Spectral and Wavefunction Statistics (II) V.E.Kravtsov, Abdus Salam ICTP.
Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s...
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Transcript of Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s...
![Page 1: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/1.jpg)
Physical Chemistry IIIExamples and Exercises
![Page 2: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/2.jpg)
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
![Page 3: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/3.jpg)
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
![Page 4: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/4.jpg)
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
![Page 5: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/5.jpg)
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
![Page 6: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/6.jpg)
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
![Page 7: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/7.jpg)
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
![Page 8: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/8.jpg)
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
![Page 9: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/9.jpg)
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
![Page 10: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/10.jpg)
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
![Page 11: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/11.jpg)
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?
![Page 12: Examples and Exercises. Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate.](https://reader036.fdocuments.in/reader036/viewer/2022082517/56649f1a5503460f94c2face/html5/thumbnails/12.jpg)
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