Unit II:
Wave function and its physical significance
Schrodinger time dependent equation
Separation in time dependent and time independent parts
Operators in quantum Mechanics
Eigen functions and Eigen values.
Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative
analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative
analysis of Zero point energy)
Syllabus:

The Schrödinger equation is a linear partial differential equation that describes
the wave function or state function of a quantum-mechanical system. It is a key
result in quantum mechanics, and its discovery was a significant landmark in the
development of the subject.
The equation is named after Erwin Schrödinger, who postulated the equation in
1925, and published it in 1926, forming the basis for the work that resulted in
his Nobel Prize in Physics in 1933.
Erwin Schrodinger or Erwin Schroedinger, was a Nobel Prize-winning
Austrian-Irish physicist who developed a number of fundamental results
in quantum theory: the Schrödinger equation provides a way to calculate
the wave function of a system and how it changes dynamically in time.
Schrödinger equation

In addition, he was the author of many works on various aspects
of physics: statistical mechanics and thermodynamics, physics of
dielectrics, colour theory, electrodynamics, general relativity,
and cosmology, and he made several attempts to construct a unified
field theory.
In his book What Is Life? Schrödinger addressed the problems of
genetics, looking at the phenomenon of life from the point of view of
physics. He paid great attention to the philosophical aspects of
science, ancient and oriental philosophical concepts, ethics, and
religion.[4] He also wrote on philosophy and theoretical biology. He is
also known for his "Schrödinger's cat" thought-experiment.

In classical mechanics, Newton's second law (F = ma)is used to make
a mathematical prediction as to what path a given physical system will
take over time following a set of known initial conditions.
Solving this equation gives the position and the momentum of the
physical system as a function of the external force F on the system.
Those two parameters are sufficient to describe its state at each time
instant. In quantum mechanics, the analogue of Newton's law is
Schrödinger's equation.
The concept of a wave function is a fundamental postulate of quantum
mechanics; the wave function defines the state of the system at each
spatial position and time.

Schrodinger time dependent wave equation
The general equation of a wave propagating along X axis for a particle is given by;
𝒚 = 𝑨𝒆𝒊 𝒌𝒙−𝒘𝒕
In quantum mechanics, the wave function Ψ corresponds to the variable y of general wave;
𝝍 = 𝑨𝒆𝒊 𝒌𝒙−𝒘𝒕 (1)
The debroglie’s wavelength is;
𝒑 =
𝒉
𝝀
=
𝒉
𝟐𝝅
∙
𝟐𝝅
𝝀
= ℏ𝒌
And Einstein energy;
𝑬 = 𝒉𝒗 =
𝒉
𝟐𝝅
∙ 𝟐𝝅𝒗 = ℏ𝝎 ∵ 𝝎 = 𝟐𝝅𝒗
∵ 𝒌 = 𝟐𝝅/𝝀

∴ 𝒌 =
𝒑
ℏ
and 𝝎 =
𝑬
ℏ
Putting in eqn (1)
𝝍 = 𝑨𝒆
𝒊
𝒑
ℏ
𝒙−
𝑬
ℏ
𝒕
𝝍 = 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
(2)
Eqn (2) represents the wave equivalent of a free particle of total energy E and momentum p.
Differentiating eqn (2) w.r.t. x
𝝏𝝍
𝝏𝒙
=
𝒊𝒑
ℏ
∙ 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
Again differentiating w.r.t. x
𝝏𝟐𝝍
𝝏𝒙𝟐
= −
𝒑𝟐
ℏ𝟐
∙ 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕

𝝏𝟐𝝍
𝝏𝒙𝟐
= −
𝒑𝟐
ℏ𝟐
∙ 𝝍 (𝟑)
𝒑𝟐𝝍 = −ℏ𝟐
𝝏𝟐𝝍
𝝏𝒙𝟐
(𝟒)
Now, differentiating eqn 2 w.r.t. t
𝝏𝝍
𝝏𝒕
= −
𝒊𝑬
ℏ
∙ 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
𝝏𝝍
𝝏𝒕
= −
𝒊𝑬
ℏ
∙ 𝝍
∴ −
ℏ
𝒊
𝝏𝝍
𝝏𝒕
= 𝑬𝝍
∴ 𝑬𝝍 = 𝒊ℏ
𝝏𝝍
𝝏𝒕
(𝟓)

When speed of the particle is small compared to the velocity of light the total energy of a particle is some of KE and PE. i.e.
𝑬 =
𝒑𝟐
𝟐𝒎
+ 𝑽 𝟔
Multiplying by 𝜓 on both sides
𝑬𝝍 =
𝒑𝟐
𝝍
𝟐𝒎
+ 𝑽𝝍 𝟕
Putting values of eqn 4 and 6 in above equation
𝒊ℏ
𝝏𝝍
𝝏𝒕
= −
ℏ𝟐
𝟐𝒎
𝝏𝟐
𝝍
𝝏𝒙𝟐 + 𝑽𝝍 (8)
Above eqn is obtained by Schrodinger in 1926, therefore it is called Schrodinger wave eqn. and it is time dependent eqn.

In 3 D we can write;
𝒊ℏ
𝝏𝝍
𝝏𝒕
= −
ℏ𝟐
𝟐𝒎
𝝏𝟐
𝝍
𝝏𝒙𝟐 +
𝝏𝟐
𝝍
𝝏𝒚𝟐 +
𝝏𝟐
𝝍
𝝏𝒛𝟐 + 𝑽𝝍
𝒊ℏ
𝝏𝝍
𝝏𝒕
= −
ℏ𝟐
𝟐𝒎
𝜵𝟐
𝝍 + 𝑽𝝍 (𝟗)
𝒘𝒉𝒆𝒓𝒆, 𝜵𝟐
=
𝝏𝟐
𝝍
𝝏𝒙𝟐 +
𝝏𝟐
𝝍
𝝏𝒚𝟐 +
𝝏𝟐
𝝍
𝝏𝒛𝟐

THE WAVE FUNCTION
Waves on a string are described by the displacement y (x, t) of the string. In the case of a
sound wave in air, the pressure p (x, t) varies in space and time.
In electromagnetic waves, the fields, E and B, are the ones that vary in space and time.
Therefore, to characterize the de Broglie wave associated with a material particle, we require
a quantity that varies in space and time.
The variable quantity is called the wave function for the particle and is usually designated by
Ψ which is a function of the co-ordinates (x, y, z) and time t.
The displacement of a wave can either be positive or negative. In analogy, the wave function
Ψ (x, y, z, t) can have positive as well as negative values. The uncertainty principle tells us
that we can only get the probability of finding the particle at (x, y, z) at time t. As probability
cannot be negative, Ψ (x, y, z, t) cannot be a direct measure of the presence of the particle.
Hence, Ψ as such is not observable. But it must in some way indicate the presence of the
particle, as it represents the wave associated with the particle in motion. For a particle having
a well defined momentum, the uncertainty in co-ordinate is infinite. Hence, the wave
associated with it will be of infinite extent. Then, a free particle moving along x-axis with a
definite momentum is described by the plane wave;

Schrodinger’s equation : Steady-state form Time independent
equation
In a great many situations the potential energy of a particle does not depend upon time explicitly.
The forces that act upon it, and hence V, vary with the position of the particle only. When this is true,
Schrodinger’s equation may be simplified by removing all reference to t.
The one-dimensional wave function Ψ of an unrestricted particle may be written in the form;

TIME INDEPENDENT SCHRÖDINGER EQUATION

SIMPLE APPLICATIONS OF SCHRODINGER’S EQUATION
The Particle in a Box : Infinite Square Well Potential
x = 0 x = L
V = ꚙ V = ꚙ
V = ꚙ V = ꚙ
V = 0
Graphical representation of infinite potential well
Ψ(x)=0 Ψ(x)=0

𝒅𝟐
𝝍
𝒅𝒙𝟐
+
𝟐𝒎
ℏ𝟐
𝑬 − 𝑽 𝝍 = 𝟎 (𝟏)
∵ 𝑉 = 0

The wave functions are given by:
∵ 𝑠𝑖𝑛2
𝜃 =
1
2
1 − cos 2𝜃

Definition of an Operator
1. The mathematical operation like differentiation, integration, multiplication, division, addition,
substation, etc. can be represented by certain symbols known as operators.
2. A operator is a rule by means of which a given function is changed into another function.
3. In other words an operator 𝑶 is a mathematical operation which may be applied to function f(x) which
changes the function f(x) to other function g(x).
4. This can be represented as;
𝑂𝑓 𝑥 = 𝑔(𝑥)
For example:
𝑑
𝑑𝑥
4𝑥2
+ 2𝑥 = 8𝑥 + 2
In operator language 𝑶 =
𝒅
𝒅𝒙
operates on the function 𝑓 𝑥 = 4𝑥2
+ 2𝑥 and changes the
function 𝑓 𝑥 to the function 𝐠 𝒙 = 𝟖𝒙 + 𝟐

Operators in QM
The wave function for 1 D motion of a free particle along X-axis is given as;
𝝍(𝒙, 𝒕) = 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
(1)
Differentiating equation 1 w.r.t. x;
𝝏𝝍
𝝏𝒙
=
𝒊𝒑
ℏ
∙ 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
𝝏𝝍
𝝏𝒙
=
𝒊𝒑
ℏ
∙ 𝝍
𝝏𝝍
𝝏𝒙
ℏ
𝒊
= 𝒑 𝝍
∴ 𝒑 𝝍 = −𝒊ℏ
𝝏𝝍
𝝏𝒙
(2)

Now differentiating equation 1 w.r.t. t;
𝝏𝝍
𝝏𝒕
= −
𝒊𝑬
ℏ
∙ 𝑨𝒆
𝒊
ℏ
𝒑𝒙−𝑬𝒕
𝝏𝝍
𝝏𝒕
= −
𝒊𝑬
ℏ
∙ 𝝍
∴ −
ℏ
𝒊
𝝏𝝍
𝝏𝒕
= 𝑬𝝍
∴ 𝑬𝝍 = 𝒊ℏ
𝝏𝝍
𝝏𝒕
(𝟑)
Equation 2 indicates that there is an association between the dynamical quantity p and the
differential operator −𝒊ℏ
𝝏
𝝏𝒙
is called the momentum operator.
It can be written as;
𝒑 = −𝒊ℏ
𝝏
𝝏𝒙
(4)

As it is related to variable x, we have;
𝒑𝒚 = −𝒊ℏ
𝝏
𝝏𝒚
𝒑𝒙 = −𝒊ℏ
𝝏
𝝏𝒙
𝒑𝒛 = −𝒊ℏ
𝝏
𝝏𝒛
𝒑 = −𝒊ℏ𝛁
From eqn 3, a similar association can be found between dynamical variable E and the differential
operator 𝒊ℏ
𝝏
𝝏𝒕
∴ 𝑬 = 𝒊ℏ
𝝏
𝝏𝒕
(𝟓)

We have Schrodinger’s time independent equation;
𝛁𝟐𝝍 +
𝟐𝒎
ℏ𝟐
𝑬 − 𝑽 𝝍 = 𝟎
−
ℏ𝟐
𝟐𝒎
𝛁𝟐𝝍 − 𝑬𝝍 + 𝑽𝝍 = 𝟎
−
ℏ𝟐
𝟐𝒎
𝛁𝟐𝝍 +
𝟐𝒎
ℏ𝟐
× −
ℏ𝟐
𝟐𝒎
𝑬 − 𝑽 𝝍 = 𝟎
[−
ℏ𝟐
𝟐𝒎
𝛁𝟐 + 𝑽]𝝍 = 𝑬𝝍
∴ 𝑯𝝍 = 𝑬𝝍
𝑯 = −
ℏ𝟐
𝟐𝒎
𝛁𝟐
+ 𝑽 is called as Hamiltonian operator, Since, H=K.E.+P.E. is the Hamiltonian of the system.
Where,

Eigen functions and Eigen values;
Let 𝝍 be a well behaved function of the state of the system and an operator 𝑨
operates on this function such that it satisfies the equation;
𝐴 Ψ 𝑥 = 𝑎 Ψ 𝑥 (1)
Where, a is a scalar.
Then we say that a is a eigen value of the operator 𝑨 and the operand 𝝍(x) is called the eigen
function of 𝑨.
The total energy operator H is given by;
𝑯 = −
ℏ𝟐
𝟐𝒎
𝛁𝟐 + 𝑽 is called the Hamiltonian operator.
If this Hamiltonian operator H operates on a wave function 𝝍n , we get
−
ℏ𝟐
𝟐𝒎
𝛁𝟐 + 𝑽 𝝍𝒏 = 𝑬𝒏𝝍𝒏

H𝝍𝒏 = 𝑬𝒏𝝍𝒏
The wave function 𝝍n is called the Eigen function and En corresponding energy eigen
value of the Hamiltonian operator H.
𝒅𝟐
𝒅𝒙𝟐 𝒆𝟒𝒙
= 𝟏𝟔 𝒆𝟒𝒙
Example;
In this case
𝒅𝟐
𝒅𝒙𝟐 is the operator;
𝒆𝟒𝒙 is eigen function;
And 16 is the eigen value.

Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position,
experiences a restoring force F proportional to the displacement x:
F = -kx
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it
undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a
constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as
a damped oscillator.
Mass-spring
harmonic oscillator
Simple harmonic motion

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