This document provides an overview of quantum mechanics and modern physics concepts. It discusses: 1) How classical theories could not fully explain phenomena like the hydrogen spectrum and blackbody radiation. Max Planck proposed quantizing energy in 1900, originating ideas like the photoelectric effect and line spectra. 2) Planck's radiation law successfully explained blackbody radiation across all wavelengths and temperatures by assuming oscillators could only absorb and emit discrete energy packets called quanta. 3) In 1923, de Broglie postulated that all matter has both wave and particle properties, inspired by photons. The de Broglie wavelength equation relates a particle's momentum to its wavelength. 4) Experiments in the 1920s demonstrated

1. MODERN PHYSICS
PHY1005

2. QUANTUM MECHANICS
Module - 5

3. Physics is the ‘Queen’ of science and forms the foundation of engineering and
technology.
The real understanding of new discoveries, the latest developments in the technical
field are a sound knowledge of basic physics and its principle.
Quantum Mechanics is one of the most important fundamental concepts discovered
in the 20tℎ century.
Quantum physics is the theory which describes phenomenon on the atomic and
molecular scale systems. The new discoveries of physics at the end of 19th century are
Introduction
 Black body radiation spectrum
 Stability of atom (Hydrogen spectrum
 Photoelectric effect
 Line spectrum
 Compton effect
 Specific heat of solids…

4. The above physical problems could not explain by the classical theory.
As per classical theory hydrogen spectrum is a continuous but practically it is
discrete
To explain these discrepancies of blackbody radiation spectrum by Quantum
theory which was proposed by Max Plank in 1900.
The Planck’s idea of quantum, from which everything is originated such as
 Stability of atom (Hydrogen spectrum)
 Photoelectric effect
 Line spectrum
 Compton effect
 Specific heat of solids…
Applications of quantum theory give us transistors, computer chips, lasers, and
optoelectronic devices, etc. Thus, quantum theory encompasses a large fraction of
modern science and technology.

5. Energy emitted and absorbed by material objects in the form of
electromagnetic waves is generally termed electromagnetic (EM)
radiation or simply radiation.
A specific distribution of EM radiation, such as the colors of the
rainbow, is termed a spectrum.
Distribution of EM radiation comprises a continuous region of
frequencies or wavelengths, it is a continuous spectrum; if it
comprises a series or group of discrete frequencies, it is a line
spectrum.
Radiation

6. Common materials or objects do not absorb the entire radiation incident upon
them; they are not perfect absorbers of radiation.
Imagine a ideal body which does absorb all the EM radiation that strikes on it,
whatever its wavelength or intensity, such a body is called a black body.
Definition: A black body is a perfect absorber, it must also be a perfect
emitter (i.e., it must be able to emit radiation of every wavelength at any intensity).
Heat radiation emitted by a black-body is called black body radiation.
Perfect black body - absorbs and emits all the radiation that fall on it. This radiation
is black body radiation
Independent of : 1) the Material 2) shape of the black body Depends only on
Temperature
Black body Radiation

7. Cu
Lamp black
Radiation pass thro hole undergoes
multiple reflection and completely
absorbed.
While place in a bath at T, heat radiation
come out only from the hole not through
wall of sphere
1. The distribution of radiation intensity E
is not uniform at given T
2. Intensity of radiation E increases w.r.t 
and maximize at particular point then
decreases
3. When T is increased max decreases
4. For all ’s an increase in T causes
increase in energy

8. The concept of black body radiator was introduced by Kirchhoff in 1859.
The Kirchhoff’s law state that the emissive power, 𝜀𝜆, of a medium is
equal to the absorptivity, 𝑎𝜆, of this medium under thermodynamic
equilibrium i.e., 𝜀𝜆 = 𝑎𝜆
For a black-body 𝜀𝜆 = 𝑎𝜆 = 1 and for a non-black-body 𝜀𝜆 = 𝑎𝜆 < 1.
The Kirchhoff’s law applies to gases, liquids and solids if they are in
thermodynamic equilibrium.
Classical Laws of black body radiation
1. Kirchhoff’s law (1859)

9. 2. Stefan-Boltzmann law (1879)
The EM radiation power (Energy per unit time) emitted from a black body is a
function of wavelength which is called the blackbody spectrum.
The total radiation power density E (total energy per unit time per unit area) is
simply an integral over all wavelengths:
Stefan constant
This law was first proposed by Josef Stefan in 1879 and theoretically studied by
Boltzmann a few years later, so it is named after both of them.
The total radiation power density is directionally proportional to the fourth power
of temperature in Kelvin. This is the Stefan-Boltzmann law:
𝐸 =
0
∞
𝐸𝜆 𝑑𝜆
𝐸 =
0
∞
𝐸𝜆 𝑑𝜆 = 𝜎𝑇4
𝜎 = 5.6703 × 10−8
𝑊/𝑚2
𝐾4

10. 3. Wein’s displacement law (1893)
The black body spectrum is different
for different temperatures.
Maximum radiation power is increased
by increasing the temperature of the
black body.
The peak wavelength of the spectrum,
𝜆𝑚, shifts to shorter wavelengths.
The wavelength for maximum
radiation
𝜆𝑚 𝛼
1
𝑇
𝜆𝑚 𝑇 = 𝑐 = 2.898 × 10−3
𝑚𝐾
which is known as Wiens’ displacement law.

11. 4. Wein’s radiation (1983)
In 1893, Wein also attempted to fit
an empirical relation to explain
black-body spectrum it is called
Wein’s radiation law. The radiation
density in black body radiation is
given by
Where 𝐶1 and 𝐶2 are empirical
constants. By proper choice of these
constants Wein’s law can be made to
fit the experimental curve in the
shorter wavelength region
alone but fails in the longer
wavelength region.
𝐸𝜆𝑑𝜆 =
𝐶1
𝜆5
𝑒−𝐶2/𝜆𝑇
𝑑𝜆
5. Rayleigh-Jeans law (1900)
𝐸𝜆𝑑𝜆 =
8𝜋𝑘𝑇
𝜆4
𝑑𝜆
In 1900, actually some months before
Planck’s breakthrough work, Lord
Rayleigh was taking a more direct
approach to the radiation inside the
oven to be a electromagnetic oscillators.
Without going into any detail, it turns
out that classical EM theory predicts
that
This is known as Rayleigh-Jeans law
and it agrees well with the
experimental curve for longer
wavelength region

12. The disagreement between the
observed curves and the curve
predicted by the Rayleigh-Jeans
formula worsens as period decreases
this is called the ultraviolet
catastrophe.
“Ultraviolet” because the difficulty
occurred at shorter wavelength
beyond the violet end of the visible
spectrum.
“Catastrophe” because the energy
intensity that was actually observed
was very much less than predicted by
theory.
Ultraviolet Catastrophe

13. Planck’s radiation Law (1900)
(Idea of Quantization)
The Rayleigh-Jeans law is the continuous nature of EM
oscillators
Classically, an electron oscillator may vibrate at any frequency
or have any energy up to some maximum value.
Max Planck
1858 – 1947
Planck assumed that the walls of the cavity consists of microscopic (Planck’s or
quantum) oscillators.
The absorption and emission of radiation by an oscillators take place in the form of
discrete packets of energy, called quanta.
The quantum oscillator can have only discrete, or specific, amounts of energy is called
quantization of energy which is also called photon.

14. This concept of quantization of energy is foreign to
the classical physics.
The energy (E) of an oscillator depends on frequency
(𝜐) in accordance with the following equation
𝜀 = 𝑛ℎ𝜈 𝑛 = 0, 1, 2, …
The average energy per oscillator is given by 𝜀 =
𝐸
𝑁
=
ℎ𝜈
𝑒ℎ𝜈/𝑘𝑇 − 1
The number of oscillators in the frequency range 𝜈 and 𝜈 + 𝑑𝜈 is
𝑁 𝜈 𝑑𝜈 =
8𝜋𝜈2
𝑐3
𝑑𝜈
The energy density in the frequency range 𝜈 and 𝜈 + 𝑑𝜈 is
𝐸𝜈𝑑𝜈 = 𝑁 𝜈 𝑑𝜈 𝜀 =
8𝜋ℎ𝜈3
𝑐3
1
𝑒ℎ𝜈/𝑘𝑇 − 1
𝑑𝜈

15. This can also be expressed in terms of
wavelength
This is known as Planck’s radiation
law. It significantly explained the
entire black-body spectrum for all
wavelength and at all temperature
which is shown in figure
𝜈 =
𝑐
𝜆
⇒ 𝑑𝜈 = −
𝑐
𝜆2
𝑑𝜆
𝐸𝜆𝑑𝜆 =
8𝜋ℎ𝑐
𝜆5
1
𝑒ℎ𝑐/𝜆𝑘𝑇 − 1
𝑑𝜆

16. Planck derived the theoretical shape of the blackbody spectrum.
Planck did not realize how radical and far-reaching his proposals
were.
He viewed his strange assumptions as mathematical
constructions to provide a formula that fit the experimental data.
It was not until later, when Einstein used very similar ideas to
explain the Photoelectric Effect in 1905, that it was realized that
these assumptions described “real Physics” and were much more
than mathematical constructions to provide the right formula.
Finally, Planck, Einstein won the Nobel prize on 1918, 1921,
respectively.

17. de-Broglie matter wave (1923)
In his 1923 doctoral dissertation, Louis de Broglie postulated that “because
photons have both wave and particle characteristics, perhaps all
forms of matter have both properties”.
This highly revolutionary idea had no experimental confirmation at the time.
In 1927, two important experiments established the wave property of matter
particles.
1. Davisson-Germer Electron scattering from a nickel target
2. G.P. Thompson electron diffraction
The calculation of the wavelength observed from diffraction is the same as
the de Broglie wavelength.

18. Expression of the wavelength associated with a matter particle can be derived on
the analogy of radiation.
According to Planck’s hypothesis, the energy of a photon of frequency 𝜈 is given by
Derivation
According to Einstein’s mass-energy relation
1
𝐸 = ℎ𝜈 =
ℎ𝑐
𝜆
2
𝐸 = 𝑚𝑐2
= 𝑝𝑐
Comparing the above two equations, we get
𝑝𝑐 =
ℎ𝑐
𝜆
𝜆 =
ℎ
𝑝
3
de Broglie postulated that “photons have both wave and particle
characteristics, perhaps all forms of matter have both properties”.

19. where 𝑝 = 𝑚𝑣 is the momentum of the particle. Equation (4) is called the de Broglie
waves for the matter particles.
Non-relativistic particles: For non-relativistic particles having mass 𝑚 and
moving with a velocity 𝑣 and kinetic energy 𝐸𝑘 = 𝑚𝑣2
2 = 𝑝2
2𝑚, the de Broglie
wavelength is
𝜆 =
ℎ
𝑝
=
ℎ
𝑚𝑣
4
Since the nature of moving particle is just like photon, hence for a matter particle
moving with velocity (𝑣) and mass 𝑚 then the equation (3) can be written as
𝜆 =
ℎ
𝑝
=
ℎ
2𝑚𝐸𝑘
5
Relativistic particles: For high energy particles, 𝐸2
= 𝑝2
𝑐2
+ 𝑚0
2
𝑐4
, having
kinetic energy E = 𝐸𝑘 + 𝑚0𝑐2
, the momentum can be calculated by
𝐸𝑘 + 𝑚0𝑐2 2
= 𝑝2
𝑐2
+ 𝑚0
2
𝑐4

20. The de Broglie wavelength is,
𝐸𝑘
2
+ 2𝐸𝑘𝑚0𝑐2
= 𝑝2
𝑐2
⇒ 𝑝𝑐 = 𝐸𝑘 𝐸𝑘 + 2𝑚0𝑐2
𝜆 =
ℎ𝑐
𝑝𝑐
=
ℎ𝑐
𝐸𝑘 𝐸𝑘 + 2𝑚0𝑐2
6
Accelerated charged particle: Let us consider the case of an electron of rest
mass 𝑚0 and charge 𝑒 which is accelerated by a potential 𝑉 volts from rest to
velocity 𝑣, then
1
2
𝑚0𝑣2
=
𝑝2
2𝑚0
= 𝑒𝑉 ⇒ 𝑝 = 2𝑚0𝑒𝑉
The de Broglie wavelength is,
𝜆 =
ℎ
𝑝
=
ℎ
2𝑚0𝑒𝑉
=
12.27
𝑉
Å 7
The above equation indicates that the wavelength associated with an electron
accelerated by a potential 𝑉 volts.

21. All moving objects that we see around us (e.g., a car, a ball thrown in the air etc...) is
along definite paths.
Hence their position and velocity can be measured accurately at any instant of time.
Is it possible for subatomic particle also?
This idea of a fundamental limit of moving quantum particle was put forth by the
physicist Werner Heisenberg in 1927.
His principle is now one of the fundamental postulates of quantum mechanics and
is known as the Heisenberg uncertainty principle (HUP).
Heisenberg principle states: “It is impossible to measure simultaneously
the position and momentum of a small microscopic moving particle
with absolute accuracy”
Heisenberg uncertainty principle (1927)

22. We made to measure any one of these two quantities with higher accuracy, the
other becomes less accurate.
The product of the uncertainty in position (∆𝑥) and the uncertainty in momentum
(∆𝑝𝑥 = 𝑚∆𝑣𝑥 ) is equal to or greater than ℏ 2 where ℏ is the reduced Planck
constant.
The mathematical expression for the Heisenberg uncertainty principle is simply
written as
∆𝑝𝑥 ∆𝑥 ≥
ℏ
2
Dimensionally, this product has units of J-s. Another form of the uncertainty
principle can be written as
∆𝐸 ∆𝑡 ≥
ℏ
2
∆𝐿 ∆𝜃 ≥
ℏ
2

23. Explaining HUP With An Example
Electromagnetic radiations and microscopic matter waves exhibit a dual nature of
particle (momentum) and wave (wavelength) character.
Position and momentum of macroscopic matter waves can be determined
accurately, simultaneously.
For example, the location and speed of a moving car can be determined at the same
time, with minimum error.
But, in microscopic particles, it will not be possible to fix the position and measure
the momentum of the particle simultaneously.
An electron in an atom will not see such small particles by our naked eyes.
A powerful light may collide with the electron and illuminate it.

24. Illumination helps in identifying and measuring the position of the electron.
The collision of the powerful light source, while helping in identification increases
the momentum of the electron and makes it move away from the initial position.
Thus, when fixing the position, momentum of the particle would have changed from
the original value.
When the position is exact, error occurs in the measurement of momentum. In the
same way, the measurement of momentum accurately will change the position.
Hence, at any point in time, either position or momentum can only be measured
accurately. Simultaneous measurement of both of them will have an error.
Heisenberg quantified the error in the measurement of both position and
momentum at the same time.
∆𝑝𝑥 ∆𝑥 ≥
ℏ
2

25. Applications of Uncertainty principle
1. Ground state energy of hydrogen atom:
The classical expression for the total energy of an electron in the ground state of
hydrogen atom is given by
where 𝑎 is the radius of the first orbit. Let the uncertainty in position of the electron
Δ𝑥 be the order of 𝑎, therefore
1
𝐸 =
𝑝2
2𝑚
−
𝑍𝑒2
4𝜋𝜀0𝑎
Δ𝑝 Δ𝑥 = Δ𝑝 𝑎 ≅ ℏ ⇒ 𝑝 = Δ𝑝 =
ℏ
𝑎
2
Substituting eqn. (2) in (1), we get
3
𝐸 =
ℏ2
2𝑚𝑎2
−
𝑍𝑒2
4𝜋𝜀0𝑎

26. For the ground state energy 𝐸 has to be minimum when
Form eqn. (3), we have
𝑑𝐸
𝑑𝑎 𝑎=𝑎0
= 0
𝑑𝐸
𝑑𝑎 𝑎=𝑎0
= −
ℏ2
𝑚𝑎0
3 +
𝑍𝑒2
4𝜋𝜀0𝑎0
2 = 0
𝑍𝑒2
4𝜋𝜀0𝑎0
2 =
ℏ2
𝑚𝑎0
3 ⇒ 𝑎0 =
4𝜋𝜀0ℏ2
𝑚𝑍𝑒2
4
This is the radius of Bohr’s orbit of hydrogen atom 𝑍 = 1. Substituting eqn. (4) in
(3) becomes
This is the required expression for ground state energy of an electron in the
hydrogen atom.
5
𝐸 =
ℏ2
2𝑚
𝑚𝑍𝑒2
4𝜋𝜀0ℏ2
2
−
𝑍𝑒2
4𝜋𝜀0
𝑚𝑍𝑒2
4𝜋𝜀0ℏ2
=
𝑚𝑍2
𝑒4
16𝜋2𝜀0
2
ℏ2
1
2
− 1
𝐸 = −
𝑚𝑍2
𝑒4
32𝜋2𝜀0
2
ℏ2

27. 2. Non-existence of electrons in nucleus:
We consider, an electron is exist inside a nucleus, the maximum uncertainty in its
position ∆𝑥 = 2𝑟0, 𝑟0 being the radius of the nucleus.
The minimum uncertainty momentum of electron is equal its momentum, therefore
𝑝 = ∆𝑝 =
ℏ
Δ𝑥
=
ℏ
2𝑟0
For a typical nucleus 𝑟0 = 10−14
𝑚 . hence
𝑝 = Δ𝑝 =
1.055 × 10−34
2 × 10−14
= 5.28 × 10−21
𝑘𝑔 𝑚/𝑠𝑒𝑐
The kinetic energy of the electron
𝐸𝑘 =
𝑝2
2𝑚
=
5.28 × 10−21 2
2 × 9.1 × 10−31 × 1.6 × 10−13
= 95.7 𝑀𝑒𝑉
If electron exists inside the
nucleus its energy should be
of the order of 97 MeV.
In 𝛽 -decay, the maximum
kinetic energy of the emitted
electron is few 4 MeV .
Therefore one does not
expect electron to be a
constituent of the nucleus.

28. Other applications
3. Ground state energy of harmonic oscillator
4. Ground state energy of a particle in a box
5. Existence of electron in an atom
6. Existence of protons in nucleus
7. Binding energy of an electron in an atom
Prepare Answer for the above applications of HUP

29. The mathematical model of quantum mechanics (systems) is the Schrödinger wave
equation.
Our observable universe is entirely composed of matter and energy. Light is one of
the various forms of energy.
When light propagate through free space, its motion in space and time can be
represented by the electromagnetic wave equation.
Again the concept of symmetry of nature one can expect a similar equation to
represent the motion of a wave-particle in space and time.
In 1926, Erwin Schrödinger developed an equation known as Schrödinger equation
which represents the motion of the matter-wave associated with free particle.
Schrodinger wave equations

30. Time Dependent Schrödinger equation: in which time explicitly appears,
and so describes how the wave function of a particle will evolve in time.
Time Independent Schrödinger equation: which is describe the allowed
energy levels of the free particle.
Time Dependent Schrödinger equation
According to de-Broglie theory, a particle is always associated with a wave whose
wavelength is given by 𝜆 = ℎ 𝑝.
A particle of mass 𝑚 is in motion along the 𝑥-direction.
The wave function 𝜓 be the dependent variable of the de Broglie wave which is a
function of the coordinates 𝑥 and 𝑡.
As 𝑣 = 𝑥 𝑡 = 𝜔 𝑘, the wave function may be written as a function of (𝑘𝑥 − 𝜔𝑡) i.e.
𝜓 = 𝑓 𝑘𝑥 − 𝜔𝑡 . Using the relations

31. 𝑝 =
ℎ
𝜆
=
2𝜋ℏ
𝜆
= ℏk
𝐸 = ℎ𝜈 = 2𝜋ℏ𝜈 = ℏ𝜔
𝑘 =
2𝜋
𝜆
𝜔 = 2𝜋𝜈
1
𝜓 = 𝑓 𝑘𝑥 − 𝜔𝑡 = 𝑓
𝑝𝑥 − 𝐸𝑡
ℏ
More general, wave would be a sum of a sine and cosine waves. Then the equation
(1) in an exponential form as follows:
We assume that the energy and momentum of the particle are constant.
Differentiating the above equation with respect to 𝑥,
2
𝜓 = 𝐴 𝑒𝑥𝑝
𝑖
ℏ
𝑝𝑥 − 𝐸𝑡
𝜕𝜓
𝜕𝑥
=
𝑖𝑝
ℏ
𝐴 exp
𝑖
ℏ
𝑝𝑥 − 𝐸𝑡 =
𝑖𝑝
ℏ
𝜓

32. Eqn. (3) represents the momentum operator. Differentiating equation (2) with
respect to 𝑡 gives
𝑝𝜓 + 𝑖ℏ
𝜕𝜓
𝜕𝑥
= 0
𝑝 + 𝑖ℏ
𝜕
𝜕𝑥
𝜓 = 0 𝑝 = −𝑖ℏ
𝜕
𝜕𝑥
𝑝𝜓 =
ℏ
𝑖
𝜕𝜓
𝜕𝑥
= −𝑖ℏ
𝜕𝜓
𝜕𝑥
3
𝜕𝜓
𝜕𝑡
= −
𝑖𝐸
ℏ
𝐴 𝑒𝑥𝑝
𝑖
ℏ
𝑝𝑥 − 𝐸𝑡 = −
𝑖𝐸
ℏ
𝜓
𝐸𝜓 − 𝑖ℏ
𝜕𝜓
𝜕𝑥
= 0
𝐸𝜓 = −
ℏ
𝑖
𝜕𝜓
𝜕𝑡
= 𝑖ℏ
𝜕𝜓
𝜕𝑡
𝐸 − 𝑖ℏ
𝜕
𝜕𝑡
𝜓 = 0 𝐸 = 𝑖ℏ
𝜕
𝜕𝑡
4
Eqn. (3) denotes the energy operator.

33. Partial derivatives with respect to 𝑥 and 𝑡 are connected by means of the relation
between the momentum and energy.
The total energy of the de-Broglie wave associated with particle of mass 𝑚 and
velocity 𝑣 is
𝑝2
2𝑚
+ 𝑉 = 𝐸 5
where 𝑉 is the potential energy. Multiplying the eqn. (5) with 𝜓, we obtain
𝑝2
𝜓
2𝑚
+ 𝑉𝜓 = 𝐸𝜓 6
Substituting the eqn. (3) and (4) in eqn. (6), we get
which is the famous 1-D time dependent Schrödinger wave equation.
−
ℏ2
2𝑚
𝜕2
𝜓
𝜕𝑥2
+ 𝑉𝜓 = 𝑖ℏ
𝜕𝜓
𝜕𝑡
7

34. Here 𝜓 = 𝜓 𝑥, 𝑦, 𝑧, 𝑡 = 𝜓 𝑟, 𝑡 , 𝑟 = 𝑥 𝑖 + 𝑦𝑗 + 𝑧𝑘
Eqn. (8) is the famous 3-D time dependent Schrödinger wave equation.
We extend the equation (7) to the 3-dimensional case, we find that
−
ℏ2
2𝑚
𝜕2
𝜓
𝜕𝑥2
+
𝜕2
𝜓
𝜕𝑦2
+
𝜕2
𝜓
𝜕𝑧2
+ 𝑉𝜓 = 𝑖ℏ
𝜕𝜓
𝜕𝑡
−
ℏ2
2𝑚
𝜕2
𝜕𝑥2
+
𝜕2
𝜕𝑦2
+
𝜕2
𝜕𝑧2
𝜓 + 𝑉𝜓 = 𝑖ℏ
𝜕𝜓
𝜕𝑡
−
ℏ2
2𝑚
∇2
𝜓 + 𝑉𝜓 = 𝑖ℏ
𝜕𝜓
𝜕𝑡
8
∇2
=
𝜕2
𝜕𝑥2
+
𝜕2
𝜕𝑦2
+
𝜕2
𝜕𝑧2
𝑖𝑠 𝑎 𝐿𝑎𝑝𝑙𝑎𝑐𝑖𝑎𝑛 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑠

35. Time Independent Schrödinger equation
The 1-D time dependent Schrödinger equation can be written as
Potential energy 𝑉 of a particle does not depend on time, it varies with the position
of the particle only and the field is said to be stationary.
For stationary problems Schrödinger equation can be simplified by separation of
time and position dependent parts.
Accordingly, we can write the solution of eqn. (1) 𝜓 𝑥, 𝑡 as a product of 𝜓 𝑥 and
𝜙 𝑡 . Then the eqn. (1) takes the form
−
ℏ2
2𝑚
𝜕2
𝜓 𝑥, 𝑡
𝜕𝑥2
+ 𝑉𝜓 𝑥, 𝑡 = 𝑖ℏ
𝜕𝜓 𝑥, 𝑡
𝜕𝑡
1
𝜓 𝑥, 𝑡 = 𝜓 𝑥 𝜙 𝑡 = 𝜓 𝜙
Substituting equation (2) in (1), we get
2

36. Dividing the above equation by 𝜓𝜙, we
obtain
−
ℏ2
2𝑚
𝜕2
𝜕𝑥2
𝜓 𝜙 + 𝑉 𝜓 𝜙 = 𝑖ℏ
𝜕
𝜕𝑡
𝜓 𝜙
−
ℏ2
2𝑚
𝜙
𝜕2
𝜓
𝜕𝑥2
+ 𝑉 𝜓 𝜙 = 𝑖ℏ 𝜓
𝜕𝜙
𝜕𝑡
−
ℏ2
2𝑚
1
𝜓
𝜕2
𝜓
𝜕𝑥2
+ 𝑉 = 𝑖ℏ
1
𝜙
𝜕𝜙
𝜕𝑡
3
Here, potential energy 𝑉 is a function of 𝑥 only, the entire left hand side of eqn. (3)
is a function of 𝑥 while the right hand side is a function of 𝑡.
Since 𝑥 and 𝑡 are independent variables, both sides must be equal to constant,
which we will call 𝐸. The right side of eqn. (3) yields
𝑖ℏ
1
𝜙
𝜕𝜙
𝜕𝑡
= 𝐸

37. Therefore, left side of eqn. (3) can be written as
−
ℏ2
2𝑚
1
𝜓
𝜕2
𝜓
𝜕𝑥2
+ 𝑉 = 𝐸
−
ℏ2
2𝑚
𝜕2
𝜓
𝜕𝑥2
+ 𝑉𝜓 = 𝐸𝜓 4
𝜕2
𝜓
𝜕𝑥2
+
2𝑚
ℏ2
𝐸 − 𝑉 𝜓 = 0 5
Eqn. (5) represents time independent Schrödinger equation. Eqn. (4) can be
written as
−𝑖ℏ
𝜕
𝜕𝑥
2
2𝑚
+ 𝑉 𝜓 = 𝐸𝜓
−
ℏ2
2𝑚
𝜕2
𝜕𝑥2
+ 𝑉 𝜓 = 𝐸𝜓
Where 𝐻 is the Hamiltonian
of the system which means
that the total energy of the
system. The Eqn. (6) another
form of time independent
Schrödinger wave equation.
𝐻 =
−𝑖ℏ
𝜕
𝜕𝑥
2
2𝑚
+ 𝑉
𝐻𝜓 = 𝐸𝜓 6

38. Properties of wave function 𝝍 𝒙, 𝒕
The solution of the Schrödinger wave equation associated with a particle is wave
function 𝜓 𝑥, 𝑡 became a complex number.
The complex form of wave function itself has no physical interpretation i.e., It is not
measurable.
The wave function, at a particular time, contains all the information about the
particle.
What does the wave function mean?
The physical interpretation of the wave function is possible by the product of a
complex number with its complex conjugate is a real and positive number.
The wave function can be written as
𝜓 𝑟, 𝑡 = 𝐴 + 𝑖𝐵 𝑟 = 𝑥 𝑖 + 𝑦𝑗 + 𝑧𝑘

39. where 𝐴 and 𝐵 are real number. The complex conjugate of wave function is
𝜓 𝑟, 𝑡 𝜓∗
𝑟, 𝑡 = 𝜓 𝑟, 𝑡 2
= 𝐴 + 𝑖𝐵 𝐴 − 𝑖𝐵 = 𝐴2
+ 𝐵2
Thus, the square of the absolute value of the wave function 𝜓 𝒓, 𝑡 2
is a measure of
the particle or probability density.
Probability density: A particle will be found is equal to the square of
the absolute value of the wave function.
The probability density that the particle will be found in the small volume element
𝑑𝜏 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 about any point 𝒓 at time 𝑡 is expressed as
𝜓∗
𝑟, 𝑡 = 𝐴 − 𝑖𝐵
𝑃 𝑟 𝑑𝜏 = 𝜓 𝑟, 𝑡 2
𝑑𝜏
Total probability of the particle is obtained by integrating 𝑃 𝑟 𝑑𝜏 over the whole
space is a unity.
𝑃 =
−∞
∞
𝑃 𝑟 𝑑𝜏 =
−∞
∞
𝜓 𝑟, 𝑡 𝜓∗
𝒓, 𝑡 𝑑𝜏 =
−∞
∞
𝜓 𝑟, 𝑡 2
𝑑𝜏 = 1

40. Normalization and orthogonality of wave function
Let us consider two wave function 𝜓𝑚 𝒓, 𝑡 and 𝜓𝑛 𝒓, 𝑡 . Two wave functions are
over the entire space, i.e.,
where 𝛿𝑚𝑛 is kronecker delta function. If 𝑚 = 𝑛. 𝛿𝑚𝑛 = 1 then the function satisfy
the normalization condition i.e.,
−∞
∞
𝜓𝑛
∗
𝑟, 𝑡 𝜓𝑛 𝑟, 𝑡 𝑑𝜏 =
−∞
∞
𝜓𝑛 𝑟, 𝑡 2
𝑑𝜏 = 1
−∞
∞
𝜓𝑚
∗
𝑟, 𝑡 𝜓𝑛 𝑟, 𝑡 𝑑𝜏 = 𝛿𝑚𝑛
If 𝑚 ≠ 𝑛, and 𝛿𝑚𝑛 = 0 then the function obey orthogonality condition i.e.,
−∞
∞
𝜓𝑚
∗
𝑟, 𝑡 𝜓𝑛 𝑟, 𝑡 𝑑𝜏 = 0