. n(V D n ~\t~.'TEST 2b Apr~l 6, 2001 NAME: '~~~~'\~"'" PHY242 -- 242H page 1

If you have a test impediment (e.g flue, etc) check here and see me afterStandard personal cheat sheet allowed, to be handed in with test ---- Avoid calculators!

All students please attempt all problems; Honors problems add up to 20%+3% bonusnon-H add up to 25%+3% bonus (so skipping one problem is possible but not advisable) .

---------------------------------------------------------------------

Inside front cover of the book you will find that:a) hc= 1,239.8 nrn eVb) larnbda_c= 2,426.3 10~-6 nrnc) a_o=52,918 10~-6 nrnd) l/R_oo= 91.13 nrn

............................... ' .[H4/5] Problem 1) PART A: State in turn the name and the physical meaning of

quantities shown, why were these introduced?

Ad) Rei)

~,.' P[<Ain<.b 1(A.{L'VLtv\'Y}'y' iM ctc'X'11.pi.-- 1~{G\.~~ J? h',1SJu. •. I~ I2r(jftk""~

c: v'e_WCl~J i t,,\h+-- ) ~ ,.£&\.!,'v1 ~~ ('rUt ,'••1.Lt.V1 vdcu,\ 7J ~R-l,J 6or;Qc...

!(.G"' c.o """p h.JV1 W!V\/-tiRffi U ~~ i W'V-V't,te..116 (..L~ h...f1-"":- Uh1-v(J l.-e;; 6(,tlbfvr : 1.). t"). ~ I). c. l \-Li.~ C-.:] )

.. ~

" dilv;r"'\-\ 6-A. "\,"'-LlLu.I.'~ ~ ~~;l.l;k'\.'\ .~ q L-l£?'nA '&,i

"'t)ct'lC~ r ~"'1)...Ut/), I It"',) oL/...t ~La v't1-

r<~JbG'tJcv-.v.J~J-: rdt:k1 w'tVi't Le,nd(1i. ~ IL~h1-0..\. , . , , . , .. , ... , ... , . , , , .. , , .... , , ... , , , .~~~ .~/1J'~tktA 60 ~Y1/.) J,v 9~» v},4) [H4/5] Problem 1) PART B:· HINTS ,~~m_e=O.511MeV, m-p=938.3MeV 137/(2 pi)=21.8, 13.6eV is electron binding in H

Aa)

Ab)

Ac)A: Il...\. (3 ,

EXPLAIN/ STATE quantitative FORMULAS how the quantity a) (hc)influences the lengths given in b) ,c),d)

(sho~ explicitly how the units arise and what quantity controls the magnitude) .

'- h .~ ~\C.Bb)j\ C - m(! c -\'nQ... C1.

Lf ii£-L1

~-Bc) . Ct c -= ~ e. L mC

Bd) R ryJ--

· ~i~t~k~TEST 2b Apr1l 6, 2001 NAME: \ PHY242 -- 242H page 2

[H4/5%] Problem 2: Number your formulas if you refer to them

Formulas stated without derivation do NOT COUNT

NOTE: You begin at the level of Newton's equation and Bohr's quantization postulate.

ATOMIC MODEL OF BORN for a one electron ion of Carbon (A=12):OBTAIN (means DERIVE, not state) elementary formulas fora) the orbital speed,b) energy of orbits,d) Paschen series wavelengths and energies, ~anuition ~:~~Ql~n~rn~ and the series limit.

ZolC*~ ~.~~,~

F-vt

_..,L. 7' L_I l.""X_ 2- fY) L- 0-.. L.-- LY\

- '\ = L-

e~ ---V\::. l

· . ~ ~e~~'U~TEST 2b Apr~l 6, 2001 NAME ~ PHY242 -- 242H

[H4/5] Problem 3: ANSWER IN TERMS OF THE SCHROEDINGER WAVE FUNCTION

page 3

lA~ (T V1 ~) Q.-cl "1 VY1./~ L.e . ~ c7C-~l t-, X Ld c

tv~ r0t-~Mv~ .t'~ ~t~ q/\-\)=~~r~)

I' /< > I I t ~(.t{(... \0 = 0Yl O-(r ~ ' ki i\'l v~ .c..; VWl __

~ \-e-vt ~~> =0 ~ ~I;

a) What is the probability P(x) to find a particle within interval dxat position x? What does this imply for a property of the wave function?

~

r>c;.") cl y.. ~ \ ~u< J \ leA f.. ~ > J::: ~ \'t'C":'J \ '2 ot )<..

b) What is an expectation value, and why does it vanishfor the momentum of a particle in a stationary state?

OvioeJv'L ~...t. +O\!J ~ .

~r "/. tJ >- Sc\)< LV \:.CXJ C/ t v()-o--J

-. I I·~ _ '..1- U~ ik F>-ot'-.V\ c/ -~ P - - (;~;F1.

c) Consider a infinite square well potential in 1 dimension with walls

a~ x=O, and x=L. EXPLAIN WHY THE wave function is a sine and not a cosinew~th argument kx. '

L{)CL'1.- "t Fry. '-,~ nu.t-J ("""(;/)'\(..<.)(\ c{; i< :::..{.'J CL-1-Pc "."1-~__ry(!,'>, lex C..cl>C0 fk-1'

J.11,'"') f/1L-&<-A..~ c{.~k.v ",v{~ Sctvt--J ~J1A. &~t:U~~_ ij LAVL. I t.;(..

'f¥L.GI.-# h.~ ~J uJd. j;~ c.<.~U6\. 1*-1,.--t~'YI-eP-,& C/Yldt ( .

d) Show how the above argument also fixes (quantizes) the valuesallowed for the wave number k and thus also the energy of the quantum state.

-

TEST 2b April 6, 2001 NAME:. ~\Jok~............. PHY242 -- 242H page 4

[H4/5%] Problem 4: PART A------------------------------------------------------------------------------

Compare quantum particles bound in infinite' and finite square well potentials,-------------------------------------------------------------------------------------

a) Sketch three lowest energy eigenvalues and wave function in a figure below.Make the range of the wave function in the well clearly visible,show with a dot where the wave function vanishes. Relate the quantumnumber n (what range of values?) to a feature of your picture:Among comments: Using physical arguments, compare the eigeneneries forfor infinite and finite (depth/height V_a) well potentials of width L:

I Figure: INFINITE WELL------------------------------------------------------------------------

I ~ A I n\ '--:-_V,

I ,- 'lY1..C-t'lt/V .t:ivrt:./~\-'_0.t.l~t - "~d3 . 1 s.~tt:] d/Y\. ~-S.~DyUty

r ,,--/ . I .M-(ftv\C,YlO.-V\ Otl\ t'~"\'~hkAf\'i-.

I I QA~-I"V01) l' ~~~ c~-'OtJ~ -wvt<..I I J\ ~ lio!4t f"t~t..1VY\r'~\ waG

I I k,n'<.Y\ - J.- I~t<} ,1/1t"u&. £,.If,t(I I ~{,w \;\.I-~~ ~vF=" rt~1,1.~\.(;tA- b~vftJ

~~l J ~

-I -. IVII'- ~~'-'Lh hi.-tR J~ e~c<tv'-\

I I ~v1;{.,\/'ttvr1.6 U. t\ CjL-'-tC/yVlr.k1 tv~I I ~v\.~L- C"/(.:~r (,\'\~Q-11~ Joe .~/ e/)1J1- ';J

n ¢L I 'J - _ ~ .,: . ,: •.

-I .. , r:6..t-;-v,...L{", ] W P vnA-o c£6m,~•...lI)I L ~ I~;): Cf& '" ~'-, \I••.• ~ f)., A1;C~ ~v'f.~~b) Using the formula for the eigenenergy of a quantum state 'n' i~Jan (V'

infinite well potential estimate the number of bound states you expect

to lOcalize. in a well of width L=4Angstrom and dePth~he;.ightV_O=12eV: ..,.,(;/, _

£,,\~ t\1.I\Lrr-l <.. i./ J\::!. Y'1?-<:: ~\Y\Vol-\ ~~I~P( 41\)'212<:1/= ~~~I~'~I~»M'

• .t-IY\ [2.- O .. -i\ LlL 1. . ~,,~ C:.) 2. -~ fI'1\ -l. U,l '\~~wlf~~-:::? h =-1 (t,V'(,\ .V\ :::..2 _~~V'-<I\ ~.~'IO'r\ yrC~J~) =- 5"

'1'\ -=- ~ CYY-i'-- ~ h./,-, u/).'\.lJ&"'C1-~\.. 'BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUS BONUSFOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL FOR ALL

ICOMPARISON COMMENTS

--. -

Figure: FINITE WELL

, ..

====================================================================================

PROBLEM 5 [3% for all]Consider an infinite well (square box) of sides L in 2-dimensions. Presenta) normalized wave function and b) eigenenergies, c) show / explain the quantum numbers,and d) discuss

o L

\~t.L=r\ 'f.. ( f"

~::GL== ~\J\1f\~J(\1 ~A, LI -~

'1=- ()