Quantum chemistry-B SC III-SEM-VI

1. Dr. Y. S. THAKARE
M.Sc. (CHE) Ph D, NET, SET
Assistant Professor in Chemistry,
Shri Shivaji Science College, Amravati
Email: yogitathakare_2007@rediffmail.com
UNIT- V
ELEMENTARY QUANTUM MECHANICS

2. Quantum mechanics is a branch of science of motion of micro like
atoms, protons electrons (Subatomic Particles) particles etc. Its
study is important because of physicochemical properties atoms
and molecules, structures, spectral behaviour and even reactions
can be interpreted in terms of the motion of micro particles.
The branch of science based on Newton’s laws of motion
and Maxwell’s electromagnetic wave theory to explain phenomena
related to motion and energy is known as classical mechanics.
Classical mechanics could not explain the phenomena like black
body radiation spectrum, heat capacity of solids, photoelectric
effect, atomic and molecular spectra etc.
Development of quantum mechanics starts with Planck’s
quantum theory which was following by de Broglie’s hypothesis of
matter waves and Heisenberg uncertainty principle. Schrodinger’s
wave equation leads to the formulation of wave mechanics.

8. Classical Mechanics Quantum Mechanics
i) It deals with macroscopic particles i) It deals with microscopic particles
ii) It is based on the Newton’s laws of ii) It is based on de Broglie’s concept
and motion Heisenberg’s
uncertainty principle
iii) It is based on Maxwell’s electromagnetic iii) It is based on Planck’s quantum
wave theory according to which energy theory according to which energy
is emitted or absorbed continuously is emitted or absorbed not
continuously but discretely
iv)The position and velocity (or momentum) iv) It gives the probability of finding
of the particle is defined precisely and a particle at various point in space
simultaneously
Comparison of Classical Mechanics with Quantum Mechanics

9. Atomic Orbital

10. Atomic Orbital

11. Electron shell

12. Atomic Structure

13. Black Body Radiations
Radiation (electromagnetic wave) is emitted by any solid at any
temperature as a result of vibrations of its particles. Different solids
emit radiation at different rates at the same temperature; the rate is
the maximum when the solid is perfectly black. A black body absorbs
all the radiation that falls on it and retains all the radiant energy. For
experimental purposes, a black body is generally a blackened
metallic surface, a hallow sphere, blackened inside with a hole. All
the radiations entering through the hole are absorbed completely by
successive reflections inside the enclosure.

15. Fig. 5.1 Spectral distribution of radiation from a black body at various temperatures

20. A black body is not only a perfect absorber of radiant energy, but
also an idealized radiator. The radiation energy emerging from the
hole will also be very nearly equal to that of the black body. The
energy emitted from the black body depends only on temperature
(T) and is independent of the nature of the solid.
The dependence of radiant energy density on temperature and
wave length is shown in Fig. 1.1. The classical mechanics fails to
explain the energy distribution of black body radiation. Max Planck
in 1900 explained the energy distribution of black body radiation
and was awarded a Nobel prize in Physics in 1918 for proposing the
quantum theory of radiation.

21. Characteristics of the curves
For each temperature there is a particular wavelength at which
energy radiated is the maximum.
With increase of temperature, the position of the maximum shift
towards the lower wavelength.
Higher the temperature, more pronounced is the maximum.

22. Planck’s Quantum Theory of Radiation
In 1900, Max Planck studied the spectral lines obtained from
radiation emitted by hot black body at different temperatures and
put forward a theory known as Planck’s quantum theory of
radiation, The various postulates of this theory are:
1. A hot body emits radiant energy not continuously (not as
continuous wave) but discontinuously or discretely. In other words,
a hot body emits radiant energy not as continuous wave but as
small packets or bundles or discrete unit of waves. Each of this unit
is called a quantum (plural quanta) which can exist independently.

23. Fig. 5.2 Emission of radiant energy from heated metal ball
a) Continuous waves (according to Classical old theory)
b) Discontinuous waves (according to Planck’s quantum theory)

24. The energy associated with each quantum or photon (in case of light) is
proportional to the frequency (ν) of emitted radiation, i.e.
E α ν
or E = h ν ---------------------- (1)
Where h = Planck’s constant = 6.626 Χ 10−34
J s
ν = Frequency of emitted radiation or light in Hz (or CPS)
But ν =
𝑐
𝜆
Where c = Velocity of light (ms-1)
𝜆 = Wavelength of radiation (m)
 E =
ℎ 𝑐
𝜆
------------------------------- (2)
The energy emitted or absorbed by a can be either equal to one quantum of
energy (hν) or any whole number multiple of it, i.e.
E = n h ν ----------------------------------- (3)
Where n = 1, 2, 3 …
Thus energy emitted or absorbed by a body cab be equal to hν, 2 hν, 3 hν
etc. but never equal to fractional value of hν like 1.2 hν, 2.7 hν, 3.6 hν etc. Thus
the energy emitted or absorbed is quantized and this is called the concept of
quantization of energy.

25. Applications: It is used
To explain the photoelectric effect (1905)
To develop Bohr’s structure of atomic model (1913)
To explain the Compton effect, i.e. scattering of x- rays (1923).

26. Photoelectric Effect
The phenomenon of emission of electrons from the metal
surface when light of suitable wavelength falls on it is known as
photoelectric effect. The emitted electrons are called
photoelectrons.

27. his became known as the photoelectric effect, and it
would be understood in 1905 by a young scientist named
Albert Einstein. Einstein's fascination with science began
when he was 4 or 5, and first saw a magnetic compass.

28. Fig. 5.3 Photoelectric effect Experiment

29. Fig.1.3 illustrates an apparatus for demonstrating
the photoelectric effect. In a highly evacuated tube,
when a beam of UV light falls on the metal surface M,
electrons are ejected from the surface. These electrons
are pulled to the anode P by high voltage V, thus
completing the circuit. The flow of current is measured
by a galvanometer G.

31. This current is known as photoelectric current. After studying the
photoelectric effect carefully, the following observations are made:
1. For each metal there is a minimum frequency of incident light
below which no electrons are ejected or emitted. This is known as
threshold frequency, υ0. The threshold frequency is different for
different metals.
2. The kinetic energy (or velocity) of the emitted electrons is
independent on the intensity of incident light but is directly
proportional its frequency.
3. The number of ejected electrons from the metal surface
depends on the intensity of incident light. The greater the
intensity, the larger is the number of ejected electrons.
The photoelectric effect was explained by Einstein (1905) on the
basis of Planck’s quantum theory of radiation. According to
quantum theory, the radiation consists of packets of energy called
quanta or photons of energy hν, where h is the Planck’s constant
and ν is the frequency of radiation.

36. When a photon of energy hν is incident upon a metal surface, its energy is utilized in
two ways:
A part of its energy is used up in ejecting the electrons from the metal
surface. This energy depends on the nature of the metal and is called work function
(φ) of the metal.
The remaining part of the photon energy (hν – φ) is converted into kinetic
energy of the ejected electrons. Thus
h ν = φ +
1
2
𝑚𝑣2
Where m = Mass of the electron and v = Velocity of the electron
But φ=h ν0 ---------------------- (2)
Where ν0 is the threshold frequency. Therefore equation (1) becomes
h ν =h ν0+
1
2
𝑚𝑣2
h ν =h ν0+
1
2
𝑚𝑣2
h ν −h ν0 =
1
2
𝑚𝑣2
1
2
𝑚𝑣2 = h ν −h ν0
1
2
𝑚𝑣2 = h( ν − ν0 ) ----------------- -- (3)
Equation (3) is called the Einstein photoelectric equation. Einstein won Nobel prize in1921 in

37. Compton Effect (1923)
A.H. Compton found that if monochromatic x-rays are allowed
to fall on carbon or other light element, the scattered x-rays have
wavelength larger than the incident x-rays. The phenomenon in which
there is a change in wavelength of scattered x-rays is called Compton
Effect. The angle through which the incident x-rays are deflected from
their original path is known as angle of scattering and is denoted by θ.
Fig. 5.5 The Compton Effect

38. By applying the laws of conservation of energy and momentum, and
by assuming that x-rays are composed of photon each of which has
energy hν, Compton gave the relation
∆𝜆 = 𝜆′ − 𝜆 =
ℎ
𝑚𝑐
1 − cos 𝜃 =
2ℎ
𝑚𝑐
sin2 𝜃
2
Where ∆𝜆 = 𝜆′ − 𝜆 = Increase in wavelength
= Compton wavelength shift
ℎ = Planck’s constant
c = Velocity of light
𝜃 = Angle between incident and scattered x-rays, i.e. angle of
scattering
Here 𝜆′ > 𝜆, i.e. wavelength of scattered x-rays > wavelength of
incident x-rays
The above equation clearly shows that ∆𝜆 depends on the
magnitude of the angle of scattering,𝜃.

39. Case 1: When 𝜃 = 00, ∆𝜆 = 0, i.e. the scattered light is parallel to incident light.
Case 2: When 𝜃 = 900, ∆𝜆 =
ℎ
𝑚𝑐
=
6.625 × 10−34
9.1 × 10−31× 3 ×108 = 2.42 pm (∵ cos 90 = 0)
Case 3: When 𝜃 = 1800 , ∆𝜆 =
2ℎ
𝑚𝑐
=
2 × 6.625 × 10−34
9.1 × 10−31× 3 ×108 = 4.84 pm
(∵ cos 180 = −1)

40. De Broglie’s Hypothesis
In 1905, Einstein suggested that all electromagnetic
radiations including light have a dual character, i.e. particle and
wave nature. The diffraction, interference and polarization support
the wave nature and photoelectric effect and black body radiation
support the particle nature of radiation.
In 1924, de Broglie extended the idea of wave- particle
duality of light to matter particles. He suggested that all moving
material particles like electron, proton, neutron, atom, molecule
etc. in motion possesses particle as well as wave characteristics, i.e.
dual characteristics. The wave associated with matter particle is
called de Broglie’s matter wave.

41. De Broglie’s Equation
In 1924, de Broglie suggested that the momentum p of a moving
microscopic particle associated with a wavelength λ such that
𝜆 =
ℎ
𝑚 𝑢
=
ℎ
𝑝
--------------------- (1)
where, h = Planck’s constant
m = Mass of the material particle
u = Velocity of material particle
p = m u = Momentum of the particle
Equation (1) is called de Broglie’s Equation.

42. Derivation of de Broglie’s Equation
According to Planck’s quantum theory, energy of the photon (light ) is given by
E = h ν =
ℎ𝑐
𝜆
----------------------(1)
Where ν = Freuency of photon
According to Einstein mass energy relationship , energy of photon is given by
E = mc
2
------------------------ (2)
Where , m = Mass of the photon
c = Velocity of the photon
From equation (1) & (2) ,we get
ℎ𝑐
𝜆
= mc
2
or mc =
ℎ
𝜆
------------- (3)
According to de Broglie, a moving material particle (mass = m & velocity = u) exhibits dual
character like a photon and hence equation (3) can be applied to it.
Momentum (mu) of the particle is given by,
mu =
ℎ
𝜆
or
𝜆 =
ℎ
𝑚 𝑢
=
ℎ
𝑝
--------------- (4)
h = Planck’s constant
𝜆 = Wavelength associated with the particle.
Equation (4) is called de Broglie’s Equation.
The de Broglie’s equation is significant for microscope particles like electron, proton,
atom etc. It is insignificant for macroscopic (very large) particle.

43. Heisenberg’s Uncertainty Principle
The electron has wave nature. Hence, it is not possible to
measure simultaneously its exact position and exact velocity (hence
momentum) in space at a given time because wave is extending
throughout a region of space. Thus, Heisenberg in 1927 introduced
a principle called as Heisenberg’s Uncertainty Principle.
Statement: According to Heisenberg, “It is impossible to determine
simultaneously both the position & momentum (or velocity) of a
small moving microscopic particle like electron with perfect
accuracy.” It means that both position and momentum associated
with the microscopic particle can not be measured accurately and
simultaneously.

44. Mathematical Expression
Mathematically , it can be stated as
Δx Δp 
h
4π
-------------------- (1)
Where, h = Planck’s constant
Δx = Uncertainty in the measurement of position of the microscopic particle
Δp = Uncertainty in the measurement of the momentum of the microscopic particle
From equation (1), it is evident that, if Δx is small i.e., if the position of the particle is measured
with high accuracy then Δp will be large i.e., the momentum will be measured less accurately
& vice versa. If the position of a particle is measured accurately, Δx should be zero and hence.
Δp =
h
Δx ×4π
=
h
0 ×4π
= ∞
This means that error in the determination of the momentum is infinite.
If m = Rest mass of the particle
Δv = Uncertainty in the measurement of the velocity of the microscopic particle
Then Δp = m Δv
Hence equation (1) becomes
Δx Δv 
h
4πm
------------------- (2)
A microscopic particle like electron can not have a definite path due to its wave nature.
Some times instead of measuring the position and momentum of the microscopic
particle, we measure its energy (E) and time (t) for which the particle remains in that energy
state. In such a case, uncertainty principle can be express as
ΔE Δt 
h
4π
-----------------------(3)
Where ΔE & Δt = Uncertainty in the measurement of energy & time respectively.

45. Bohr theory versus uncertainty principle
Uncertainty principle tells us that due to wave nature of
electron, we can not described the exact path or position of the
electron in an atom at any time. Thus Bohr concept of definite orbits
electrons is no valid.
In order to describe the position of the electron in an atom
the maximum we can do is that we can only predict the probability
(or relative chance) of finding the electron in a particular region of
space round the nucleus i.e. we can only predict where an electron
is most likely to be found. The probability density of finding the
electron at any given time is given by 𝜓2, where 𝜓 is the wave
function.
Uncertainty principle is significant only for microscopic
particles like electron, neutron etc. & has no significance for large
(macroscopic) particles.

46. Sinusoidal Wave Equation
There are two types of waves, running waves (e.g. water waves) and stationary
waves (e.g. vibrating string fixed at two ends). Electron waves in atoms or molecules are of
stationary type. For stationary waves ψ does not vary with time.
The equation for stationary wave fixed at x = 0 and x = a is given by
𝑑2𝜓
𝑑𝑥2 =
− 4 𝜋2
𝜆2 ψ ----------------- (1)
The boundary conditions are ψ = 0, at x = 0 and x = a.
The general solution of equation (1) is given by
𝜓 = 𝐴 sin
2 𝜋 𝑥
𝜆
+ 𝐵 cos
2 𝜋 𝑥
𝜆
------------------------ (2)
Where A and B are constants.
When x = 0, 𝜓 = 0, thus we have
0 = 𝐴 sin
2 𝜋 ×0
𝜆
+ 𝐵 cos
2 𝜋 ×0
𝜆
0 = A × 0 + 𝐵 × 1
∴ B = 0
When x = a, 𝜓 = 0, thus we have
0 = 𝐴 sin
2 𝜋 𝑎
𝜆
∵ A ≠ 0, thus sin
2 𝜋 𝑎
𝜆
= 0
This is true only
2 𝜋 𝑎
𝜆
= n𝜋, where n = 0, ±1, ±2, ±3 …
Thus λ =
2 𝑎
𝑛
or 𝜈 =
𝑛 𝑐
2 𝑎
since 𝜈 =
𝐶
λ
𝑜𝑟
𝐶
𝜈
= λ
X=0 x=a
𝜓 = 0 𝜓 = 0

47. If n = 0, then 𝜈 = 0 i.e. there is no vibration. Only 𝜈 =
𝑐
2 𝑎
,
2 𝑐
2 𝑎
,
3 𝑐
2 𝑎
,
4 𝑐
2 𝑎
etc. vibrations are
allowed.
Thus by putting the value of B = 0 and λ =
2 𝑎
𝑛
in equation (2), we get
𝜓 = 𝐴 sin
2 𝜋 𝑥
𝜆
+ 𝐵 cos
2 𝜋 𝑥
𝜆
------------------------ (2)
𝜓 = 𝐴 sin
2 𝜋 𝑥
𝜆
------------------------ (2)
𝜓 = 𝐴 sin
𝑛 𝜋 𝑥
𝑎
------------ (3)

48. Schrodinger wave Equation
In 1926, Schrodinger, an Austrian physicist using de Broglie’s idea of matter wave
developed the wave equation for electron wave. This model describes the electron as 3-D
standing wave (stationary wave) around the nucleus.
Schrodinger proposed that since electron has wave like nature, it should obey the
same classical wave equation of motion as the other waves. On the basis of this idea he
derived an equation which describes the wave motion of electron & is known as
Schrodinger’s wave equation.
∂2
∂𝑥2
+
∂ 2
∂𝑦2
+
∂ 2
∂𝑧2
+
8𝑚 2
ℎ2
𝐸 − 𝑉  = 0
𝛻2
 +
8𝑚 2
ℎ2
𝐸 − 𝑉  = 0
Where  = Mathematical function called wave function
E = Total energy of the electron
V = Potential energy of the electron
m = Mass of the electron
h = Planck’s constant
𝛻2
=
∂
2
∂𝑥2 +
∂
2
∂𝑦2 +
∂
2
∂𝑧2 = Laplacian operator ( read as nabla operator)
Schrodinger equation is a differential equation of second order.

49. Derivation of Schrodinger Wave Equation in one Dimension
The wave motion of a wave of a vibrating string can be described by
the equation
𝜓 = 𝐴 sin
2 𝜋 𝑥
𝜆
---------------------------- (1)
Where 𝜓 = Amplitude of a wave called wave function
A = Constant
x = Displacement of the wave
𝜆 = Wavelength of the wave
Differentiating equation (1) w. r. t. x, we get
𝑑𝜓
𝑑𝑥
=
2 𝜋 𝐴
𝜆
cos
2 𝜋 𝑥
𝜆
------------------------ (2)
Differentiating again equation (2), we get
𝑑2𝜓
𝑑𝑥2 = −
2 𝜋
𝜆
2
𝐴 sin
2 𝜋 𝑥
𝜆
𝑑2𝜓
𝑑𝑥2 +
4 𝜋2𝜓
𝜆2 = 0 -------------------------- (3)

50. Since an electron in an atom described as a standing wave round the nucleus, the
classical wave equation (3) which describes the wave motion of any particle
vibrating along x-direction should be applicable to the standing wave of an
electron.
From de Broglie’s hypothesis, λ =
ℎ
𝑚 𝑢
∴
𝑑2𝜓
𝑑𝑥2 +
4 𝜋2𝑚2𝑢2
ℎ2 𝜓 = 0 --------------------- (4)
The total energy of the electron is given by
E = Kinetic energy + Potential energy
=
1
2
𝑚 𝑢2+ V
i.e. m2u2 = 2 m (E – V) ----------------------- (5)
Substituting the value of m2u2 from equation (5) in equation (4), we get
𝑑2𝜓
𝑑𝑥2 +
8 𝜋2𝑚
ℎ2 𝐸 − 𝑉 𝜓 = 0 -------------- (6)
Equation (6) is the Schrodinger’s wave equation of a wave moving in one
dimensional space (along x-axis)

51. For the wave motion in a three dimensional space (3-D), the Schrodinger equation take
the form.
∂
2

∂𝑥2 +
∂
2

∂𝑦2 +
∂
2

∂𝑧2 +
8𝑚 2
ℎ2 𝐸 − 𝑉  = 0 ------ (7)
𝛻2 +
8𝑚 2
ℎ2 𝐸 − 𝑉  = 0 --------------- (8)
Equation (8) can be also written as
−
ℎ2
8𝑚𝜋2 𝛻2
+ 𝑉 𝜓 = 𝐸 𝜓 -------------------- (9)
The Hamiltonian operator 𝑯 is defined as
𝐻 = −
ℎ2
8𝑚𝜋2 𝛻2 + 𝑉 -------------------- (10)
In terms of Hamiltonian operator𝐻, the Schrodinger’s equation can be written as
H 𝜓 = 𝐸 𝜓 -------------------------------- (11)
Schrodinger’s wave equation is the second order differential equation.
Therefore it has several solutions for 𝜓 but many of these solutions are imaginary and
hence are not valid. Only those values for 𝜓 are acceptable which satisfy following
conditions:

52. The wave function 𝜓 must be single valued i.e. it must have one
and only one value at a particular point.
The wave function 𝜓 and its first derivative
𝑑𝜓
𝑑𝑥
,
𝑑𝜓
𝑑𝑦
,
𝑑𝜓
𝑑𝑧
must be
finite and continuous functions.
The value of E obtained by substituting the acceptable values of 𝜓
in the Schrodinger wave equation are called eigen values of the
function 𝜓′𝑠 which are called eigen functions.
Thus in Schrodinder’s wave equation E values are called eigen
values while 𝜓 values are called eigen function. The eigen function
for an electron in an atom is called as atomic orbital.
The eigen functions are normalized and orthogonal. The wave
function 𝜓 is called normalized if 𝜓𝑖
∗
𝜓𝑗 𝑑𝜏 = 1 (i = j) and
orthogonal if 𝜓𝑖
∗
𝜓𝑗 𝑑𝜏 = 0 (i ≠ j).

53. Physical Significance of 𝝍 (the wave function):
First Interpretation: This interpretation was given by Schrodiger himself. The wave
function 𝜓, itslf has no physical significance except that it represents the amplitude
of the wave. In studying (dealing) with all forms of wave motion such as light
waves, sound waves or matter waves, the square of amplitude of the wave (i.e. 𝜓2)
at any point represent the density or intensity of wave i.e. the number of waves per
unit volume. Therefore, the value of 𝜓2 (𝜓∗𝜓) at any point around the nucleus is
the measure of charge density ( particle density) for electron.
Second Interpretation: Born in1926 gave statistical interpretation of 𝜓. He
awarded a Nobel prize in 1954 for his interpretation of 𝜓. According to this
interpretation, | 𝜓2 | represents the probability per unit volume of finding the
particle at that point at a given instant.
If 𝜓 is the wave function of a particle, then the probability of finding the particle
within the range from r to r + dr is given by
𝜓∗𝜓 dr = 𝜓 2 dr
Where 𝜓∗ represents the complex conjugate of ψ. 𝜓2 is always real.
If 𝜓 = a +ib, then
𝜓∗ = a – ib. Therefore, 𝜓∗𝜓 = 𝑎2 + 𝑏2, 𝑖 = −1 .

54. Applications of Schrodinger’s Wave Equation
This equation has been used to calculate energy & wave function of
a freely moving particle in a one dimensional box
This equation has been used to calculate energy & wave function of
a freely moving particle in three dimensional box
This equation has been used to derive an equation for energy of an
electron in hydrogen & H – like atoms.

55. Energy of a Particle in a One Dimensional Box
Let us consider a particle (say e- of mass m ) is confined to move in a
one dimensional box of length a & having hard walls of infinite
height. Suppose move along x-direction &
is confined between x = 0 & x = a. Potential energy (V) of the
particle inside the box is zero. Thus, there is no restriction on
movement of the particle within the box. Outside the box V = ∞, i.e.
V = ∞, at x ≤ 0, x ≥ a. V = 0 in x = 0 to x = a. Therefore ψ2, i.e. the
probability of finding the particle is

56. Schrodinger’s wave equation for 1-D box is given by is
d2
d𝑥2 +
8𝑚 2
ℎ2 𝐸 − 𝑉  = 0 ------------- (1)
As V = 0, we have
d2
d𝑥2 +
8𝑚 2
ℎ2 𝐸 = 0 ---------------------- (2)
Or
d2
d𝑥2 + 𝑘2 = 0 ------------------ (3)
Where 𝑘2 =
8𝑚 2
ℎ2 𝐸 ---------------- (4)
The solution of 2nd order differential equation (3) is given by
Ψ = A sin kx + B cos kx ------------ (5)
Where A and B are constants
Now applying the boundary condition when x = 0, ψ = 0, hence
from equation (5), we get
B = 0
Putting B = 0 in equation (5), we get
Ψ = A sin kx ----------------- (6)

57. Applying other boundary condition for equation (6), we get
When x = a, ψ = 0
Thus 0 = A sin ka
Since A ≠ 0 (if A = 0, ψ2 = 0 every where in the box which is not possible)
 sin ka = 0 . This is possible if ka = n (n is an integer)
Thus ka = n
or k =
n
𝑎
-------------------------- (7)
k2 =
𝑛2 𝜋2
𝑎2 ------------------------------- (8) 𝑘2 =
8𝑚 2
ℎ2 𝐸 -------- (4)
Comparing equations (4) and (8), we get
8𝑚 2
ℎ2 𝐸=
𝑛2 𝜋2
𝑎2
8𝑚
ℎ2 𝐸=
𝑛2
𝑎2
E = En =
𝑛2 ℎ2
8 𝑚𝑎2 ----------------------- (9)
Equation (9) gives the energy of the particle in 1-D box. From equation (9), it is observed
that energy of a particle is quantized. All values of E are not allowed. Again Enα n2 & En≠ 0
as n ≠ 0. The En are the eigen values.
From equations (6) and (7), we get
Ψ = Ψn = A sin(
nx
𝑎
) ----------------- (10)
Ψn is called the eigen function.

58. From equation (9),
E min =
ℎ2
8 𝑚𝑎2 = Zero Point energy (since n = 1)
E values for the particle are 1, 4, 9, 16 etc. times the minimum value.
The plots of Ψn & Ψn
2 against x are shown in fig (5.8) & fig (5.9) for the particle in 1-D
box. From these plots it is observed that

63. The probability of finding the particle (Ψn
2) depends on x & E
& is always positive.
For the lowest energy state (g. s.) half wavelength covers the
box & hence there is no internal node, i.e. there is no point
where Ψn
2= 0.
In second energy level the wave function has one node &
third has two nodes. In general, the number of nodes in the
nth energy level = (n – 1) nodes.
Greater the number of nodes more is the curvature in
the wave function & greater is the total (K.E.) energy

64. Fig. 5.10 Particle in a 3-D box
Energy of a particle in a Three Dimensional Box
Expression for energy of a particle in a one dimensional box can be extended to
a three dimensional box with some modifications. Energy of a particle in a three
dimensional box is given by
𝐸𝑛 =
ℎ2
8 𝑚
𝑛𝑥
2
𝑎2 +
𝑛𝑦
2
𝑏2 +
𝑛𝑧
2
𝑐2
Where 𝑛𝑥 → Quantum number along side ‘a’ (x-axis)
𝑛𝑦 → Quantum number alon side ‘b’ (y-axis)
𝑛𝑧 → Quantum number alon side ‘c’ (z-axis)