orosz laszlo-kvantummechanika munkafuzet
TRANSCRIPT
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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)
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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 \ ' ! { -
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- 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 ?
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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
)
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''-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 (",
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+'-------.\.-- -----
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lPl'or"
^f
,.{) -- R(.) Y. (,},.{)
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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
$
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lelies \aegoto{ds \^idvoqi\s\evu isq ese\€.^ :
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hETLEL v
hAGNESES -
tsalra \a. cr{.q i\
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^ ( 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
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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 { \ " - ^ )
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. t . i .1, ' r . ; Af o.tsr"^ol. c\cl.{kot^ s+trt eaetc-
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Y ( t - i ) H lkv i ioh A\avo\
t_
E-,
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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
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-+ (s cr h.\\48
IJ
o+ 4\41{sLoL
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\ )
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cleLtvo<,s,1qv \rer-\e- (1)
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.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?)/ /
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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$lt vs
{ =ou " =O
t ^
\ ?. (u++) = i'tte.ai "fn,"," = ft",. t')
i aq \ t { ' kqa \v iL" r
\ l "Y I
IL
31 ! o,3
t=1J
t (.f,f)Pn" cor$.
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a'' '4,-L
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?. ",^
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\ 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 \ )
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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"
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,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)
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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\ :
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!€v t^qtriscivc. q! *.ev9iq*i!"
lcL.
Er
| ' , - l l l ,,/ \
- ( lee t reu ,{ t{aLtls .}
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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';
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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$vrs)
, " : :
otsr*\6vr.s I lerov\ \ { \4L)
qA o.!s!'\ \qd5\^e\es r'^ruq\\"- o( s.)
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A w+45 €"\e\\ !e1a\. i o\ e\eLtYolr^dL( o w r r q , i s o { " \ { , q q c { l e u u \ ) ' . - . 9 5
s o l o t h ^ d q q R \ e S \ q ! u e l ^ 1 v \ & o \ .
\ , o ^ sc , i , i \ Pe ta t ^ l e te i s :
).1
{ i \ e
kat
Mr.-*Lr --+=--1- L t Z q
f'4., = t pro I r.a-' l - i -+ "sl = _ e
( - 1 ^ l ( \ 4,r = r z .J -+
Ar e\eLt"o,^ sr,"tr^ie es so]. i \ w.o1t r:e:{ \ ,1\o. \q\"^ v-a9qqd t \-{A Seuqi l iL\".ssti\.v s Jir^ o rqi-Gvo,t
.1.Spix s\1o< ct , ivrqrq i lcq i vn\{6ai ,
q6
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( A'\",-^ ,- --> qi! L{[ !.*.r 1 ).t
e\wiieti uo"ra! le\.)
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',
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s+
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: I E !,r.1-
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lct ! iY let
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GovDSt\ iT - vHl le i , r tE cK
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$
L
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a\e\.tro ^ - o\\qeotsL
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F35 e\eL{v"."c\\oe.t ( ' , .*. r seiN-riLYA)
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t .5.2 A I{A|\rrl.E€ Lo\c\i{is rs o+ S(+ \abd$eF
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