or , r )/ul / ^ \
I,I
I
/u."r
?(")
,1
[r.t.a.+oul
t(",q{) = I ?n," G\.Tu'* ' ' t*. .*). "J'- t ' . s-!- h : ,tra3, .-.
t . I e t{+r),-
t =o ra ,1 r ^ . - , ( h -a )
\ , 4 (= o ,1 , { , 4L , . . - ! L
B o\.Yt
h { \ " - ^ )
FIZIKA III - (vantumrnechanika - 0R0SZ L. Fizika Tanszdk
. t . i .1, ' r . ; Af o.tsr"^ol. c\cl.{kot^ s+trt eaetc-
A1 €reLt"o*.I. L-n\itriL6\c! '6h\*t c, \{ el \^o\( j i t i " l -
Y ( t - i ) H lkv i ioh A\avo\
t_
E-,
Hef --> h = j
t=oP
?q : I'ro, , (ir)' tzof 1.9;'(3.) (o =na)
t . r8
?-19
H q
N q
k
( .1.) '
( r+11 ( )a f [1 . )G
(rr1' lrrJ [zp; c (ra1'1:p1c
LA"l( \ .)r . i r - . . l a r . . ? , . (L A r l ( < r J \ L t a ) \ $ F )
-+ (s cr h.\\48
IJ
o+ 4\41{sLoL
L6\6i L"\a\d\
\ )
.Ini iv l i \ \aq
" . .<qcs1d\ .L
cleLtvo<,s,1qv \rer-\e- (1)
Ua\a
.Y
L
I' t
= '11r," (") l-"t(...r1 .x-"{
t =o \1 \? r - . - L ! . - 1 )
r^^r = or-!1 ,+-t \ - -- t L
@-1
- 1
(2e+. ) J r
FIZIKA III - Kvanturnmechanika - 0R0SZ L. Fizika Tanszdk
l . r Y { ,
I / , I
ti / J; . , / (s F\ LGI
----------------J\-- - - (\ r) irol/ \ l c ^ \ l a i
/ / (a?)/ /
/ /, . . , . , . . . . . . t , { , + p ) [ 6 ]t ' l \ l \ \ r )) . |+a)(+i \ / _ __-1-' - t3" r ) L ,o l
Lnrl ) (e 1= 5 \
[',s] l ( " 1= 36
)/-_ ('+)) Lrl
(K)
(l ro) LGIh = j
( i \ ) l r i r )(3a) z/r_ (3a) L2 l LBI Ar A =19
(2e) LGI(ar) il\c) l t l Ne l=to
(u.) Lzl
! '1^) (.1^) Lrl I r"r, f
t( \ { \ .€r \a5es
aa q\ \^e iq\ . Iq \s \ \ i \ i
\ e\{\}!.q!+.\\^^)
\!rveqdiet !ua.lqd.s
4
3
+
1
FIZIKA I11 - KvantLrmmcchanika - 0R0SZ L. Fizika Tansz6k
, ,1- j , i .5. A+ o\\o1,s{i,9g"ir"1c\ ( o\sr^^po\5.iL) g.o$lt vs
{ =ou " =O
t ^
\ ?. (u++) = i'tte.ai "fn,"," = ft",. t')
i aq \ t { ' kqa \v iL" r
\ l "Y I
IL
31 ! o,3
t=1J
t (.f,f)Pn" cor$.
A^"^f.
^?,.,a,_r-
a'' '4,-L
Q"",n (*). r^"" r)
?. ",^
(") c.rJ'
9.',^ (") ,.-.:"
- J ' (e
+i Ye '
\ c i t { v i . , l . . s g g :L - - h a 1 } r
L. 'o = L. -Q+ r !'^.\ I r
Q=e' I_ I
Ll = t',^n.' Ic J
\i\e.;.'is I^.\-\i\.i (i" ! "\ ( *.':f
".'to-)
( .1 , { r )q I L . 1L"- , n J
--- . l r \uq,
c ' i \ { )
( re \q " r (+ :6 i \ .q \ , i \ )
FIZIKA 1I1 - Kvantummechanika - 0R052 L. Fizika Tanszdk
9u,,,-r = 'D,n,n t') l.;^ rl . .o- g
-- fl,n,n [') .,-' ri'
-- rtt",^ (") -|,i^J ^i^^ q
!o\ .,b ( q . qbt) L" . " "\i
(il o\r-
x -- Y )L;1> c$y
T= \ ' { } ,x.^-{+ = k C,s) 1J\
+
+n{r,n,- , - P^ atoioo{
\,.,*, o P1 '\\ '""tqt,n,o + P1 o\ l 'P' t
x
i"
3
^1, - T)
^ } - 0
FIZIKA III - KvantumrEchanika - 0R0SZ L. Fizika Tanszdk
,t. \..1. |'l. .1.
ii ts \^^ o t< t""dq\(sQS tirbcw
ffi d!\i,\
l,(\qssti los \^^ec[^qqi\aq :
_:_L
B t g'q-t\ r^r"*"tiL)
CLL
fl,- oa. dsr, ibi \ : c lL
.^ \^a1jdsag\{q\a{b\t : dL
L rq^ $ <,r9l ' 1 , . 8 . )1 , - J , . \ t
L .q^.$
c\ pve <es31i i sa igsc\<ss ege: ;;
= HL . l3 ' ) i , -O J t
M , . t 3L -
L AQnon- L;," feeL!e\^c{q . , ets2 r,!
Kv o.{t \,\q\*e <\quiL d :
B -ot - -
lBI
IiI
I
II
IIIIIIIi
t,f' r .J
1++ r$rlr
It lH
H.
B (o, o, B+)
IZr'\
+ v (r): -M r ) - +;=
l-l ^! -- t ".1.
t' tv. , \p
{1.e. e'
FIZIKA TII - l(vantummechanika - 0R0SZ L. Fizika Tansz6k
t ^
[H" + u,. L1 ) \*
Hs*, + \r! Lr t-
F^+ |- n r v ! I,t
, *
E . * l -
^ i l - uR.e. jq er
' ' = ,-.i r-. ' ^f,--\/- I u.-{
tw . , . ea _ '
te- -- E{ +q(nL \^^r ( u^ . - o , I 1 , ! r - , . . . !a )
A l"i,hb \^^oqr^<res
kil ir \r l i { [r;vra\ :
B= o
!€v t^qtriscivc. q! *.ev9iq*i!"
lcL.
Er
| ' , - l l l ,,/ \
- ( lee t reu ,{ t{aLtls .}
Ju^as e{{rLtosot.
\e\\dpueL,
c!rr\ +uL
A vo.l.\s(tq\ou q \"etgr-t hou.5ot"\!c\b, r-ctt r.ui1
1.+. Z. A+ e\(\L{to \^ sq\d+ \^'\dq\^'ses '^^' !' th{v\^^ q'
ts qa <\q!\voh!y,ie\
?. . \ ] AX
luf 1r.sf 1"r';
als\a e\e L{h\s1er \.e+c\t :
t (.o)' (. pf 1r",.1t" 1*4' lopf 1++\t"{ ('}{) (r-r; """..:t
t (s,,) (t-o) L^". = o
(v(q\ {Y\ .L e\ r l . rvsq)
*^.=** .1** = f*'hu = 1i.,.n,.r,..;.1"1/
(d{.6$vrs)
, " : :
otsr*\6vr.s I lerov\ \ { \4L)
qA o.!s!'\ \qd5\^e\es r'^ruq\\"- o( s.)
FIZIKA lII - Kvantummechanika - 0R0SZ L, Fizika Tansz6k
A w+45 €"\e\\ !e1a\. i o\ e\eLtYolr^dL( o w r r q , i s o { " \ { , q q c { l e u u \ ) ' . - . 9 5
s o l o t h ^ d q q R \ e S \ q ! u e l ^ 1 v \ & o \ .
\ , o ^ sc , i , i \ Pe ta t ^ l e te i s :
).1
{ i \ e
kat
Mr.-*Lr --+=--1- L t Z q
f'4., = t pro I r.a-' l - i -+ "sl = _ e
( - 1 ^ l ( \ 4,r = r z .J -+
Ar e\eLt"o,^ sr,"tr^ie es so]. i \ w.o1t r:e:{ \ ,1\o. \q\"^ v-a9qqd t \-{A Seuqi l iL\".ssti\.v s Jir^ o rqi-Gvo,t
.1.Spix s\1o< ct , ivrqrq i lcq i vn\{6ai ,
q6
\ \ \^ ( 1eq _ Lrusst\\("1 \.^^es{e\a\8is
( A'\",-^ ,- --> qi! L{[ !.*.r 1 ).t
e\wiieti uo"ra! le\.)
A! Ag a{s\q L).\i\
',
A luivis e"ed\M;\^.te I
MA$ _-- t lB
M.*=lfs
A+ q\eLtvs\^ !^o i .
( " . , " , . p i ^ )
s+
I
F l N s r E i N - D E h A A \
: I E !,r.1-
GEh.LAC H
lct ! iY let
(ovet \<e+!-$s
FIZIKA III - Kventummechaniko 0R0SZ L. Fizika Tanszdk
GovDSt\ iT - vHl le i , r tE cK
?d\tqq"+tastAi "Ji "to L , i . s"!dl (\"\\5 ) EFs
^&J
L. - t - t t t+a )
- , + 0
^ l- l
I
= - *. g" ' r .
9e=1= -*. x. i*"
I
I
II
l -a . , - , - . +5g'-'-\-.+
ta^+r ) J to
*f ".\,.*.
e\qe\(t i \ascr{ll
sTE h-N - C.E\!!Ncq
d \ . S .
- : ; Xr ' )+
-* x, l*.l_-_-w-
- f s X. 'h. '
sTE\r|J - cE|\!ACu (tf B)
Ei r.Js.rEiN- DE uons (r !)
\e. u.)
MM1
M,
M.
(: f.)vr*=tfsa.+t = -t_
* 9r '^r = + 1 t= 2I
z -r '^ . .5=ri -+ sr: T 5
I*:*::*):5"--5:*"r. ;
I[ ; ,
---'- Ms
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
! )
lsq\! \ . l "asa\"; \qi )
d s\.i.^ \^ogtsd3a ur* ,^fvl"ef[ )
( {o '*n\ is so1<i{ iv l iL eelr- l<+ )
,^)( , =at -
---5 qlq\ I *. = fL 1-r = t'l
L l
( r&+ s"\.,{'\',rj1,i '^js;\ .i\i
I I u,q, spi \ v i" l \$aa
X [!) u.". sPi'. \ 'r33r'iutr
I
d. tl)
r}r 9r---r-+>
+ l
539 i!
A spir"- Fdl\o. ldi\tsii r^\^a{ds
x (1)
-h
s.)
S.o(=+d
s.F -- iF
Tovq. i \ i !qq b<r,e.retr . t l$
$
L
Oa\y<1r
,{ . \ .3,
S=
s. 'Y
o
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
a\e\.tro ^ - o\\qeotsL
no . V ( \ . \ - +eu 4 ( r ' uoss r " " )* f ( " r
F35 e\eL{v"."c\\oe.t ( ' , .*. r seiN-riLYA)
,|
rh (r r \ +- ch , (t,1)' 1 - r r , o , u . , + 1 , ' \ 5 ' ' - I h . r r h . r - !
Aa "a\.L\t!\^ {e\\^S ^
{iqq qf e\e\''ts\^ !\'i\\'\;\
.t,seiN_o^LYA Li l<\ ' i ,uuq{d!
V s L - - N L . l . 1 s - - L S
\ r , I )
\ . i \cs i* \^ \dr \ e \ \^ou\d.q5\ \ !k
^x (1)
d t1)F 11)
legfet\ e[b all e\t\rtroq \Pi"ct
/ t f 'no'll*rn
Al e\V-i ve!!qr" l \ev.
Q,',.,*.,*. (-c \ s F \ h - P d ' \<
'1)=Y- (F).
.-__.v--
t 'o \ \ aq \ o p o !
Id\L- \dbc! .
Y (F)
3l*!-:!t 1E95 st? ih- \ .d t \q d{qP"{ \o*
A{s*c( c\{L+s!\s+cyk<\e{<
I e \ q d 6 { i I A S . s i r T )
Y.r,--rr1|
^ V t -
V." = - " - rS
rl t t tI/ t l
r IZ I ^A I I I - ( van rJn rechanr^a - 0R052 L . f i z r "a l anszdk
4"
I .
SoL ( qr".^os\ visrecs\.d bb\ o\\d vc!.ct:+ev-
l(\oss1i\(!s (^^<<\nq\^iLc. I
L vi 1 Pt I i=,
(. tcp urrurasi: i j<h)
A ftqrai l \oq +iqgv;uj :N r t r f {
H=>[l--]:',\.+>rE\2 l . .
,nTr " ! . i / . T r j * n' ," 1r; - v;.\
Lia.\Ri lLn\.si,q-.t .q\dr (co"l ! \a\)!i:l:::^."irssrys:
( l t . A i i . u . a : ) \ x . , . * t , } i , F r . ,
1, Ixt' t' ?
J aYi
n N . + 1 - a - l| ' \ - \ l"=31-"^Tt-\-1.", l-
.1,2r )\ ?12x,. )c1 )r 1
P1i , t ' r i J1=,
N N
qr ' r " \ \ - iJ \
A^!
( sp i r - p , i t5c \^ ; \ ( ) ;qL.q!d\ te\ \^d\ iqSsl j !L )
(11r. Axihu"\ A v(qor\ \ev ; \sr{qe"9,".1q
i rY =EY --J Y e,E,. . . . ,FN )+ t (q) ' \ ( i , ) . . - q*r \ )
A c ? 1 o \ o ( c i ! c . t ^ ^ a { o q q \ i L - i \ o q q d * o \ L l u t o l - r t
.t,I r ' i tEL\TdsE\L L. \ \ { q a (
.L€9! - vi\\.cst.e |(i,1e\i{€-r
cil\<y,f{!3qvir^3 {i\iLqi !.*{<\q". l+
FL I rt to, ,o., , "., , n,) \Lav, dv\ c,v! . - . d!i,
4. A H ap{,vd\or^ is qr d,, 'lln\/'ot'\ w qq!et1q
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
t .5.2 A I{A|\rrl.E€ Lo\c\i{is rs o+ S(+ \abd$eF
; y'-.-r-''
/ -)tEd--v,
+ 2 e -
Hq o t " . * ( t = r , u -z )
(sqL q pd\\q- \\o,trn\s[L^\ {!g\.\\.or"\t .A spiul- 1o.ilj c. L;\<!6u\"qt.i\l e\\" ocja3 ol i"l. .
- l
9.n1. \r , -F1l\_.-vr
Lircsi:u!.-\ri s
a V,t (Yr-)
H = L-u,^ a,
H t [ t ,q) =E
Dvw- Li :e\ i l " l :
d L ! \< \ ' i6 \ - tds t { ! * {es
H -w i \ \oq oer "d fouo. )
c\{t{v.q v e q.r\ar
"v.t.,r l * I-$ ^.-!.(Y,)l +t-
- t
2 e r
6i1o\^tl\c'r-)
+i (Fi I' f
\ (FJ I
t(r.;L-) ----5 tqsso\d\d\ai .)91ti5_!.iJ$'Jl
A+ {e\{rcqoL \(s}6\ i \<(\<r i*q\d\\e\U.q\qlo\rJ!\ : \x,/ (rlr) =tl
u. t (..,r1) = p. t (a, '-l' ')Sr{poe;\\^-{ i ! .aqo\d.i \ I Y tr i ,r,) = *. tn\. ^f j (. !)
jal;\is\ u^{ q a\\ o.V. o\o \ I
"tt tr'.l7 \ ar e\oLlvor- -\.*.tllii""
o+ d,\\q\r bf+.ss! a\als[!\i\i\q
AaicriL q L:n'.+\.iL (E{\.d-t :
\ * - : - - A^ +V- ( r . ) \ + . (aJ =€ .L Z s I t
[- l ' l- +v-(v.]l ^f, (-Y, \ = erL z h . - I J - 4
l-$ o* \r.(F\ I "t (F\ = t ^+(i)
l r { ! { i ;qqc\t( \
tl.\cq-{iLqt otoLi'l
Yro!\!\o\ tr1,jd{o\5o.(!
JHdf!\{d{i\^dil(.J e+
e\."ivd\eqs c\\ <1\e(c\e\
{ 3
€ L ( v {E \ * ( i \
FIZIKA III - KvantLrmmechanika - 0R0SZ L. Fizika Tanszdk
A I:S:s: k61e\t\is
r .^a1o\dq\d6 q :
tr
t-r.,n ) ' q (F,,t,\ =
= ( " | (F . . i , \ l H l ^+
z
.t
E
9; (r.) Yl (t,)
(n,nJ ) --+ Eu.."!
"H t ( r - . ,Y.) = E t (q,o,)
E = ("Y (. \ , i ' ) i l r I t tn ," ' 1 ; =
!n\ci;\\"-!d\t
L
u1 (F^)] qi (Fr) = ?. f . ru) 1
ur(r.)]qi(FJ - ir ,r , to') J
A\ + \c Lt\) +
AL ! v. ( t1) +
L U q
l - : : -
) \
+i
(i,
" * l
L l ! ,
6t i
' l )
l.?i
w( \
e'
\!-
\".
; \
a
TI
J
i L
('
nE
)
Y!'I
tE
.t,
, t1L\,\r r" \r. -fr\
dv.
€\aLtro\ \^ ,ut { i , r f . l \ " fG )(?r'
:-
qa
z\^
' l e
(?r'
+.!i-
qa
Z\^
9;
t-L
(n)
(i,)
( . . )
(r. )
] ' '1,
l*,J9j (FJ I Jxr
1 q;.teo\ Jtur
cso{ . \ { i \ \qSye-di .$. \ "<qcu\ <$. \ \c{ veurr \1{u.
Sc= (S. t4 co\ \ t<\ T i . \e) s .9.1adloLet \ .cv. r i !^ t<.
Heqolaas \^dr l \av( ' . i { { !dd; (H.. Louv(vqd' l ' " soLqs\ NEn)
. 4 i
'l_ i
,.p, (r,)(') --+ {, (?,f"\
.i jrJ"'-- q (l \('\d _ l
grioi) t"---o tPu tc)"'
I t h . . . S q F u qr r . ' <@
FIZIKA III - Kvantummechanika - 0R0SZ L_ Fizika Tanszdk
- l
N
. s-)- z_(l +i.)
e2'lPj (1)lt- ai I+$i" lq --1 \
aLr; I. J {1( r r ) = € i t1 .( \ )
--=-=n,-
N
( r= r , r , f , \ r . - - s t )
)-) u1 tr.1j r i
J . AoN\oLv L -F I
r.i, r e,-lr { <: L{(Fi) hiLorcr c\cL{^.,". 9o*1,ro.^
lir<lRor\o^ d\du rai{ai(\esig5 <s-L <q1e$r< Rqr{dct \* ."^.1t ( !^t i j .ho< N aLo b\1ere* .
cot5-fo,tq"- a"1.l{qtr\! \L !d\ )F ,I.\ 'I
l - ; - A * \ ( ( r ) + u ( t ) I 9r ( t ) - r i q . ( t )
=V(n) ( , . r . .^. ws.rert Fs!.qc\at r c1l . t \ { l ,Ct. - L61e\ i{ i !' j i rde . " I )
t r
9 i t i )
tn-r.rq<\t tqt \\a.l)J_;lgg ( . s1'i\<ivot{cs{rlildbqq l iwuo"rtt\)
q\ i . c l .Lr"eq
r] (ri)
N\ dlo (\, qq\av\iL\ e\4Lr*qbo\d$ q veqd*{v.
N- - v / ; , ; . \z < . c \ \ ) F l ,
J
+ 2 Ui (F. l
, + i . -J
* lr (iJ
T
.tSlHNEIO.T ELD
.q.l{u\c!hii\ d\t6
( i - ' , . , ' r , . . - N.6. r i r )
FIZIKA III - Kvantur nechanika - 0R0SZ L. Fizlka Taflsz6k
.{ . 5. 3. A ?AW\ eW
a,J
, l \
- fir rrt . /- / \ q
/ " \/ ( l-- r,r, i, ' t ,. -. s)
N ollo qr"r^os l"ilaecsl.;b6\ ci\\d
A Vc\d)\<v a\ev3i{ i }c. :
uQ = =$Q tl^,;.,;.,...,
r{e-n r.l tiiss;+T€T HeT;k T E G i
H
r. ' . . l ' ).t& = ( i ,1;)
r l L
I (" 1" I," )l
9(...^j. xi..) =5J] 9, ; i I ,-)c'=r .+l i{
0( r. l ; ' ) - - . '
_ryg_3: ( at+qti,^..1{,.. L )
- _t -4 {etes spir^\ rii{<<sr..i ( --_., {evqr.qo[ __, TEvI.r\i-DilrA( s]ot.'-
lr I eqirr spi" l r. i !r€<s!a! - 'r horsuot -. RbsE-E\NSrEi\\ s. lct.
li:::i!.*s:t^".\1. F ej ,qlo\^ '"e\d\ae. ( \ . , [ N=Z je\etr*." i - l+=t!)
I r l t tlQt n t; )l=l
Atr"rau.sirrr f,
Qtr,,,.1 = $ ti., i") = $.,o.rq,o,r +-Q. F,)$.gr,; =_$to.,";\^^+/+{%
$, lO. S, 6- (Er ieY NE\ td1l:it-T -r-r Ij -lSfL iJ I *l.6tl-l I .ffi |
Fti',r.# f0,r;.rO.ter-S,ti.rdt,,r]
=,(rP - r lQ.( i . \ o(i ,) l
Itz lS,u-.t S.tr,t
Qi(i,\ 0iGJ ". o.F")..;; I t"$ tt, r,. ,o")= # lo'&') s.G'\
('^ i:,1:r\;::,r) lo"1'r 6,1'r "' ourr,y(-{)
Te\it I <r.r._<q) vis+<<sr.e ler^.t uq\q*a.$q.^ d spin-p;\!q <i\\avrtcq ,
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
2. Brro.vruasrqr I r t , H=z
Q. (i ) S.(?,)
(E+ lc\ lEh 1! I )
GJs, (r,)l- t
tEH€T qa d\d\ \ i i !
( r , ' x ! ) = 9, i7 ' )Qi( t r )
li[\**ts-!t$= v'11<c!\( i\ irr'^t{ r:q5ou','hoc'
I Al\"r,i\"\i{.is
(i.,\, ..,;\,) = g Ig_(0.f,Feru^in\ :^ bols\ urr^dsrcc o\qp-\o \rotdi qq(^ o+ 6ssleUorc-l i{ci I ..
+ $,ix,t$.. tatx. . - - . i , - - .( , l r i , x r ) : v t x i , r a ) = L , - -
t-I Q,t,,t,. =\.-
16.(i.\0,(t.). Q
lt
Y
D E
IR\1 (.,1.)
AY d.trJ- ' - Qr"tr ,1)
.-{I,TFT:I]-,,-fl_---
A\ o\i,d\\.ao{ (T=a).1,
3 e v l . 1 t t a \
Er-Ti|t6 nAveosll .o?iurs u\\EL\.€sdS -- t lE\1.esSeL,t
Nogsstdwl vi \ \q<sL( <\a\ i ! . (S o e'rot t)
St6! ' r \ t iL! ! ql ,ds\Q'. \ L. \ \ {q. f -- kvAlulvt \<TA-I i (}Ti \(A1(
Ardol\oo.t (T=o\.t
9<ui . ! t t i !
FIZIKA III - Kvantunn'Echanika - 0R0SZ L. Fizika Tansz6k
/ .5 . t r .
N .llo
(vo\\v*^s!d\i sali L,ir(
vis\kgLibt\ q\\d rru.{\lee 1
i1 , . . i r r . . . x* , )
d; i 0.',
r - l . l l ' l l i l . l - - - d- r . -- z I I t . t t . t I
I oaaiq9\
( i . , i . r . . . ,1, . . .=rQQtt, ,o:. . , . .
Q tr",r.,..i.,... i,, ) 3 C Z & ({,4*,r Q,g,r.. {.-t'"1)
a9!-vitta< gl^e ;\\dv.i\5\L
N'D{ ---.' stqliialiLo.i \^^Ad11\wL -
Bose E tNlsrE I NFE F.HI -D\I IAC
r, -l:r;l-_- d. r" €, -tr]=-- dn h.' Lrl
I
Tev6" diqcqi\ tqi c\ \opot + uroLv" d\\oeot ' I t^ ' , " ' , ^" -"
(r-) 0l!\iLY! d[o!'ot
{cvwldtr^dqiLqi u '\! s\i '^ij si ?
/ d l \ d . r '
\.,J= I \ ' ld.-\)r
(cl i +rr i -. t) | rr (di+,ni) i
: n -hrl (di-^) I
' r ' hir- di \ .
{ , "$! t5(uc{ i , { . \q \ r ld \ -e: (d i - l )
."r\'""1,t
(T.D )Ir\x/( , , ' . '^ ' , , . r , - ." ' . ,^ ' ) = {
-
lTL.
H oj5o";.{<^t I tt ttu[ir"-{sviL*i
I d L
( r -b ) l . l ' \ l ' l I l r . )
ri;}al
FIZIKA 1II - Kvantummechanika - 0R052 L. Fizika Tansz6k
5il:{9!,\ X / t h , , q . , . . . , \ i , . . ) : \ ^ ^ o - \
; { e \ { i ' e , 1 a 1 5 1 Z h ' - N
Z t i u i 'Ei
\ r\qoLo.\ '--) \.i$.^siies s\i\'i: iu\t\ \n\o'r''f.-----
LAe !-AP te $q.d!rqc )
z
Te\\i\e\es sti\:iiivli!.
= L-. 1."^u/ --'' S = tu\i,/
- + rn . =o L *e \a r \e , \e {c \ c r' l- ) - € : \ ^ . =O I
S
N
w t,.,Z,nt
T t t" '
^ i . - - )
= E
l ( ( rn. , ,^ . , . . ' i , . . . ) = s + " ( [N-z '^ i ) ' [ (=-1 t : - t ) . .
i'.!G\h ucau't(vD\ua{i 6\\d*"r'iL
(V"i) L;{;usdles (\rr{ i \e\ r" i \r.n\ i) s\ i \ \ ' i 'v\ i \
*,** * (( . l - l ) + l l (6-t i ) =
&: - /-Ft i =o
(ai- '^1\ t^ (di-.^i) - '^rt-"i I
+ d\\dqd{
(ai+t^i) t- lai+'^.) -ui t''^ u; f
+ ci\\ dqcto
2\{'d t^ i
(r-s)
(R-€)
(B.E) \ 'cq l (F-b)
\ Zt-o. l '2[-u^1a,- ' i1t-u'"rr l \a-D)n Jq^\v- \ - ur l -2ua; \ (B-E)
I L l { - l d i +u . ) l - u ^
I i
: r i b -L iNG \ . \ r " \ i \ e ! L \ ^^ ' { t *q rc iL^ \ I t qx la x t \ ^ t - x uq x : : t
( t i .>>{ ; \ \d \ r i !oh ) \ " . N- ) "L )
\ I \- tai-".) 0,^ l"r;-u;) + [a,-r";\ - ui lr^ra. +v';
\ r Z t- a' I
t\\U 1, J\ ) | (aL*" i ; r ,^ (a ;+ r^ ; ) - (a i+ . . ) -u i tura. ' " .
I -Z r -a i i
LTT
t-t
zz
i
FIZIKA III - Kvantumnechanika - 0R0SZ L. Fizi i<a Ton.izdk
{.^^w P
l-LTiI3,
2 \ i
* t^^*
{"^\J:
t ( ! : -a\
, , ^ [ * - , )
i l
l^r -1
^ i l
( t -b\
t+ -0)
( t -E) l
t; ::::-'.i':h."^".\d\i\i\$"qo! \,n\c\i\a\v1
q,^ l=*{ \\ u L J
XV?h i
r l d i _ , \ ^ -= l l l ; : + I l - a ( _ l ! t , =t t -
c\i
h i -
otl -_h i
J=
d.Urvi( tr.rn Lr
[ : . rqleqevaci i orcr o
l. "* a\.o.1.'l; "i; ro O ;
t t+o\t
sF iq^ \D. i \ \ { < i \ \op ! \ r \ r \dqq )
- ) ? (e)E ds- .M(r )
\ - . , - . J
l .
I Fo\ \o\ drrdvot
sFi\a A{dp.!o\. s\<iuq
= a, . { ( t l ;a ,p)
\__-_w+
ehs+\qs.\ !93!eu!
tr)
o
e
q
4+ l l€ .
4+ f t i!4
4 + B i :e ' -+c f
x (l)
tvq tt\
xt . \J t
FIZIK,A II l - Kvantumme:hanika - 0R0SZ L. Fizika Tanszdk
$^. .v j id !a i \ t (L
d i
t i
(L" n+i\ {o\st"uts .urvlia!+i*{rt
t ( t ) dtr^ (t\ <i t
rnt=di4(tr ta,p)
,l^ Lt) = ? (€)
E =E(r) --5 F-r(r)
tto=;rf,-- -->
4 ^ | t ) d t
I4+ B t
e I .t d
I t.r -- 4n'u t.)-a - Fz "
F .B: -- l -=4 t t)=
= X rtr . { ( t ; a, p)
F-^5-.r\-r:"-u'r ,Zts ra i =E (= . i \ q \ "1 { )
fI {. , (r\ dL
. t=--F-
4{ ct,rl =
+d
=J3:+isqa"ui
l . ( j r r c { ) e l t
l k q t > > ( . \\ /€.- i -
n=ErJl " l
Lr**.+r-.*^lr.^ )
N: d\\ oqdi -+f
_ | *n
o = 1-"n
ttAYvEl't: BnLT+t{ArN
'?<k(a/{{ ta! 4 I{ ; l ; ! . ib"d\
A vasl.aqLelL e\ee5tc.-s\ev,L{i e\.s\\dr {l qqvt r^. o
e-iE=r- + J
A^t r ) : ? ( t )
FTZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
A u- \<!v- io;
: E ( H l - D i R A C
,! [t)
Bo:E :E*!!sl,EjS- sLd{i\\+iL.\ qs€{c\ ..
T1
-- ' ?({) |
!r^{3 U qt dro a d\ I { I
q h^i\clrqLavi {iai\qi v.q"r\1.1-"( 1"ttr*rS]
/^ ('t) =- t F (T) (L^oe, ret tri <-r"3,-a)
' t to= Xt t r . {Dt t , r1
( { , - r ) " l t =
aJ
tF ( r )
= Itrr\ .+J
i)
t
c \ 6 iuo{a( t . v i l i \ i l ( i
N
1. r r " * r5 ]
X(i \ .+ t i \ r )
I q ( r j { ( r r )
^ B E+-
-+ |
( r )
o u{ l ( r )
z - to
P \T)
T
J-T {To
f=1"
tN = N"(T) + Nt('r) -, d,\"*"'.;
g . t . J \ - . . . v J
"II^,....,"-ooELL,. A bol.\oL Sld\q < vaun\it
I N- N(r' l ]
--- ,t--.-.--- - _! ' L
t o
g (t) dt + .L(rt)
FIZIKA II l - Szitdrdtestf izika - 0R0SZ L. Fizika Tanszdk
,1,S.5. Bor,suoLbi\ d,\ld r(\dti,ar"
E\c\.tvo ,^^6g!^s3.9
t
q.!s6\kos \L!iegbtr.{e\uv$ \^vt\d!*oL
I
K!^ssaiL!S
I
rpe9q.*.fetli{e(rL ---- .i\i\"tld\-o[ --. j:i-tvro\r
k (L ,Lr ,Lr )
I t . :=ck I o\ii+peqiis v{{(;
J.
lrr^ { r^ .au\a^,^
.'- C 1r,"'+,"{ +"i ) r^ id ls . ) ( \^d\ \^ t \ \ \ )
L^ iaussntl)sc1 (E d,\. \ ,"f$r-J,rct)
q ((-) = c[F - -N ((,). w J - - - - 1 _ -
t L.vbr\i d\",r.\s"\llctp.loYi\a,ciiL s+dqq
v'l +,^i +'^f L r r - " ) .
C T
. - 3 L= A - L ( ! - \ ( !lf\
z
olfl
!a-t '*rtt\at
' Lr.^icallL c1t
..f,i (.,\-- A-Li. *'.u (u)
e+da
t -- -.,.- .sq.1"lt.t|.11lls 6
FTZIKA Il I - KvantumnechanikaOnOSZ L. fizit<a Tansz6k
a , > , ) . 4 . Fotot^gi+
6"oo
l-
-t-e\ . I - tvoqdghcses ,' ,-eri (e,B)
.,
{"eLva\<rd,il l.Nl dv!, {.t"ivq{ dia I
a h l d = Z a a
r
\^v\\drp.!!\ aqevSi d
---. d\d\^.l\ d \-"L +
qa (! {ret{w.nrr<;ji
t(u) c.\$ :
.t
2.a f . . '
t t-t (PLhNcl()
.t,
f
-t !r (!
EC:
IJ
Es5
6 . , q u ( \ ^ = o , r , l t . - - )
+oroN = c\.vgto. o<i\6j (L!4*{!w.)
)
]
+ (r.r).-5 J ((!,r )
(- {"r.U"r\"iiii \-; <r{ s " t.. (.ri) FotsNr ta.tq\\'aa "
X(s\d(^) - -+ o+ t ' *
{ve\ teu<oi i , ( ' \e ;c l , r ,soU l \ . i \ao\
;* { c +ortr ,ot aas-s \' i ! r \ . s r \
<quc!, q qs.rllqqt
, o l { q r w g i c t 4 .
( - t r ( ! l + r ! b
E\!E -EiilliEi \, r!-\i\\\iL{ ( r"rsuorr = \6\o,e\ a\,iqqvi\ta{rL ..
l= o I c r-\dgouti aq{!rtid\.;\ e\rau'q$q!-
)
/\t i s$e*("gro. ..
r ,E =
J ( h b ) . v \ ( u ) " \ a
= I n \ \ ! . 1 A L - \ ! - .J
B E
4 tu,r) c\r^) --
.cL " l c t \ - ) d ! !
J-
r: I ( b ,b \ .9 rs )
I
!Y1
5(u) = zA!:!-
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tansz6k
1. 5 .S. 2 . T or^ ora q ri+
c\ ( u^ -L. . ' ' : .L ip i t rur ; ' \c t lqh.qt )
qa\sr: !6rtg --5 .i\\i\^"\.i uol. -! \ai du:ok
I
q
AlOtti tlEllli.€!€\
f-/L--.*
v ' )N\ J,r olaq --i
L !D
3 Nj t \
!Ialodsd1{o!i' v.k4r\tvt{ii[ir\.i,\; e.\gi\id\\wot [&:"\"s\ \d\.\:.\
\, Go
- aI ( w ) . \ w = j A L t l b r . [ E"J
I
J
\ { vd * I , kqc q r " q q : t . \
\ ( .q-r . . i \d. ! . .
Eq\ 6 +"el . ! . - (r di i u;arqs
I r . - - ! -q* 'nnde\ |
- * - - :__
t -
I t^v\ \o!5.ur i {ua" ' r io .
vEs\ t- {"c\"eq.i"rj! \a; d! s
: 3AL- - - - x - - iA :5
9 N
tt -o'
I t t . t \ < - I r u t t ,
1 (e) ab
f
I \ \ :E -Ei \ rsrE\ N !h l i \ l \ i l ( 1
( tsssNotr : bolou.L \ \ak{
d \ "u\ \6.0s*! i rurv l t . i { i \
r " . \ r r ( \ ^ = o \ a . 1 , 1 , . . - )
" t o r - a N = I ' e u c v l i < q d ( j ( t " " ' t - - i '
'L\ (d! \ jqy!{r . {or\-r*qr
--5 q! "r- {ve!'ructall"' r^^A"r'trol' c\-d'- "\
-+ ,1^ ( ! - )dqr (c . laNoNa\L !1 i iqq
i,:!Erc* )
.qkaL .. un .r'^h(\.- d \ r \ lwg io lA .
r r . \\ ; q ! ) + : - . w
::""*l
\ , , i t { . \ i \ - : , /q.o 'el{.\i*i{ )
FIZIKA III Kvantummechanika - 0R0SZ l. Fizika Tanszdk
1. S.S. 2 . Tor^staq i1
v*rvrv {o\ (t-. t trrt^ipitnur i . t€t1q|>tqi)
d Aroni $Eq{€}er
ruqotun: Lirrg -s d\i'L"\dho\ ---., \A' d!:ol<
^,
3 N
JN
ti. I
( - , N Jlr o!^\* -5 l]:c\ddsd31.t\.: v.rcl(\sw
L;\titb.i)+{ \..\dri.i\\o{e\ (\{:"r's) \rft\i\
r!!
.r .t s ) c l w = j A L ' l \ ! - d a
J- l
k a * \ , i \ d , ! :
Eq5 b {"e]""r-ql aii |A^i ar.qs
I r !Hu . +nnde\\ I
'**
I u,\\"1p"*\i '*r"1t<J
E95 .- ,l"eL"ucrcli \u; d! !
t q J
= sAL- ----r -- 3A:9 N
*-i
* , - r = {
t * * \ ^ a $ ' - a D
' L o l ^ * . - > u b
l { ! c ! I . sGec \q q : \ ^ \
I \ \ !€ -eiv!r€\ N \ | \U$\ iLq 1
( +ot"ruott : bor"-o! r'rquq
^ u*\ \cps*l i r \^a$l i ' i \ ' \
r " . \ . - ( q = o \ . { . r \ 1 . - - - )
';1y\ = " c*cv"1ir oo{cX (t'"u-t"*)"
'v\ (dlo) s! !Jtr . {q ' lc\qcr
q1 "u {ue\ 'e"< ic i i l " \ * ;o t ' ! ro \ ( \a \^q
--t {^(b).lu.r (.. I'or'oNo\'_ ('r'i*"
i!!,\tsas )
aqkeU q cn aliud\L
' d \ auev l r o la - .
l l
\ - u - ) + : q ' "
::*"""1B E I E
r n - - \+ (rJ) .-5 + tur ' )
v d l l " \ i \ - : , / u . O i
",.a"q,-1i.^\ )
*- l
t
FIZIKA III - Kvantummechanlka - 0R0sZ L. Fizika Tanszdk
f ,E-= I th*
f .J
er 6ss+eqe"gi d, j . " I
( ,) q r u t c r r o = I ( t s ) . n ( b )
) -
i E
. 1 .
3N - | -------_ .rar . . r I ! , ![.*$)-f-'* =
g*T - 'l\.-----..-------
z
- - - a*=5a*
t *o : L€o bEBrE \^ l ;wivl iLL-!
9oT
, , . - . t + f iF=qNr j - l : -L \ ' .1 x J .x.-r. " . 1 \ r I I :z J e
- 1
o
N vi\a\\€v k\i.\ (v.tli''i)
eq.vs id jar iq \ Fq1 q
\^3 wi" r{i L(. l.\$ t .
T<'€t d\o-+g p
T
I^ f
l e - l
ti
{;
l l _ELr )= gNLT \ ; -( \_ / r \vo
Tr,^ilu!i B{q\i\|:'1
A = N \ L
--------) E^tr) -- 3R-r ( : )t.
a*5
eD LEI
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
T
+.!t"$ir (
-F
t^ljL"\pq.i {d,s )
+
f = 6
711. = $ u r -
LlaFBAb th\€A!)
(e\". i p." t i . iA ) -+
A v . \ d $ e w e \ € v l i d i q I E = N A
A {.i\^i (\-"\,i.ir L\ L'\'" 't\'ir)
(r).= 1 u-T
t : ) i + Nq <a )< = rsor-r l r* 1) = RT (- t- : )
.tq -+ J\! = PdV Jq=
ctE '
.rE + PdU<"M J.,F
(6 i1 t r ! ilueh \ \ i \a l
) oL[ = c,K ",.tr = C. 'lI -->
L\ossriLus J+ilcr q"Jrl{ :
E = qr (jr 1) -'-' q" = t 'r ' i) ru.1l * - - ' :< a 'eo i T : ro "kl "= " ' " * " " - t
DILING eEi i l + ;v "u}z \-=-.. r oo 4^'lq^ i
.\.c*t!qu{(5{u'\ci _$:Y:^;.
^ i , f(<y = l a , j ; . - s vq
I i I
{ - -
\fls
l l \3 [u- r -oo ' l\Bs r
I ,-r 4su
(.1'";o'"t i 'rs l 'r ' l) Eo= j P.i- LT =) 9Dl
(": (",, - re ctr'1*l-'L=, . ' 3\.%) r. 'Q s
in- I5 Q r,
lG,.* o I --(z)
stG\6d .\{Ll!'rg'i}
L# l"I
(i* )
MERd,S
FIZIKA III - Kvantummechanika0R052 L. Fizika Tansz€k
r.6. A Lov<rttr.rs Li.r^^tc,t L'ir{er
A. A tr,,s\ c L"t o, y',i\q d L
,{.6. 1.,1. €19dit-1.\^\ i ; ! q^sole\t
\ teEl s lo{eucid,lv6tqb t.. . l i t- t i \ ,
A t\ \seLL dQ\iqri\d\( :
IF.\e\( td\ ui \3\ I
5q1s+le\ d{L\,".. L{ l \S\ \^\r{ucid\ ! t \g\brq:
B
- 1 l , ' \o, z\ ,,
" r. i i t\" oi.yr
"Laair i" 9iL\A
\rite\i\\ s\d.\q!\d! : t. a ^lr - tg
"*L:, "*1 + "tg
AL'rAqF\irAS i'-/- \--V---./
lrJ drsqPir\d\(
FIZIKA III - Kvantunnechanika - 0R052 L, Fizika lanszdk
- v( (r^) -v< (YB) + v. (!.)
(6de\- \ar . ; r \ ;L)
^l (?)
, . - v r { v r I"=
--;- I
. - '^i^ | -- + 1'',,1',,.i1t = - ; - |
IYJ
E1+aLl-\ \" aro\a\-\i, t 1{. t, q";.\'i s)
a t",1..{) = N(v).\Yr.f).4tp)
[ '" = r,',,r, .. -
1 t . o r1 \ ? .3 ! . . .
L r^^ r = o , t ,1 , : r \ - -_ r t e L+
{ 162 i62 P6Z 6 11-
;
@.S-j-4ffir
t r,l)
Ar r\eLt!.\,i\\d!€{sl^
H"a = E"*
lELi;LEseI I
1 o , l , r , . - . Ir r ,p, d, . . I
It o r . \ 1 r . - . IL 5 r l t r s1 . - . J
---5 latili ?iLyA r
-+ v;rf, ,rA...r I
\T t . r
l ,6. r.2. A t^ t dvoq i !^ r^^ o\ cLvl q
D -
. -+ -S- - -O- - l
tl
\
ffi
s
" , i "^.,/ \'s
tsa_bts .
,L
\'*
FIZIKA III - Kvantummechanika - 0R0SZ L. Fizika Tanszdk
1.6.2- , dt .t6,r"!isa" il..dLb.!l
L t
Q = , t
6
(s)
kor o tk )
(s)
LAlird tL)
(x)
I-@-l+@>**oI
( ' )
(r)
(L-)
(t.)
(L)
FIZIKA I l I - KvanturTnechanika0R0SZ L, Fizika Tansz€k
,1 ,6 . i . A h id uoq a.\^ {-o t e\.vl 6\ ( !tr)
(ou\e1&\ tL6\ (\ii!"\^dtd\ot ,
V, (t\ = ---:- ' -l-- +tt. r
- -:- - \rc (ta{) -v" GB{) - s ar
l {Qtr, ,r . ; = E Q(i , t . )x_. = ( - r t , I i ) I
IQta,r.) = -Q(\,r,) J
-\\,.(r^J -v.(rB1) + v< (r,r)
= Ho ('{ )
tL. + \\ H r I
lvwd Li,\.\i!'is r q1 €\eLt\-oL
r^ ^ II \.{" tt) .{- U$t!)+!.(a) IL I
sbi\rp,i\\d L;\ (\'a !. \^dl d s \€ \(^ d\ a-1o\'t!\
}i !.\,ilev!aL e1.14urcl racpa';.\L^lo
I ^ , . 1
l la" (^ \ +Ho(r ) | t , ( r , .v , \ - \E-v( (q \ \ . Ys( i , , r , )L I
Y\. (F.) Yr._(r')
A+ euL{,.hoL Li\6\i !-i\.Grr-tris (Liacti\o.) li\\.resbrva.ta\q
E = ( Ys (F,,?,) \ \r l*r(F,,t ' \)t
"lLtY'\"tr (.v1) --+ scF \a'1.\.rdsoL r.
EG\dB t\i \stEvlE t.------
{u.+\
1 v. (h)
r----v,J !a#
+++ts
r^aq \^-uaL <q5q; sva v. (trl)
Qtr.,,-.t= rSt"-,,4)
$ t..,i- t . "l (q, ,r. ) . ) [1^ ,1, )
I t - tn ' t ' t 'J, 11"1. ' ,_lQ (?,,i\) = 1
L a. (q,r").'x.(1,,1.)
( F l ,
FIZIKA 1I1 - Kvantummechanlka - 0R0SZL. Fizika Tanszek