Schrodinger Wave Equation
By Sameen Fatima

Why do we need Schrodinger wave equation?
 The Schrodinger equation plays the role of Newton's laws and conservation of energy
in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is
a wave equation in terms of the wave function which predicts analytically and
precisely the probability of events or outcome. Schrodinger's equation cannot be
derived from anything. It is as fundamental and axiomatic in Quantum Mechanics
as Newton's Laws is in classical mechanics.

In physical chemistry, a
dynamical system is described as
a "particle or ensemble of
particles whose state varies over
time and thus obeys differential
equations involving time
derivatives". In order to make a
prediction about the system's
future behavior, an analytical
solution of such equations or
their integration over time
through computer simulation is
realized.

Introduction:
Erwin Schrodinger, was an Austrian physicist
who developed the wave mechanical model of
atom in 1926.
This model takes into account the wave and
particle nature of the electron. In his model,
he visualized the atom as a positively charged
nucleus surrounded by a standing electron
wave which extends around the nucleus.

Stationary waves
1. These waves do not travel and remain
confined between two boundaries in
the medium.
2. These wave do not transmit energy in
the medium.
3. The phase of all the particles in
between two nodes is always same.
But particle of two adjacent node
differ in phase by 180.
4. Particles at nodes are permanently at
rest.
5. The amplitude of vibration change
from particle to particle. the
amplitude is zero for all at nodes and
maximum at antinodes.
6. All the particle attain the maximum
displacement.
Progressive wave
1. These waves travels in a medium with
definite velocity.
2. These wave transmit energy in the
medium.
3. The phase of vibration varies
continuously from particle to particle.
4. No particles of the medium vibrate and
amplitude of vibration is same.
5. All particles of the medium vibrate and
amplitude of vibration is same.
6. All the particles do not attain the
maximum displacement position
simultaneously.

Standing waves vs progressive waves

Explanation:
 Let us consider a particle P
moving with uniform angular
velocity ω rad/s in a circular
path of radius A, which is
executing simple harmonic
motion. We measure time
from instant when P passes O
and after a time t second,
we imagine P to have θ = ωt
radians. The variation of
displacement with time can
be represented by simple
harmonic wave.

Derivation:
 A simple harmonic wave may be produced in a medium by a body executing
simple harmonic motion. By considering the right angled triangle PBC, we can
write
PC/PB = ψ/A = sinθ
ψ = A sin θ = A sin ωt (since θ = ωt) …….(1)
Where ψ is pronounced as psi and it is representing the vertical displacement of
the harmonic wave. We plot this displacement against time as on the right hand
side.
The angular velocity, ω = 2π ν
Where ν is the frequency
ψ = A sin 2π ν t …….(2)

 In order to consider the nature of progressive waves, we are more interested in
the distance-time relationship.
x = v t ………(3)
Where x is distance covered in time t at speed v.
Combining equation 2 and 3 , we have
Ψ = A sin 2Πνx/v (since t=x/v) …………..(4)
And wave is shown as:
This image represents the simple harmonic motion and harmonic wave.

Besides the frequency ν, we now have another property by which we can characterize the
wave-its wavelength λ , which is the distance traveled during a complete cycle.
ν = v/λ
So, we have
ψ = A sin 2Πx/λ ……………(5)
on differentiating equation 5 with respect to x, we get
d𝜳/𝒅𝒙 = 𝑨 𝐜𝐨𝐬
𝟐𝝅𝒙
𝝀
𝟐𝝅
𝝀
=
𝟐𝝅
𝝀
A 𝒄𝒐𝒔
𝟐𝝅𝒙
𝝀
Differentiating again ,we get
𝒅
𝒅𝒙𝟐
𝟐
𝜳 =
𝟐𝝅𝑨
𝝀
− 𝐬𝐢𝐧
𝟐𝝅𝒙
𝝀
𝟐𝝅
𝝀
=
−𝟒𝝅𝟐
𝝀𝟐 𝐀 𝒔𝒊𝒏
𝟐𝝅𝒙
𝝀
=
−𝟒𝝅𝟐
𝝀𝟐 𝜳 ∗ 𝜳 = 𝐀 𝒔𝒊𝒏
𝟐𝝅𝒙
𝝀
or
𝒅
𝒅𝒙𝟐
𝟐
𝜳 +
𝟒𝝅𝟐
𝝀𝟐 𝜳 = 0

 This is the classical wave equation describing the wave motion of any particle along x-
axis. Since the electron is proved to have a wave character, let us assume that the same
behavior is shown by electron waves. To apply this equation to a particle, λ must be
replaced by the momentum of the particle using de-Broglie’s relationship i.e.,
𝜆 =
ℎ
𝑚𝑣
Taking square on both sides
𝝀𝟐
=
𝒉𝟐
𝒎𝟐𝒗𝟐
Or
𝟏
𝝀𝟐 =
𝒎𝟐𝒗𝟐
𝒉𝟐 ……….(8)
 Substituting this value in equation (7) we get,
𝒅
𝒅𝒙𝟐
𝟐
𝜳 +
𝟒𝝅𝟐𝒎𝟐𝒗𝟐
𝒉𝟐 𝜳 = 0 ………..(9)

 In order to express this equation in terms of energy, we make use of the fact that
total energy (E) is the sum of kinetic energy and potential energy (Bohr’s theory).
total energy = Kinetic energy + Potential energy
E =
1
2
𝑚𝑣2
+ V
Or
1
2
𝑚𝑣2
= E - V
𝑚𝑣2
= 2 𝐸 − 𝑉
𝑣2
=
2 𝐸 − 𝑉
𝑚
Substituting the value of velocity in equation (9).
𝒅
𝒅𝒙𝟐
𝟐
𝜳 +
𝟖𝝅𝟐𝒎
𝒉𝟐 𝐸 − 𝑉 𝜳 = 0 ………..(10)
V' stands for the
potential energy of
the particle in
Schrodinger wave
equation. Because the
applied force on the
electron or particle of
the system is in term
of voltages.

 This is the wave equation when the particle is moving in one dimensional system,
i.e., the wave is moving in one dimensional system, i.e., the wave is moving in one
direction x. for electrons which can have their wave motion along any of the three,
axes, x,y,z, we can similarly write the wave equation as:
𝝏𝟐𝜳
𝝏𝒙𝟐 +
𝝏𝟐𝜳
𝝏𝒚𝟐 +
𝝏𝟐𝜳
𝝏𝒛𝟐 +
𝟖𝝅𝟐𝒎
𝒉𝟐 𝑬 − 𝑽 𝜳 = 0 ……..(11)
Or simplify
𝛁𝟐
𝚿+
𝟖𝝅𝟐𝒎
𝒉𝟐 𝑬 − 𝑽 𝜳 = 0 ………(12)
Where Del squared is known as the Laplacian operator.
𝛁𝟐 𝚿 = −
𝟖𝝅𝟐
𝒎
𝒉𝟐
𝑬 − 𝑽 𝜳
or 𝛁𝟐 𝚿= -
𝟖𝝅𝟐𝒎
𝒉𝟐 E𝜳 +
𝟖𝝅𝟐𝒎
𝒉𝟐 V𝚿……..(13)
∂
The symbol is
variously referred to
as "partial", "curly
d", "rounded d",
"curved d", "dabba",
or "Jacobi's delta",
or as "del" (but this
name is also used for
the "nabla" symbol
∇).

 Multiply this eqaution by
𝒉𝟐
𝟖𝝅𝟐𝒎
𝒉𝟐
𝟖𝝅𝟐𝒎
𝛁𝟐 𝚿 = - E𝜳 + V𝚿
Or E𝜳 =
−𝒉𝟐
𝟖𝝅𝟐𝒎
𝛁𝟐 𝚿 + V𝚿
Or E𝜳 =
−𝒉𝟐
𝟖𝝅𝟐𝒎
𝛁𝟐 + V Ψ
Or H𝜳 = 𝑬 𝜳
 Where H =
−𝒉𝟐
𝟖𝝅𝟐𝒎
𝜵𝟐
+ V, and is known as Hamiltonian operator

Application of
Schrodinger Wave
Equation
Important applications are:
1.This equation is used to calculate the energy and wave function of
a particle such as electron in a one-dimensional box.

2.This equation is used to calculate the energy and
wave function of a particle in a three-dimensional box.

3. This equation introduces the concept of degeneracy in atomic orbitals.

4. This equation has been used to derive an expression for an electron in H-atom.
5.This equation has been used to derive various quantum numbers which represent
the postal address of an electron in an atom.

6. This equation has been used to calculate the energy of pi- electrons in
conjugated systems like benzene.
7. This is also used to calculate the resonance energy of molecules.
By applying equation