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Hu MNH (ch bin) - T DUY LITHANH - L TRNG TNG
............. ..... HUI I
BI TPA >
VT LIL THUYT
TP HAI(C HC LUNG T - VT L THNG K)
GUYNLIU
(1Ds
NH X UT BN GIO D C VIT NAM
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NGUYN HU MNH (ch bin)
T DUY LI - NH THANH - L TR N G T NG
Bi tp
VTL L THUYTTp hai
(Cd hc lng t - Vt l thng k)
(Ti bn ln th hai)
PI HC H NON
H U H C 1 IM H o c u i u
NH XUT BN GIO DC VIT NAM
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|V3r--:>y. ' . H V t
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p h n I I I
C HC LNG T
A - B I
1. NHNG C S VT L CA c HC LNG T
MU NGUYN T ROZEPHO (RUTHEFORD)
L THUYT BO (BHR)
1. Xc nh nng lng, khi lng v xung lng ca phtnc bc sng tng ng vi :
2. nh sng c bc sng X = 4,2.107m c chiu trn mt
kim loi kali. Cng thot ca lectrn t mt kim loi kali bng19 *
3,2.10 J. Xc nh vn tc cc i ca lectrn bay ra t mt
kim loi kali.
3. Tm cng thc tnh bc sng Bri (DeBroglie) cho
ht tng i tnh.
a) nh sng trng thy c
b) Bc x Rnghen c
c) Bc x gamma c
=0,7fj.m
X =0,25
. = 0 ,0 16
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4. Tm bc sng Bri cho cc trng hp sau :
a) lectron bay qua cc hiu in th IV, 100V, 1000V
b) lec trn bay vi vn tc V = 108 cm /s
c) Electrn chuyn ng vi nng lng 1 MeV.
d) Qu cu c khi lng lg chuyn ng vi vn tc
V= lm/s.
5. Dng iu kin lng t ho Bo pdq = nh (q l to
suy rng tng ng vi xung lng suy rng p, n l s nguyn n =
1, 2, 3... v h l hng s Plng) (Planck) tm :
a) Bn knh qu o Bo th nht v th hai ca lectrn trong
nguyn t hir v cc vn tc ca n trn cc qu o .
b) Cc mc nng lng ca electrn trong nguyn t hir xc
nh gi tr mc nng lng ca lectron trn qu o Bo th nht.
c) Bc sng ca vch quang ph khi lectrn trong nguyn t
hir chuyn t qu o lng t th t (n = 4) v qu o lng
t th hai (n = 2).6. Dng iu kin lng t ha Bo tm cc mc nng lng
ca dao ng t iu ho mt chiu vi tn s (0 .
7Hm sng ca ht trong ging th mt chiu c dng :
V/n (x) = Asind
trong 0 < X < d vi n = 1, 2, 3, 4... Xc nh A t iu kinchun ho hm sng.
8. Trng thi ca ht c m t bng hm sng :
2
- + ikxy(x) = A e 2a
trong A, a, k l nhng hng s.
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a) T iu kin chun ho hm sng xc nh A.
b) Xc nh X cho mt xc sut tm thy ht c tr ln
riht.
c) Tm xc sut ht nm trong khong t -a n +a trn
trc X. Cho bit :
e~ax2x = j |e - x2d x = 0 , 8 4 ^-0O 0
9. Hm sng ca lectrn on g nguyn t hir trng thi
c bn (trng thi c mc nng lng thp nht) c dng :
r(r) = A e a
trong a = 0,529.10"10m l bn knh qu o Bo th nht.
a) Dng iu kin chun ho hm sng xc nh A.
b) Xc nh r cho mt xc sut theo bn knh c gi tr
ln nht.
2. T O N T
10. Chrig minh rng :
X d 2 2 _ 2 d 2 d -a) = x i -^T + 4 x - f -+ 2 ,
dx dx dx
b)d_
dx= X - + 3x + 1.
dx dx
11. Chng minh rng nu cc ton t A v B l nhng ton
t tuyn tnh th ton t (A + B) v ton t AB cng l nhng
ton t tuyn tnh.
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12. Chng t rng nu cc ton t A v B l nhng ton t
ecmit th cc ton t (A + B) v (AB + BA) l nhng ton t
ecmit. Vi iu kin no th AB hoc BA l ton t ecmit ?
13. Chng t rng cc ton t sau y l ecmit :
a) X = X, = y, z = z, px = - i h ~ ,x
Py = aT" Pz = a"J y z
3 ^ 3 , 3L\ P x ^P y^ P z . . . Vb) H = ------ ^ -------+U(x,y ,z)2m
(m l khi lng ca ht, u l th nng c a ht)
c) Lz = xpy - px , Ly = zpx - xpz , Lx = pz - zp y
14. Chng minh rng nu A, B l nhng ton t ecmit th
[ , B ] = AB - BA = i c
trong c l ton t ecmit.
15. Chng minh rng nu A , B l nhng ton t ecm it
[ A , B] = i C v a l s thc th :J|(a -iB)\|/(x)|2 dx = J V (x )(a2A2 + aC + B2)v/(x)dx
16. Ton t tnh tin mt vect v cng b c k hiu l
T v c nh ngha nh sau :
TV(r) = V|/(r + )
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Tim dag 'ton t tnh tin T v biu din n qua ton t
xung lng
P =i
_ _
r + Z- + k -= - iW
x y dz
17. Tm ton t quay mt gc cp.rt b quay hng 0 v
biu din n qua ton t mmen xung lng L = [r Ap]. Cho bit
ton t quay mt gc b = 0Sq> c k hiu l R(cp) v c
nh ngha nh sau :
R(6q>)v[/(r) = \|/(r + r)
trong r = [ A r ] .
18. Ton t A+ c gi l ton t lin hip ecm it vi ton t
A nu :
|v/(x)(A+(x))*dx = Jcp*(x)Av|/(x)dx
Chng minh rng : ^
a) Ton t A l ton t ecmit nu A+ = A
b) (B)+ = B +A+
c) [,B]+ =[B+,A+]
19-. Chng minh rng ta c cc h thc giao hon gia cc
tn t sau :
a) ypx - pxy = > z P x - P x z = 0 > xpx - p xx = ifc .
b) XLx - Lxx = -i z, z L x - Lxz = i%
trong p x Lx = ypz - zpy .
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20. t L+ = Lx +iLy , L _ = Lx - iLy chng m inh rng :
) Lz L+ L+ Lz = ftL+
b) L ^ Z - C C = - f i C
c) Cz ( C _ Q )- (C _ C +) Q = o
d) L1 = C _ Q + } z + h z .
e) LzL2 - L 2Q = 0 , Cy L2 - L % = 0 , Q l 2 - L 2 Q = 0
trong L2 = LX + Ly + Lz .21. Chng minh rng ta c cc h thc giao hon sau :
a ) p x f ( x ) - f ( x ) p x
b) p A (x ,y ,z )-A (x ,y ,z )p = -ifcdivA
tong px = -ih , p = -ifcv , f(x) l hm ca X v A l vec tx
ph thuc vo X, y, z.
^ c) E t- tE = i/ivi E = ih v t l thi gian
t
22.Tm hm ring v tr ring ca cc ton t sau y :
a) K = - i
b) Lx = - ih - - .cp
c) i \ = - i f t j - n u hm ring ca px l V|/(x) tho mn iu
kin V|/(x) = V/(x + a) vi a = const.
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trong I = const (I l mmen qun tnh v T l ton t ngnng ca rtato phng),
23. Ton t Ham intn H ca ht trong ging th vung gc
mt chiu c dng :
Tm hm ring chun ho v tr ring ca ton t H .
24. Gi Lz l tr ring ca ton t Lz v L2 l tr ring ca
ton t L2 .
a) Chng minh rng L2 - L2Z > 0.
b) Chng t rng nu V/m((p) l hm ring ca ton t Lz tng ng
vi tr ring th L V/m(cp) v L V/m((p) cng l nhng hm ring
ca ton t Lz tng ng vi cc tr ring (m + 1)h v (m - 1)h.
c) Gi / l gi tr ln nht ca m, chng minh rng L2 = ti21(1 + 1).
25. Tm cc tr ring ca ton t L2 tng ng vi hm ring :
0 khi 0 < X< dtrong U(x) =
00 khi X> d v X< 0
Y(0, cp) = A{cos0 + 2sin0cos(p}, A = const.
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