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Quantum Mechanics Numericals
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Quantum Mechani,:,s Pa~fo/17QUANTUM MECHANICS
I. ). .. ~(or) _'.'._
2 '
E .. hy(o rfh _:
J M..Ap 2: ~
.. '
4
S.
AE.& 2: ~
Al.A0 2: ~
..
, ._.,.. f.·'·(~l
7 P (x) • JI'+'•(x) I:
0
~~':;~ :~:t~nrt}/;~;-,t~·: ~;:. ~~;u~::t:=2 t~:~~o_},t:: lO-lJ JIK.
m. - Electron ma11 - 9.11 ,. 10·" kg.
N11mfflnh
1. lfthcuoctrtainty inthepositionofthctlcctronis4A.cakulaietheunccrtainty in its
momen1um.
Sol•lioa: Jll • 4 >< 10·1• m
We have.
p•
h
Ax.Ap 2:_
h
=, Ap •
.. 6.625 >< 10""
IAp • UIB><lO"l kj - m / 1 I
1
2. Compute the small est pouible unccrlaint y in the position ohn clec:lron moving with a veloci1 y
J ,. 10' mis which is uncertain.
Sol•lioo: Av • J >< 10 1 mls
Wchavc.
But. =, Ap z m,.Av
Ap ., ( 9. 11 >< 10-l 1 )(3 >< 10' ) • 2.733>< 10"21 kg - m 11
h 6.625 >< 10""'
Therefore,
Ax • ~ =- Ax • 4{3 .l4}(2.733><l0":, )
1Ax • l .929 >< J0"11 m)
, Quantum Mtthanks Pa~2o/11
3. A miCTOKOpc 1Uin11 photons i• employed to locate an d«tron in an atom to within a distan«
of0.2 A. What is the uncertainty in the momenl\lffl oftbe el«tron louted in thi s v,7,y! What
istheuncenainty invdocily!
io-••
We have.
Ax - 0.2 " m
.
h
M .dp ;J::_
.,
6.62S x 1o•l-l
p, . ~ , , ~
Ap=2.637 x lO-"' kg - m l •
But. p=m,.v
m,
t.v =(2.637x 10", )
(9.ll xto·" )
I <lov• 2.894x lo'm /1 I
4. The natural unce rtainty in the meuurrnxnt of sp«d of an electro n in an atom is auimatcd to
he 2.2 " 10' mis con,,:cding the ideal Jet up and error fr« measurement. Estimate the minimum
widlhaboutwhichtheclectronstaysconfincdintheatom
4
l0 rn/s
We ha,·e.
But. p : m,.v
Therefore, M.m, A v ;J::~
• . - -"- . . 6.625 >< io·"
4rt(m,.t.v) 4(l. 14}(9.ll x to·" )(2.2 xl0')
I 4ll • 2.6Jlx lo-' ml
5 Cakulatetheuncenainty inlhe velocitynfana-particle whichislocatedwithin4A.
10·10 m
Weha,·efora-panicle - 2pro1onsand2neulroos
Therefore. mas.s of a-particle. m. • 2m, + lm,, • 2(l.672 >< l0"27 ) + 2(1.676• 10·1' )
m. • 6.696 >< l~~' kg
Alsowe ha,-e. M.Ap ;J::;;
But. p=m.,.v m.,.t.v
Therefore.
, Qu,mtum Ml'chonk• Pug~ l o/ 17
0> • - ' - • 6.625zlo-"
41l.41l.m, 4(3.1-4 4 x 10"" )(6.6%xl0"11 )
t,,v z l9.693m l t
6. Compare the u111;er1aint~s in ,m, vel~ity ofan electron and a prolon confined to 1nm h<ia
Sol•llon:M • l • lO"' m
Wehave.
But. p=m,.v ::::> Ap ., m, .Av
h
~fore,
lience. All.m,.(Av) ,"' 1;-andAJi.m,.(Av) , "' 1;-
Implies .
(&>), _!",_
(Av), - rn,
1
(Av)._,. (L672 z l0-l )
{ti.v) 9.ll • IO""
(ti.v), = 183S.345
(&,),
7. A proeon ii confined to a nuclcu,i of nodiuJ lO""m . Calcu late the minimum uncertainty in iii
momenlUm and also calculate minimum kinetic ener11y of proton.
Sol•llon:M • l x 10·1'm
All.Ap ,1; _h
Wehavc, ,,
:::> A h 6.625zl0·"
p • 411.Ax "' ,i(3.14)(l xl0"")
Ap =- 5.274"10"11kg - m / 1
1
Alsoforprotonweha,·c, E = m .v1
2 •
2m., 2m.,
~fore. AE = (t.p)l = (s.274xl0"l' ) = ll.3 18x l0"l! J
----
A II. 1Kx l0-
E• ~ • S l .767keV
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