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Quantum Chemistry
1. Introduction to Quantum Mechanics:
Mechanics deals with the study of mechanical phenomena of macroscopic and microscopic
particles occurring every moment in nature. In Physics, two types of mechanics are being
discussed. They are (i) Classical Mechanics and (ii) Quantum Mechanics.
(i) Classical Mechanics:
This describes the motion of macroscopic objects, whose sizes are far larger than 10
m e.g. the
particles starting from projectiles to parts of machinery and astronomical objects. This is the
oldest and largest part of the physics. It has three parts.
Statics- Study of equilibrium and its relation with force.
Dynamics- Study of motion and its relation with force.
Kinematics- Study of geometry of motion of particles without considering the forces
acting on them.
Another way of classification of classical mechanics is based on mathematical formulations used
to describe the motions of particles. These are
Newtonian Mechanics - Mathematics involved is Calculus.
Lagrangian Mechanics- Lagrangian functions are being used.
Hamiltonian Mechanics - Solutions of Hamilton's equations.
A lot of theories describe the motions of macroscopic particlesin classical mechanics. The most
important theory is the Newton's three laws of motion. The first law, which is also known as the
law of inertia describes that a body continues its state of rest or motion until an external force has
not acted on it. The second law shows the force is equal to the product of mass and acceleration
i.e. F = ma, where F= force, m = mass and a= acceleration. The third law says for any action
there is an equal and opposite reaction.
Another important part of physics is the study of light and its interactions with matter. Newton's
corpuscular theory describes light as a particle. Huygen suggested light as a wave. In other hand
Maxwell's theory of electromagnetic radiation explains light as a radiation which consists of
oscillating electric and magnetic fields perpendicular to each other and the fields are also
perpendicular to the direction of propagation.
Inadequacies of Classical Mechanics:
The three failures, which could not be explained by the classical mechanics, are
(a) Photoelectric effect - When light falls on a metal surface, it may remove one electron from
the surface with some kinetic energy. Wave nature of light could not explain this phenomena.
(b) Hydrogen atom spectrum - When hydrogen gas was electrically discharged, it emits five
different series of spectra. This was not explained by the classical mechanics.
(c) Black body radiation - A perfect black body absorbs all range of electromagnetic radiations
and when heated, it emits all type of electromagnetic radiations. Radiations emitted by a black
body depend on the temperature
with which it is heated and does
not depend on the materials it
made up of.
As the temperature increases the
intensities of all radiations
rises.Also, the maximum of
spectrum shifts towards lower
wavelength (i.e. high frequency)
regions (Fig. 1.1).The spectrum
is less intense at lower and
higher wavelength regions and
high intense at middle
wavelength region as shown in
the figure.
In order to explain the spectrum
of black body radiation different
laws were put forwarded. However none of them completely explain the spectrum of the black
body radiation. Two of them, which only partially explained it,are discussed below.
(a) Wein's distribution law - The mathematical expression of this law is
E
λ
dλ=
A
λ
5
e
λT dλ (1.1)
Where 𝐴 and 𝐵 are constants, 𝐸 is the energy and λ is the wavelength of radiation. This equation
could only explain the lower wavelength region of the spectrum.
(b) Rayleigh-Jean's distribution law - The mathematical expression of this law is
E
λ
dλ =
8πkT
λ
4
dλ (1.2)
Where k is Boltzmann constant. This equation could explain the higher wavelength region of the
spectrum.
Fig. 1.1 - Black body radiation
than the binding energy, the electron is ejected out and moves with a kinetic energy equal to the
difference in energy of the quantum and the binding energy. In other hand, if the energy of the
photon is less than the binding energy, the electron will not be removed from the metal surface.
Mathematically: 𝐾𝐸 = ℎ𝜈 − 𝐵𝐸 (1.5)
Where KE is the kinetic energy of ejected electron, BE is the binding energy of the electron to
the metal and hν is the energy of a quantum of radiation.
(c) Further, Neil Bohr, in 1913, could explain the hydrogen spectrum by considering the
quantization aspect and discrete energy levels. Different series of radiation lines seen in the
hydrogen spectrum are observed when the electrons present in different higher energy levels in
excited hydrogen atoms jumps to different lower energy levels. Five different series of lines
observed. They are Lyman Series (UV region), Balmer Series (visible Region), Paschen Series
(Infrared Region), Brackett Series (Infrared region) and Pfund Series (Infrared region). (Details
of the hydrogen spectrum are notpart of this book.) Bohr used the classical mechanics to explain
the quantization effect in hydrogen spectrum. So he could not explain the helium atom and
chemical bonds in a molecule. However, later on, the draw backs were solved by the help of
quantum mechanics.
2. Quantum Chemistry:-
When quantum mechanics is applied to describe different phenomena of chemistry, this is called
as Quantum Chemistry. Quantum Chemistry deals with different electronic energy levels in
atoms; different electronic, vibrational and rotational energy levels in molecules; formation of
different bonds and their energy; concept of bonding, antibonding and nonbonding molecular
orbitals, the stability concepts of molecules, light mater interaction and so on.
Let us discuss some fundamental aspects which will be required to understand the deeper sense
of quantum chemistry.
(i) de Broglie's Hypothesis (Dual nature of matter):
It says, every matter has both particle and wave character. In other way to say, every material
particleis associated with a wave. These waves are called as matter waves or de Broglie waves.If
this is the case then the energy of matter can be expressed in two ways.
As per particle nature, energy can be written using Einstein's mass energy relationship i.e.
ଶ
(1.6)
Where,𝑐 is the velocity of light. Again, as per wave nature, energy can be expressed by the help
of Planck's equation i.e.
Louis de Broglie combined the above two equations and derived the famous relationship
between wavelength and momentum of matter, which is known as de Broglie equation. It has
been discussed below.
From equations 1.6 and 1.7,
ଶ
orℎ𝑐/𝜆 = 𝑚𝑐
ଶ
(Since 𝑐 = 𝜈𝜆)
or𝜆 = ℎ/𝑚𝑐 (1.8)
For a particle, with non-zero rest mass, which travels at a velocity 𝑣, the above equation can be
written as
or 𝜆 = ℎ/𝑝(1.9)
Where 𝑝 = 𝑚𝑣, is the momentum of the particle. Equation 1.9 is known as de Broglie equation
and 𝜆 is the de Broglie wavelength of matter wave.
(ii) Heisenberg's Uncertainty Principle:
According this principle, for a microscopic particle, the position and momentum cannot be
measured simultaneously and accurately (certainty). In mathematically the uncertainty in
measurement of position and momentum are related as follows.
∆x ∆p ≥
ħ
2
(whereħ = ℎ/2𝜋) (1.10)
or ∆x m∆𝑣 ≥
ħ
2
h
4π
or ∆x ∆𝑣 ≥
h
4πm
Where, ∆𝑣is the uncertainty in velocity.
Another such relation with respect to uncertainty in energy (∆𝐸) and uncertainty in relaxation
time (∆𝜏) is
∆E ∆τ ≥
ħ
2
or h∆𝜈 ∆τ ≥
h
4π
or ∆𝜈 ∆τ ≥
1
4π
Differentiating the above equation with respect to x we get
dΨ
dx
= A cos ൬
2πx
λ
൰ ×
2π
λ
Or
dΨ
dx
2π
λ
A cos ቀ
2πx
λ
Differentiating again we get
d
2
dx
2
4π
2
λ
2
A sin ൬
2πx
λ
Or
d
2 Ψ
dx
2
4π
2
λ
2
Or
d
2 Ψ
dx
2
4π
2
λ
2
According to de Broglie's equation, if 𝑚 is the mass and 𝑣 is the velocity of the particle, then
λ=
h
⇒λ
2
h
2
2
𝑣
2
λ
2
2
𝑣
ଶ
h
2
λ
2
h
2
2
λ
2
h
2
KE
Where KE = Kinetic energy =
1
2
2
.
Using the above equation in the equation 1.16 we get
d
2
dx
2
2
h
2
(KE) Ψ=
Or
d
2 Ψ
dx
2
଼ π
2
h
2
(KE) Ψ=0(1.17)
This equation is valid when a particle moves with constant potential energy (V x ). If V x is not
constant then the above equation can be written as
d
2 Ψ
dx
2
଼ π
2
h
2
(E - V
x
Where Ψ is the wavefunction, which represents the state of the particle, 𝐸 = total energy of the
state of theparticle = 𝐾𝐸 + 𝑉 x
and 𝑉 x
= potential energy along x-direction..
Equation 1.18 represents the Schrödinger's time-independent wave equation for a particle
moving in x-direction. If the particle is considered to move in space i.e. along x, y and z
directions and 𝑉 is the potential energy of the particle then the equation becomes
2
∂x
2
2
∂y
2
2
∂z
2
8π
2
m
h
2
(E - V) Ψ=
Or ∇
2
Ψ+
8π
2 m
h
2
(E - V) Ψ=0(1.19)
Where ∇
2
∂
2
∂x
2
∂
2
∂y
2
∂
2
∂z
2
, is known as Laplacian operator.Equation 1.19 is known as
Schrödinger's time-independent wave equation in three dimensions.Many books write this
equation in a different form. Let us derive that.
Equation 1.19 ⇒∇
2
Ψ = -
8π
2 m
h
2
(E - V) Ψ
Or
ି h
2
8π
2 m
2
Ψ = EΨ - VΨ
Or
ି h
2
8π
2 m
2
Ψ + VΨ = EΨ
Or(
ି h
2
8π
2 m
2
Or(
ି ħ
2
2m
2
- V)Ψ = EΨ(where ħ = ℎ/2𝜋)(1.20b)
Or HΨ = EΨ(1.20c)
Where 𝑯 = H
= Hamiltonian operator =
-ħ
2
2m
2
- V = KE operator + PE operator; i.e. Kinetic
energy operator is
-ħ
2
2m
2
and Potential energy (PE) operator is V. 𝑯is also known as Total energy
operator.
Equations 1.20 a, b and c are the other forms of Schrödinger's time-independent wave equation
for a stationary state. 1.20c says when Hamiltonian operator operates on Ψ it gives the total
energy of the system and the wave functionΨ back. Hamiltonian operator is the combination of
KE operator and PE operator as described above.
Significance of Ψ:
(i) It is a wave function, which is a solution of Schrödinger wave equation.
(ii) It represents amplitude of wave at certain point and describes how this amplitude varies
with distance and direction. Its value may be positive or negative.
(iii)Schrödinger wave equation may have different values of ψ. All are not significant. Those
are significant are called Eigen Functions and the corresponding energy values are called
as Eigen Values.
Significance of 𝜳
𝟐
:
4. Postulates of Quantum Mechanics:
POSTULATE - I: The state of a microscopic particle is described as completely as possible by an
acceptable (well behaved), square-integrable wave function Ψ ( x,y,…,t). Where x, y,... are the
spatial coordinates and t is the time coordinate.
If Ψ is normalized, then 𝛹
∗
Ψ 𝑑τ represents the probability of finding the particle in the volume
element (𝑑τ) at time t. 𝑑 τ = d𝑥 𝑑𝑦 … ( Ψ also depends on the spin coordinate of the particle.
However this book does not discuss the spin coordinates.)
POSTULATE - II: To every observable physical quantity '𝑎' which can be measured
experimentally (e.g. position, momentum etc.), there can be assigned a linear hermitian quantum
mechanical operator A
(or represented as A).
In order to find a quantum mechanical operator, first write down the classical mechanical
expression for the observable, and make the following replacements. The position expression
𝑥 should be the same i,e, 𝑥. And the momentum 𝑝 ௫
should be replaced by −𝑖ħ
డ
డ௫
POSTULATE - III: The possible values of an observable physical quantity of a system is one of
the eigenvalues of the corresponding operator. The eigenvalue equation of an operator is
represented as follows:
Which says, when an operator 𝑨 is operated on the wavefunction Ψ representing the system, it
gives the same wavefunction back multiplied with some constant 𝑎. That constant term 𝑎, is an
eigenvalue of the corresponding operator 𝑨 and is the measure of the observable physical
quantity of the system. And the wavefunction Ψ is known as the eigen function of the operator.
Several such eigen functions may be possible which represent different states of a system and
hence there will be several eigenvalues corresponding to each eigenfunction of that operator.
POSTULATE - IV: When a large number of identical systems have the same state function Ψ,
the expected average (or expectation) value of the physical quantity, 𝑎ത, is given by
∫ Ψ
AΨdτ
∫ Ψ
Ψdτ
where dτ is the infinitesimal small value of volume. (In some books
is also written as
Both represents the expectation value of observable.)
When Ψ is a normalised wave function, ∫ Ψ
Ψdτ = 1. Hence the equation 1.22 becomes
A Ψdτ (1.23)
POSTULATE - V: The time dependent Schrödinger wave equation for a system which depends
on time can be written as
𝑯𝛹(𝑞, 𝑡) = 𝑖ħ
∂
∂t
Ψ(q, t) (1.24)
In other words, the state function discussed in all the postulates obeys the time dependent
Schrödinger equation written above.
5. Discussions on postulates:
Let us discuss the postulates in details.
(i) Postulate-1 describes the criteria of an acceptable (well behaved) wavefuncction, which should be
used to explain a system.
A wave function Ψ is said to be an acceptable (well behaved) wave function if it satisfies the
following conditions. i.e. Ψ should be (a) single valued, (b) continuous, (c) nowhere infinitive
and (d) its first derivative should also be continuous (may piecewise).
A function is said to be single-valued function, if it has only one value at one point (x,y,z) in
space. For example, 𝑠𝑖𝑛𝜃 is a single valued function, whereas 𝑠𝑖𝑛
ି ଵ
𝜃 is not. Such a wave
function is required because there should be one and only one probability of finding the system
at any single point, which has experimental importance. The probability of a system cannot have
more than one values at one point.
If Ψ is a function of angle θ in radians, then for Ψ to be a single valued function, it is essential
that Ψ(𝜃) = Ψ(𝜃 + 2𝑛𝜋). In this case, there will not be a single 𝜃, where Ψ will have two or
more values.
Again, a function should be finite for all values of coordinates i.e. it should be nowhere infinitive
in space. If it becomes infinitive at some point, it would mean the probability of finding the
particle will be infinity, that means one can locate the system completely. This would violate the
Heisenberg's uncertainty principle.
Further a function Ψ and its first derivative is required to be continuous in the whole space.
Continuous of Ψ means, it makes no sudden jump in its value. If there is a sudden jump in the
Another two postulates which are parts some text books are given below.
POSTULATE - VI: The wave functions for any quantum mechanical operator corresponding
to an observable variable constitute a complete orthonormal set.
POSTULATE - VII: Wave function ψ must be antisymmetric or symmetric for exchange of
identical fermions or bosons respectively.
Further Examples of acceptable wave function
(iii) Normalization and Orthogonlization processes.
(a) Normalization:
As discussed in the above point, the wave function, 𝛹 should be so chosen that it can be
normalized. And a wave function is said to be normalized wave function, 𝛹 ே
if it obeys the
following condition.
ே
∗
ே
ஶ
ିஶ
Where, 𝛹 ே
is a normalized wave function and 𝛹 ே
∗
is the complex conjugate of 𝛹 ே
So a wave function 𝛹 can be normalized in the following process. In fact 𝛹 is a square
integrable function. i.e.
∗
𝛹
ஶ
ିஶ
where N is a value in between 0 and ∞.
By dividing the Eq. 1.27 by N in both sides, we will get
ଵ
ே
∗
𝛹
ஶ
ିஶ
Or ∫ (
ଵ
√ ே
∗
ଵ
√ ே
ஶ
ିஶ
Or ∫
ே
∗
ே
ஶ
ିஶ
Where, 𝛹 ே
ଵ
√ ே
𝛹) is the normalized wave function and
ଵ
√ ே
is the normalization constant.
(b) Orthogonalization:
Let us consider two wave functions 𝛹 ଵ
and 𝛹 ଶ
of two different states of a system. That means
the two wavefunctions are independent to each other and do not overlap., this is known as
orthogonalization. Then the two wave functions are said to be orthogonal to each other and in
mathematics they should follow the condition
ଶ
∗
𝛹 ଵ
ஶ
ିஶ
If the two functions are real, the Eq. 1.28 can be written as
ଶ
ଵ
ஶ
ିஶ
As discussed above, the integral ∫ 𝛹
∗
ஶ
ିஶ
𝑑𝜏 or ∫ 𝛹
ஶ
ିஶ
𝑑𝜏 represents the overlap of two wave
functions 𝛹
and 𝛹
. For two independent wave functions this value is 0 and for the same wave
function, if it is normalised this value will be one. A set of wave functions, let us say 𝛹 ଵ
ଶ
ଷ
…, which follow both the conditions (orthogonalization and normalization) are known as
‘orthonormal’ functions in quantum mechanics.
Mathematically this can be written as
∗
ஶ
ିஶ
Where 𝛿
is known as ‘Kronceker’s delta’ and this represents a set of orthonormal functions. A
set of functions, which are not orthonormal but are linearly independent, can be made
orthonormal by different mathematical processes. One of such process is the Gram-Schimdt
orthogonalization process, which has been discussed later.
(c) Dirac bra-ket notation:
One more notation which one should know is the Dirac bra-ket notation to represent an integral.
For example, the integral ∫
𝑑𝜏 can be written as ൻ𝛹
ൿ. i.e.
Very similarly the expectation value equation (1.22) can be written as
Details of this notation are not the part of this book.
(iv) Quantum mechanical operator.
As stated in postulates II and III, a linear and hermitian quantum mechanical operator can be
defined for an observable physical quantity, which when operates on the wave function of a
system gives the quantitative value of the physical quantity. The mathematical expression has
already been shown in equation 1.21. Let us discuss what an operator is and some of its
properties.
Observable physical quantity: Observable physical quantity means the physical quantity which
can be observed through experiments. e.g. position, momentum, kinetic energy, potential energy,
total energy, angular momentum etc.
Operator: An operator may be defined as a mathematical director which directs a specific
mathematical operation to be carried out on a function to find out the necessary information out
(a) An operator operates to give the useful result, only if, it involves the same variable as the
function itself. For example, the Hamiltonian operator needs to have all the parameters 𝑥, 𝑦 and 𝑧
if the wave function has all the three parameters.
In 𝑥-direction alone the wave function is Ψ(𝑥 ) and the hamiltonian is 𝐻
ି ħ
మ
ଶ
డ
మ
డ௫
మ
௫
In 3D space the wave function will be 𝛹(𝑥, 𝑦, 𝑧) and the Hamiltonian is
−ħ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
(b) Addition –Addition of two operators, say 𝐴
and 𝐵
, operating on a function 𝛹 gives the result
as sum of the first operator operating on the function Ψ and the second operator operating on 𝛹.
Mathematically, (𝐴
(c) Operator addition is commutative –
Mathematically, 𝐴
(d) Multiplication – Multiplication of two operators, 𝐴
and 𝐵
, operating on a function Ψ is
defined as ൫𝐴
It is customary to start the operation with the operator nearest to the function i.e. on the right side
and proceed working towards the left side. For example in the above equation, at first 𝐵
operates
on 𝛹 and then 𝐴
operates on the resulted value of 𝐵
(e) Commutative relation in multiplication – Two operators, 𝐴
and 𝐵
, are said to be commute
multiplicatively with each other if
Or 𝐴
Or 𝐴
Or ൫𝐴
Or ൣ𝐴
Where ൣ𝐴
൯ and is known as commutator. (1.36)
So if ൣ𝐴
and 𝐵
commute with each other and ≠ 0
and 𝐵
do not commute with
each other. (1.37)
N.B. – 1. In quantum mechanics, if two operators commute with each other then the
corresponding observable physical quantities can be measured simultaneously through
experiment. And if the operators do not commute, which happens in general, the corresponding
observable physical quantities cannot be measured simultaneously.
N.B. – 2. Heisenberg’s uncertainty principle also comes under this relationship. i.e. the
momentum and position operator do not commute with each other, and hence, cannot be
determined simultaneously.
Mathematically, ൣ𝑃 ௫
The products of widths of simultaneous measurement of two variables (i.e. the uncertainties in
their values) satisfies the below relationship.
௫
ħ
ଶ
(f) Linear Operator – an operator is said to be a linear operator if it satisfies the following two
conditions.
(i) 𝐴
∅ and (1.40)
(ii) 𝐴
Where, 𝐴
is the operator, 𝛼 is a constant and 𝛹 and ∅ are wave functions. The condition (ii) can
also be derived from the condition (i), hence condition (i) is sufficient to explain the linearity
property of an operator.
(g) Hermitian Operator – An operator is said to be hermitian if it follows the following condition.
∗
𝐴
∗
𝐴
∗
Or ∫
∗
𝐴
∗
𝛹
∗
𝑑𝜏 (1.42)
Where, 𝐴
is the operator, 𝐴
∗
is the complex conjugate of the operator 𝐴
, 𝛹 and ∅ are two eigen
wave functions of the operator and 𝛹
∗
is the complex conjugate of wave function 𝛹.
(vi) Important theorems related to operator.
(a) Eigen values of hermitian operator are real.
(There are two types of operators which obey the eigen value equation. They are Hermitian and
Unitary operators. Here we will discuss only about hermitian operators.)
Proof: Let us take an eigen function 𝛹 of a hermitian operator 𝐴
with the eigen value 𝑎, then the
eigen value equation is 𝐴
∗
ଵ
∗
Again multiplying 𝛹 in both the sides of equation 1.52 and integrating we will get
∗
∅
∗
) 𝑑𝜏 = 𝑎 ଶ
∗
𝑑𝜏 (1.54)
Left sides of equation 1.53 and 1.54 are equal as 𝐴
is the hermitian operator. Hence the right side
of the equations will also be equal.
i.e. 𝑎 ଵ
∗
ଶ
∗
ଶ
∗
Or 𝑎 ଵ
∗
𝛹 𝑑𝜏 − 𝑎 ଶ
∗
𝛹 𝑑𝜏 = 0
Or (𝑎 ଵ
ଶ
∗
𝛹 𝑑𝜏 = 0 (1.55)
If 𝑎 ଵ
ଶ
then the integral ∫
∗
𝛹 𝑑𝜏 vanishes. This proves that nondegenerate eigenfunctions
are orthogonal. (Degenerate wave functions are two linearly independent functions, whose
eigenvalues with respect to an operator are equal. Similarly for nondegenerate functions the
eigenvalues will not be equal.)
If 𝑎 ଵ
ଶ
, the equation 1.55 is satisfied even when the integral is finite. This implies that
degenerate functions need not to be orthogonal. But they must be linearly independent or else
they are the self functions. In other hand, the linearly independent functions can be converted to
an orthogonal pair by different mathematical processes. (One of the orthogonalization process is
discussed in the next point.)
Again the functions are square integrable, hence can be normalized. This implies the eigen
functions of a hermitian operator are orthonormal.
(c) Schimdt orthogonalization process. (Not a part of UG course)
One way to orthogonalize two nonorthogonal, linearly independent functions is the Schimdt
orthogonalization process. This process is also known as Gram-Schimdt orthogonalization
process. The process is described below.
Let the functions 𝛹 and ∅ are two linearly independent normalized wave functions.
i.e. ∫
∗
𝛹
ஶ
ିஶ
∗
ஶ
ିஶ
and ∫
∗
∅
ஶ
ିஶ
Let us keep the first function 𝛹 unchanged and consider a new second function ∅
ᇱ
= ∅ − 𝑆𝛹.
Then
∗
ஶ
ିஶ
ᇱ
∗
ஶ
ିஶ
∗
ஶ
ିஶ
∗
ஶ
ିஶ
This implies that 𝛹 and the new function ∅
ᇱ
are orthogonal to each. Further ∅
ᇱ
can be normalized
by the normalization process.
Hence one can construct several such orthogonal functions from linearly independent
nonorthogonal functions and the newly created functions will be orthonormal.
(d) Let 𝛹 ଵ
ଶ
ଷ
… .. are a series of orthonormal functions, then the series can be expanded by
constructing any arbitrary functions, ∅s, where these functions are the linear combinations of
the above series of functions.
i.e. ∅ =
where, 𝑐
’s are constants.
The ∅ is exact if the summation is taken over all the functions, which may constitute an infinite
set. In this case, if 𝛹 ଵ
ଶ
ଷ
… .. are the eigen functions of an operator 𝐴
then the function ∅ will
also be an eigen function of the same operator and hence forms a complete orthonormal set of
eigen functions. This is the Postulate VI.
(e) If two operators 𝐴
and 𝐵
commute then they have the same set of eigen functions, and
conversely, if there exists a set of orthonormal functions, 𝛹
, which are eigen functions of two
operators 𝐴
and 𝐵
, then the two operators commute.
Proof: Let the operator 𝐴
has an eigen function 𝛹
, then
, where 𝑎
is the eigen value.
Since 𝐴
and 𝐵
commute, we can write
This shows that 𝐵
is an eigen function of the operator 𝐴
, with an eigen value 𝑎
. This is
possible only when 𝐵
is a multiple of 𝛹
. i.e.
, where 𝑏
is a constant.
This shows that 𝛹
is also an eigen function of operator 𝐵
