Tutorial-1

Atomic Structure
1. The Bohr model for single-electron atoms predicts that the energies of the allowed
orbitals are given by
E n  2.18 1018 z 2 / n 2 joule
Where n =1, 2, 3, ….,  and z is the nuclear charge.
i) The Balmer sericese corresponds to transitions between which energy
levels in the hydrogen atom? Does the paschen series appear at higher or
lower wavelengths than the Balmer series?
ii) Calculate the minimum energy (in J) required to ionize an electron in the n
=2 state of hydrogen atom.
iii) Calcualte the wavelenght (in nm) of the photon emitted by an electron in
making the transition from the n =3 to n = 1 level fo the single –electron
ion, Li2+,
iv) Is the photon higher in energy or lower in energy than the transition from n
=3 to n =1 in Be3+? Explain your answer.
5+5+5+5
2. The wave functions for a particle in a one-dimensional box (length L) are
2 nx
x  sin n  1, 2,3,.....
L L
Where n is the quantum number and O  x  L is the along the length of the box.
d
Find expectation values of the momentum p̂  and square of the momentum
i dx
d2
p̂ 2   2 2 for the n =1 state of the particle in the box. Interpret your answer.
dx
10
3. What do you mean by the terms Eigen functions an Eigen values? Which of these
functions is /are Eigen factions of the d/dx operator?
i) Exp (ax). ii) exp (ax2), (iii) sin (x), (iv) 25 x2
4. Draw the radial part of the wave function and radial probability distribution for
hydrogen atoms for 1s, 2s and 2p orbitals.
10
5. For a paticle confined in a one – dimensional box of length L (o  x  L), calculate
the values of the average position (<x>) and momentume (<p>). 10
6. The basic schrodinger equation for a hydrogenise (l-electron) atoms is
h2  Ze2 
 2
    E
2  4 0 r 

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 i) Explain what the terms in this equation means. 5
ii)  is the reduced mass of the electron –nucleus pair. For hydrogen, how
does it compare with me, the mass of the electron? 5
iii) Evaleuate the average electon –nucleus separation for the ground state of
a hydrogenic atom:

 r   *r 4 r 2dr
0

giventhat  r ne r dr  nn!1
0
And the waveefunction is given by
1/2
 
 Z3  Zr /0
 (r)    e

 03 

 

Where Z is the nuclear charge and a  4 2 / m e2  52.9 pm is the

0 0 e
Bohr radius. 10
iv) Why is this different from the msost probable radius (a0/Z)? 5
7. A particle of mass m is confined in a one –dimensional box if length L. the wave
functions  n ( x) of the particle are given by

2 n x
 n ( x)  sin ; n  1, 2,3,....
L L
n is the quantum number, and o  x  L is the distance along the length of the
box.
i) Sketch the waveefunctions for the states with n =1, n=2 and n=3.
ii) Describe what is meant by the Born interpretation,
Pn ( x)   n* ( x) n (x)
iii) Sketch the probability of finding the partcile within the box for the states
with n =1, n=2 and n=3.
iv) What is the averge momentum of a partile in a box? 10
Z 3/2
8. The wave function for 1s electron of Li is given by e zr / a0 , where a0 is
 a 3 1/2
0

equal to 52.9 pm. Calculate the value of radius at which the probability of finding
1s electron is a maximum. 20
9. Electromagentic radiation of wavelength 242 nm is just sufficient to ionize the sodium
atom. Calcuate the ionization energy of sodium in kJ mol-1. 12
10. Define and differentiate between the following: 4×3 = 12
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 i) Normalized and Orthogonal wave functions
ii) Eigen value and Eigen function
iii) Eigen value and Most probable value
11. i) Write down the Hamiltionian for the hydrogen molecular ion, H2+
ii) For a trial wavefunction of the form
  ca  a  cb b

Where

i is the normalized atomic orbital on atom I and ci are

coefficients, show that the normalization constnant is N = (2+2S)-1

where S = Sab = Sba is the overlap integral.
Assume that the e is all equal and that identical normalized hydrogen –like
waveffunctions are chosen.
By expanding the probability density,  * qualitatively confirm that
  (2  2S ) 1/2 ( a  b )

Represents the wavefunction for a bonding orbital of H 2 . 20

12. The wavefuctoin of an electron in a hydrogen is orbital is given by:
 1s (r) e r / a
0

Where r is the radial distance from the nucleus, and a0 = 52.9 pm is the Bohr
radius.
(i) Write down an expression for the radial distribution function 4 r 2 2 and
explain why is functions has a different physical interpretation to  2 Sketch 
and 4 r 2 2 .
(ii) Find the most probable radius at which the electorn will be found. Justify the
statement “the hydrogne atoms has a diameter of approximately one Å”.
30
13. The energy levels of the electron in a hydrogen atoms are given by
En   RH / n 2 , n  1, 2,3,......
Where RH is the Rydherg constant and n is the principal quantum number.
i) Draw a clearly labelled energy level diagram showing the first three
energy levels of the electron and all of the quantum states helonging
to each energy level.
ii) Give the degeneracies of the first three energy levels, giving the
allowed l and m1 values and the orbitals rotations. Suggest a simple
formula which relates the degeneracy of an energy level to its
principal quantum number.

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14. (i) Show that the function
 n x 
  N sin  
 L 
Satisfies the Schrödinger equation for a particle in a one – dimensional
box with a potential function V(x) equal to for O  x  L and infinity
elsewhere.
ii) What values may the quantum number n take what are the allowed
energy levels?
iii) Why is it important tht the wavefunction be normalized? Calcualte the
normalization constant, N.
15
 1  sin 2 
Given  sin  d  2   2
2

  
15. Deduce the following equation for a particle in a one dimensional box giving the
Hamiltonian explicitly:
 n x 
 n  A sin  
  
Also deduce the energy for the system. 15
16. What do you mean by quantum numbers? Discuss different types of quantum
numbers and their significance. 20
17. (a) What is meant by electron probability function? Draw the radial distribution
curves for is, 1, 2s, 2p, 3s, 3p and 3d electrons, and explain radial nodes, radius of
maximum probability and penetrating power of electrons. (10)
b) Give the schrodinger’s wave equation for H-atoms in cartesian and polar
co – ordinates. With the helps of a diagram show the relation between the
two coordinates.
18. (i) State and explain Heisenbrg’s uncreatainty principle. Why is the not applicable to
larger particles? (10)
ii) When an electorn was accelerated through a potential difference of 1.00 ±
0.01 kilovolt, what is the uncreatainty of position of the electron along its
line of projection? (10)

19. Discuss the significance of  and  2 with special reference to electron probability
function, probability density and radial probability of finding the electron at a
distance from the nucleus for 1s, 2s, 2p and 3s 3p 3d orbitals.
1  r 
20. Show that the wave function 1S  exp   for hydrogen atom is normalized.
a 0 3
 a0 

21. What is the degeneracy of the level of the hydrogen atom that has the energy.

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 (i) RH (ii) -RH/9 (iii) –RH/25?

22. A series of lines in the spectrum of atomic hydrogen lies at the wave lengths
656.46 nm, 486.27 nm, 434.17 nm, 410.29 nm. What is the 396.93 nm
wavelength of the next line in the series? What energy is required to lonize the
hydrogen atom when it is in the lower state involved in these transitions? H = 2,
3 =ev
23. Name a particle having both wave and particle properties, state uncertainty
principle. Can uncertainty principle be applied to flying Jet Plane? Why?
24. Derive an expression for the energy of particle in a box (one-dimensional only).
25. Draw the shapes of s, p and d orbitals. Show their directional characteristics.
26. What are four quantum numbers? Show that these are not same for cu atom.
27. What do you understand by the term orthogonality of wave function? Draw the
nature of 5d-orbitals.
28. Write schrodinger wave equn in three-dimensional form explaining
characteristics of wave-function.
29. What are quantum numbers? What are their significances?
For an electron in hydrogen atom moving in second orbit, whose principle
quantum no.  n  2, calculate.

30. Energy of electron in this orbit

(i) Radius of the orbit.
(ii) Frequency of the spectral line emitted when the electron jumps from second orbit
to the ground state.

(iii) For an electron in hydrogen atom moving in second orbit, whose principle
quantum no.  n  2, calculate.
31. Write notes on shapes and orientations of S, p and d atomic orbitals clearly.
32. Calculate the energy difference between first two energy levels for a particle of
mass 1 kg confined in a one dimensional box of 1 cm length.
33. Explain various terms in time independent schrodinger wave equation. Illustrate
the term Laplacian operator.
34. Reduce the following equation for a particle in a one-dimensional box giving the
Hamiltonian explicitly:

 nx 
 n  A Sin   also derive the energy for the system.
 a 

35. What do you mean by quantum numbers? Discuss different types of quantum
numbers and their significance.
36. Give the schrodinger wave equation for H-atoms in casterian and polar
coordinates. With the help of diagram show the relation between the two
coordinates.
37. What is meant by electron probability function? Draw the radial distribution
curves for 1s, 2s, 2p, 3s, 3p and 3d electrons, and explain radial nodes, radius of
maximum probability and penetrating power of electrons.
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38. When an electron was accelerated through a potential difference of 1.00 + 0.01
kilovolt, what is the uncertainly of position of the electron along its line of
projection?
39. State and explain Heinsenberg’s uncertainty principle. Why is it not applicable to
large particles?
40. Set up and solve the schrodinger equation for particle in one-dimesional box.
41. Expain Aufbau Principle, Hund’s rule of maximum multiplicity and Pauli’s
exclusion principle. What is their significance in writing electronic configuration?
Give suitable examples.
42. What do you understand by orthogonal wave function? Normalize the wave
nx 
function, n (x)  Asin   for a particle in a one-dimensional box of size ‘a’.
 a 
43. What are orthonormal wavefunctions Normalize the following wavefunction for
nx 
a particle in one dimensional box  n (x)  Bsin  .
 a 
44. Write schrodinger equation and its solution form. Interpret the equation with
respect to probability density, eigenvalues and eigenfunctions.
nx 
45. Normalize the wave function    C sin  .
 L 
46. What is the node in an atomic orbital and what role does it play in determining
broadly the relative stability of various types of atomic orbital’s with the same
principal quantum number?

47. Explain the difference between 1s and 2s orbitals and also between 2s and 2p
orbitals.

48. What are the applications of wave mechanics? Discuss particle in a one
dimensional box.
49. Calculate the kinetic energy of an electron in the ground state confined to box
5×10-8 cm in width and moving in one dimension (x-axis) only.
50. The ground state energy of an electron confined in a one-dimensional potential
box is 900 k J/mol. Determine the length of the box.
51. The function R® represents the radial wave-function for an atomic orbital. Show
the plots of R(r), R(r ) and r 2 R(r ) for 3p, 2s, and 3d orbitals.
2 2

52. An electron has been accelerated through a potential difference of 400 V. Find its
de Broglie wave length.
53. A particle of mass 1×10-30 kg is confined to a one dimensional box of length 1 nm.
Calculate the minimum uncertainly in its linear momentum in the ground state.
54. A hydrogen like atom gives a series of spectral line with the values: 41.0070,
30.3756, 27.1216, 25.6264, nm. What is the charge on the nucleus?
55. Determine the term symbols for carbon atom in its ground state. According to
Hund’s rule, which is the most stable state?

56. The electronic energy of helium atom is -79.0 ev. What is the first ionization
potential if the first ionization potential for hydrogen atom is 13.6 ev.

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57. In the ground state of hydrogen atom the wave function  is given by the
equation:   1 e r / a . What is the value of r with the highest probability
0

a 03
density? How does it differ from the expectation value for r? What will be their
corresponding values for helium ion?
58. Show that the 1s function for hydrogen atom is orthogonal to  2s and  2p .
59. In atomic units, the 2s orbital for the hydrogen atom is given by
1
2s  (2  r)e r / 2 . What is the probability of finding the electron in the 2s
32
orbital within a sphere of unit radius?
60. What is the significance of the uncertainty principle? Give a proof of this
principle. Write down the exact schrodinger equation for the motion of the
electron in the hydrogen atom. How does an orbital depend on electronic
coordinates including spin?
61. For a particle in one-dimensional box (length = a cm). Calculate the probability of
finding the particle in the middle of the box from a/4 to 3a/4 in the ground state.
62. What is the de Broglie wavelength of an oxygen molecule at 300 K? Compare this
to the average distance between 2 neighbouring oxygen molecules in a gas at 1
bar at 300K. [average speed of oxygen molecule may be estimated.]
63. What are degeneracies of the following orbitals for hydrogen-like atoms (i) 1s
(ii) 2s (iii) 2p
64. Draw the radial part of the wave-functions for hydrogen atoms with n=3, l=2;
n=3, l=0; n=2, l=1 in different ways.
65. What are the conditions on the solution of schrodinger equation for its
acceptance as a wave-functions for hydrogen atoms with n=3, l=2; n=3, l=0; n=2,
l=1 in different ways.
66. The energy levels for a particle of mass m in a one-dimensional box of length L
n2h2
are given by the equation En  2
. Considering m  1.0 1030 Kg, and L  5.0 1010 m
8mL
and h  6.6 1034 J  s, calculate the frequency of light which may raise the particle
from n=5 to n = 6. Is it possible for n to be equal to zero?
1
1 2 r
67. The wave function for the ground state of the hydrogen atom is    3  e .
 nao  a 0
Derive an expression for the probability of finding the electron inside a sphere of
radius a0 centered at the nucleus.
68. Calculate the energy of a photon of radiation of wavelength 150 nm.
69. Show that the application of schrodinger equation to the problem a particle
confined to a one dimensional box, leads to discrete energy levels.

70. Explain normalized and orthogonal wave functions.

71. Considering an electron of mass 9.1×10-28 gm moving with a velocity v for an

uncertainty of 1 A0 in its position, what is the inherent uncertainty in its velocity?
72. A particle of mass m is confined to a one-dimensional box of unit length (i.e.
0  x  1 ). Derive expressions for i) The allowed energy levels, and ii) the average
(expectation) value of x for n=3, state where n is a quantum numer.
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73. What are the important conclusions from the ‘particle in a one-dimensional box.”
74. What is a normalized wave functions?
75. Draw rough diagrams to illustrate the nature of the radial distribution functions
for the following hydrogen orbital and give the magnitude of the orbital angular
momentum of the electron in each of these orbitals: 1s, 2s, 2p, 3p and 3d.
76. A particle of mass m is confined to a three-dimensional box defined by

77. V = O for 0  x  a,0  y  b,0  z  c

78. Find the allowed energy levels of the particle

79. Explain and illustrate the meaning of the terms: Eigenfunction and eigen value:
bonding, anti bonding and non-bonding orbitals.
80. Write the Schrodinger equation for hydrogen atom, giving the meaning of the
terms.

81. What is schrodinger wave equation? Show how it can be solved for the case of a
free particle in a rectangular three-dimensional box.

82. What is the interpretation of  (wave function) and  2 (square of wave

equation).
83. Calculate the de Broglie wave length of an electron that has been accelerated by a
potential difference of 1000 volts.
84. For a hydrogen atom in a one-dimensional box, 1 nm long, calculate the value of
3
the quantum number of the energy level for which the energy is equal to kT at
2
300 K.
85. State Heisenberg’s uncertainty principle. How is the occurrence of zero-point
energy of the particle in a box is in accordance with the Heisenberg’s uncertainty
principle?
86. Write short note on: Pauli’s exclusion principle.
87. Derive the energy levels for an electron in a one-dimensional box and point out
how this model can be used to account for spectral properties of conjugated
polyenes.
88. Write notes on: Zero point energy, Hamiltonian operator, Normalization of
function.

89. Write note on: Quantum mechanical approach to Pauli’s principle.

90. (i) For a particle in one – dimensional box of length a calculate the probability of
its locating between zero and a/2.
(ii) Write the complete wave function for 1s – orbital and deduce its shape.

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