chem 482 - hw 3

7/28/2019 CHEM 482 - HW 3

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Chemistry 482Homework Set #3Due Feb. 18, 2013

1. The solutions to the Schrodinger Equation for the 1-D particle in a box are

nx

n x( ) sin

2

(a) Determine the probability of finding the particle in the left quarter of the box.

(b)

For what value of n is this probability a maximum?(c) What is the limit of this probability for n

2. Consider a particle in a box of length in the wavefunction, 330( ) (1 )

x x x

,

a. Determine the expectation values x and 2 x

b. Determine the expectation value E for a particle in a box described by this wavefunction

3. Consider a particle trapped in the following three-dimensional rectangular box:z.

yx

a. Provide the ground state wavefunction for this system

b. Determine the energies (in units2

2

h

ma) and the quantum numbers for the five lowest energy levels.

c. What are the degeneracies of each of the six lowest energy levels?

4. Acceptable wavefunctions must satisfy the following criteria:

function must be well behaved (i.e. finite, single-valued, continuous)

must satisfy boundary conditions

)()(* x x

x must be associated with probability

Consider a particle in a one-dimensional box, defined as before as V(x)=0 for 0<x<a and V(x)= for a x . Explain why

each of the following unnormalized funcions is or is not acceptable.

a.a

xn A

cos b. )( 2

x x B c. )(3a xCx d.

)sin(a

xn

D

5. Evaluate the commutator ˆ ˆ, A B for each of the following: ˆ A ˆ B

a.2

2

d

dxx

b.d

ixdx

d

ixdx

c.2

2

d ix

dx 2d

ixdx

7/28/2019 CHEM 482 - HW 3


1

(G)

Ps #-3( Q )

n 7t ~ let ,i - = Z

n= !2 j<

n:= 4k-t-1

n =4k+~

(n=3 ) (n 3-0 )(D -mo.-x

(n-"700)

n -7 00 f)}rr , ~ 0

'p. - I (n.;:;><XJ) - :tr

7/28/2019 CHEM 482 - HW 3


.

(Q)

yf)

7/28/2019 CHEM 482 - HW 3


3.

ELL Jit(rfJ.~ '6'

f1-V

/ 4-g12

"F

~

<6

JL.~

o

(C)

_ JL ( n; + _ti,; + 4n: )8m (j1 0.

2

02

_ h.2. :2 Z ~

- - : (t)?( + n~ + 4n z; ) =

(nx=-l/ n~ ::: / , n z == 2 )

( y)x = 3~ n n= 2 --'ni -= I 0r n x =-2 / n~= 3/ ~ =-1 )

01x::= 3, n~ = L, n~ =-/ rx n :x =- /, nj = 3~ Yk=f )

{n?(:::2, YlIj=2, Yli?:=-I 1( n'X =2 / n~ =-t nt =- f 8. n~ =I, n~ =-2~ n& = I )

f I 2 I 1 / 2 r 2" I

Dee~(ft~

J

( :) j 2) I ) ( 2 j -3 / I )

(6 j

I, () ((,3/ })

( 11 2/ I)

(J,L I) (/ ~ 2,1)

(I, I f I )

Q.

2

I

2

7/28/2019 CHEM 482 - HW 3


t.ea)

N D t acceptable

bOWlJ ~ CClTIditrllTlS

let X= 0 A 00;n:x = A

I rnu>t ::::.0

if A to , nat o.cceptrWie i::ecoAAse f I:ourdruy wndJ-t,

rf A ~ , no In;tU'e6"O'fl&

(b) B( 9( +'XJ.)

(\jot o.CCe,ptn1Je bec!LLl>f? '1 bmwd'~ Grmc!JfrivT/s

(C)

C.t.X 3 ( ?[-0 )

Acce{Jio.bJe ~ sl't)le - VcMlAecf fl C<mtfnIACUS: I finJte MJ

~iJQdm;h'aJ./~ tnterorolk We{' (0, oJ

D

7/28/2019 CHEM 482 - HW 3


/\ II

[ A' B J

" 1\ ~A B jl'X)=

1\ f\

/3 A fex) =- ?( ~ fc ] == ?( 1f£C(:lI'X 'X ) d rx2.

.1\ /\ 1\ 1\ J!f _(A B - B A ) i(IX) = .2'~

1\ 1\ ~ /\

[ A / BJ;=. 2F I

A g J('X);c C}~- i'X )[(-1;- + ,''>( ) jew) J

- (-j'». -1'1)() [ -1£- + j'<x fl'X ) J

di ~Yif '-C ,~. 2f -a7 - 1'1\ F + 1J ~.'X j + v)( dl)( + rx (~)

t -t ifh) +,)(zfl'¥l

B A~'li):=- (i- -H'X ) [ ( 5:x -1'')1) f (')l ) J

- r:h- -h''X) [ f -l'?ff(W) l

_4 T i?'l ~ - ~')( - l' ]1'X) + ,~/ fc'X)

J:JL )

- w - l'fC-X) + 'X Jr'X)

/\ 1\ 1\ l \ ..p. . f(A B -13A )f(~) = < ~1( -X) + 1)'- ( If)

-'" 1\ ,~ 1"\

[Ar B] ::- (12.~~) I

--

7/28/2019 CHEM 482 - HW 3


C) 1\ 1\

A £> f(?1) = I ~: - i?() [ ( i- -r i'X:Z 1ft?: ) J

- (JJ!2 __ t')( ) [ L -t i;lf ]~ of/)(

A _if)( -#-

T i [ .2+-1-2)(

#-+ :2')( off +)(2d~3 u~ uX db(

t ){tJf

- ~ 'Ai'M- -I- ,d-f , ~ ' 3f -(Jx.I~- + 1'", of'X:2 -o1:X~ + 2 1

J + w'

BAr'X):= ri-+o"i! [ Ii;- -~'x )t(?()]

- c-ix + 1;7< '-) [ ~ ,- 1''Xf J

_83(' ~ JZL " [ f d-t.] 3L-:= ~' -t l?< ~ - t + ~~ T?( J

r- J3J: +" .2 J,l( '"p J..c'f + ?>L~ ,/X -1L1+;- -1'/1 ~ - '];' 0(-j

J'X otx ~I)( ,

(A B - BA lr'X) = 4-1?1-f- + 31f

/\/\. JL A[A / 8J::( 4-X1' oI~ + 3} )1

chem 482 - hw 3

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