Physical Chemistry IIIExamples and Exercises

Normalizing a WavefunctionFind a normalizing factor of the hydrogen’s electron wavefunction

01

!n

axn

a

ndxex

1sin0

2

0

2

0

/22* 0

dddrreNd ar

0/ arNe

220

/222 0 drerN ar

1224

302 a

N

2/1

30

1

aN

0/

2/1

30

1 area

Function of r, using spherical coordinate

Eigen eqaution

Show if eax is an eigenfunction of the operator d/dx

Show if eax2 is an eigenfunction of the operator d/dx

axax

aedx

de

OperatorConstantFunction

2

2

2 axax

axedx

de

Orthogonal wavefunction

Both sinx and sin2x are eigenfunction of d/dx, show if sinx and sin2x are orthogonal.

0* dji

Cba

xba

ba

xbadxbxax

)(2

)sin(

)(2

)sin(sinsin

0)21(2

)21sin(

)21(2

)21sin(2sinsin

2

0

Cxx

dxxx

Expectation Value

Calculate the average value of the distance of an e- from the n of H-atom

dXX ji *

0 0

2

0

/2330

* sin1

0

dddrera

drr arji

0/

2/1

30

1 areH e

a

pm 4.792

322

2

!3104

40

30

aa

a

01

!n

axn

a

ndxex

Uncertainty Principle

Calculate the minimum uncertainty in the position of mass 1.0 g and the speed is known within 1 mm s-1.

2

1 qp

Uncertainty in position along an axisUncenrtianty in linear momentum

m105

ms101kg100.12

Js10055.1

22

26

163

34

vmpq

Probability (Particle in a box)Wave function of conjugated electron of polyene can be

approximated by PAB. Find the probability of locating electron between x=0 and x=0.2 nm in the lowest state in conjugated molecule of length 1.0 nm

ll

n dxL

xn

Ldx

0

2

0

2 sin2

L

nl

nL

2sin

2

11

05.0 when n=1 L=1.0 nm and l = 0.2 nm

Harmonic Oscillator

Find the normalizing factor of Harmonic Oscillator wavefunction

4/12

2/ 2

mk

xyeyHNx y

' if!2

' if0 2/1''

2

dyeHH y

1!2

2/12

2** 2

N

dyeyHNdydx y

!2

12/1

2

N2/1

2/1 !2

1

N

Harmonic Oscillator

The bending motion of CO2 molecule can be considered as a

harmonic oscillator, find the mean displacement of the oscillator

4/12

2/ 2

mk

xyeyHNx y

0 2

1

222

11'

dyeHHdyeHHdyeyHH yyy

dyeyHHN

dyexHeHNdxxx

y

yy

2

22

22

2/2/2*

11 HHyH

ExercisesCalculate the speed of an electron of wavelength 3.0 cmCalculate the Brogile wavelength of a mass of 1.0 g travelling at

1.0 cm s-1Calculate the probability of a particle in ground state between

x=4.0 and 5.0 cm in a box of 10.0 cm lengthCalculate the probability of a hydrogen’s electron in ground state

to be found within radius a0/2 from the nucleus

ExercisesIdentifiy which functions are eigenfunctions of the operator d/dx

eikx

coskxe-ax3

Calculate the energy separation between the levels n=2 and n=6 of an electron in a box of length 1.0 nm

What are the most likely locations of a particle in a box of length L in the state n=3?

ExercisesWhat are the most likely locations of a particle in a harmonic

oscillator well of lin the state =5 ?Confirm that the wavefunction for the ground state of a one-

dimention linear harmonic oscillator is a solution of the Schrödinger eqaution

Write down the Harmonic Oscillator wavefunction in the state n=0 and 4

Write down the Rigid Roter wavefunction Y0,0 , Y1,2 , Y2,1 and Y2,-2

and calculate their energies