7/28/2019 CHEM 482 - HW 2

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Chemistry 482Homework Set #2Due Feb. 11, 2013

1. Show that ikx ikxAe Be and cos( ) sin( )C kx D kx are equivalent ways of writing the samefunction of x, and determine the constants C and C in terms of A and B.

2. Any sum of the stationary-state solutions for the one-dimensional string,

xntFtxu nnn

sin)cos(),( , is also a solution to the transverse wave equation. Consider a

string for which 1 = 2 radians/sec, 1=2=0, F1= F2=1 and =1. Using a spreadsheet or some other

graphing program, plot u(x,t) at intervals of 0.1 sec for t = 0 sec to t = 1 sec. Explain your results.

3. Determine in each of the following cases if the function in the first column is an eigenfunction of the

operator in the second column. If so, what is the eigenvalue:

a. 3 3x ixAe Be 2

2

d

dx

b. cossin

2sin6)(sinsin d

d

d

d

c. 22 yx x

yxx

)(

1 22

d. 2 2 2x y z 2 2 2

2 2 2x y z

4. Find the operator A2

when

(a) Ad

dx

d

dx

2

2(b) A x

d

dx 2 (c) A

d

dx x

1

5. The function ( ) (1 )x

x Ax , where A is a constant, is an acceptable wavefunction for the particle in

a one dimensional box of length .

a. Is this ( )x an eigenfunction of the one-dimensional particle-in-a box operator? Justify your answer

by direct operation.

b. Determine the normalization constant A

c. Determine the expectation values x and 2x

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

7/28/2019 CHEM 482 - HW 2

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," e19::: CJJSf} +isi'ne

.', elk?< == CJ;5(fkX)+ isfn (fl

7/28/2019 CHEM 482 - HW 2

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_?i+ '0 [ 2.. 2. 2 1. 6.,< .::: 8nv~ aJSe) - Sine - 1sfne --1si'n@] + G~lnB~e

2 2 J 3 IZJ- SI\t'I e me [ cJJ; {j'- 5 stfl (j + 6sme$v

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3:3 \ :2 ~.2-

__ s[\nBCiJ56

+ si'nf} caJ3=

smBCJJS(J [me 1-

SIne]

-)

4.

Let)

7/28/2019 CHEM 482 - HW 2

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1\ 1. ot_A==~+ 7?(

/\ ) J .1 rlbl)/1'). ==- ( 01 +F) ('X-jL')() + o{~

:l Jfc?() z$ + ~:::- 'l-f?l') +- ;;lXf(~) T ~ ci?r t- :x o-lx o!?I'

4- ~d ,L).-+,::. (?f +;) f:2 ~ -~;- ~ ~ J'-. ~ .

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5 . (cA ) ~ ( ?1) ~ A 'X (/- t)1\ .-+.:.2 L-H:= - .-lL- -J. :l, I :lrYl (/IM

(1)

nut ~ ~e4mato-n.'J: !{)t;) l/J ("4) 0191 = i

r;- l" ')I;). ( j- ft (hI'" IA~ JJ ~2(I-t)2oi'X~1...2. J ~r 011- 2')[2> .!2

A 0 r:r: - T + ?1 )ofw=f

A2(~,f;? -

f.e?>+d13)=1

A - J,30 .- 13

7/28/2019 CHEM 482 - HW 2

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,...;

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