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Chemistry 482Homework Set #4Due Feb. 22, 2013

1. A particle confined to a 1-D box of length a has the wavefunction:

{a. Make a sketch of the particle-in-a-box potential and this wavefunction.

b. Determine the Constant A needed to normalize the wavefunction.

c. If you make one energy measurement on one particle, what is the probability that you will measurethis average energy? Provide a brief explanation of your answer.

2. Suppose that the wavefunction for a system can be written as 1 2 31 1 3 2

( ) ( ) ( ) ( ),2 4 4

i

x x x x

where 1(x), 2 (x) and 3 (x) are normalized eigenfuctions of the kinetic energy operator T witheignevalues E1, 3E1, and 7E1, respectively.

a. Verify that )(x is normalized.

b. What are the possible values that you could obtain in measuring the kinetic energy on this system?

c. What is the probability of measuring each of these eigenvalues?

d. If you were to perform a large number of measurements of the kinetic energy for this system, whataverage value would you obtain?

3. For the simple harmonic oscillator determine:

a. [ , ]H x b. [ , ]T V

c. T Vand for the harmonic oscillator eigenstate2

1/43

21

4( ) ,

x kx xe

where

4. In the infrared spectrum of Li7H1 there is an intense line at 1406.6 cm-1

a. Determine the force constant k in units of N/m and in units of dyn/cm (1 dyn = 1 gcm/s)

b. Find the rms (root mean square) displacement of Li7H1 in the ground state, for which =159.5 pm

c. Compare the rms displacement to . By what percentage is the bond compressed?

( ) 0 / 2

( ) ( ) / 2

x Ax x a

x A a x a x a

for

for

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2

v. =- 0

v:::: co

(b) tV urm cJilxtioYJ'\0 11r* , aft 1f 0 0/ ('X) l{F ex) x:::

a. r.l ,

f- J -tit' fV

\ 'l/' "I.lf- 71r j 1. ~ /f(f)f) L/1('X) c;if)d + ~ ~(?4) 1(?() {AX =

(A rIAS; A:l9(? Jx + Jt A2 (rA-ex/ J?( = I

]1 ().3 .2.( 2(). . It" I 3z_A "-~ (z:) + A Q 2 + a ~ a - 1~ - 4 a ) - (

i-A2(l= I

~ A= T/2-7 ~J(fJ

(e) To -rnd (f> , 901fe~ toke Ip;tt~I'N " fo,i.fi~H r oleif We s1(l11fi~Me {n.i~foJ~ 0

D

Jll *2 dt ('" ~oI' -

':E> == ; rjg; X) (- 2m fi) iff, -x ) d?t + l~,[g; (o.-vf-ld~ )j(a-we 0bt~() < E- > =0 , l+,{l; Vic{aie U!JcertaJ1 6nd -fAj, oJso

Ulolccte pvsiuh-te 3 (rrM~t meCt5ure e~e,nuVues)

.' lt1D tro~olJi'Utfet7 1- meosurJ~ =O~ vi'cJ Nete : /?x? cause 1:- (s not con;t111ilOUs @ x", ~ ,the a vtu

irU~ 1- [lJriN;;h: ~'; Sfectal h~,J11'nU

Fur. mlAi0, crfiCi'fJI)ado bRAe is what 1'5 nmded 10 -treat #.e J~.

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--

e 13erwOv//Aeh 1- eJJ~ ~ .

H~kr8 UfIcerin.!'n1y

(?A'SeJIIJcJi,{ e.s /

Su ':0 ~!dfJe6n't- Violate

bvLt IT 51Jr'11 fsn~ one 01 ~

PNJbOvbIU~ I~ Si1'/I ~e10

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2

b.

o.

-Jir = .ltD I 10 ,.j2-}''f ['X) .2.'11(1)() t 4,:)(rx)+ 4- tp3l~)

J-o/:) tfc'X) cix =- J [t p~'X) t Ii r~'X) ++ if(.(?(} ] 0/",/\

(redA'3e -the ei6e1ltnctPn5 ~ T rw Dr~~ here)

J'1fr~ 11(" ..~ - r J ::2 .L . S' I ~ . I/1~ (~) 'f (~) C{~ - J 41j71 ('/()(}.?(T Iblf/:z ('X)d~ + 16 fa

- 1..J.......L..J-lL - I-f" 16 I 16 -(reoJAi eLlen e{{rfunctfon /5 rorma1J~ here)

The un~ po$ible ualue!.> VU1LIoI' obto.ln is E" af3./ OJlJ 1E.1which ()i'e ~ eI'aer;xJlAw 1 eaen @jenf'lMlctton

G FOT G, P = (v = i)'

For ?:EI I

p= (f)2= tc:

fur 1E/ 1)=( ..9Wl' ) ( 3-,M _ ..!.L

I

4- 4-) - tZ;

d ::: P, TI + P2. T.2 -+ P3 T3

_ .-l r +- .L?r t 1L .:J..r - 1!LE.- 2./ E4- . C-,. IhJt:. ) if? I c r - 16 I - T I

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3.

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.2 -2

== /-0/8 x /0 k,g s2 /0-000 ern I

- {,of8 x [0 f~8 I k tl 'em '5