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Core Theme
This lecture introduces the fundamental concepts of quantum mechanical operators, including their definition, algebra, properties like linearity and Hermiticity, and their crucial role in defining eigenvalue problems. It culminates in demonstrating how the time-independent Schrödinger equation can be formulated as an eigenvalue equation, linking the Hamiltonian operator to energy.
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[Music]
dear students hope all you are doing
very fine
today i doctor pushpal hosh assistant
professor of doctor holy shiny goat
centre university
we'll deliver a lecture on very
important topic quantum mechanical operators
[Music]
and the
subject expert or content expert of this
topic is dr moto shimano senior
scientist ca cmc live nazar after this
lecture you will learn
that what are the quantum mechanical operators
operators
what are the eigen functions eigen values
values
what are hermitian operators
and how schrodinger equation can be formulated
formulated
as an eigenvalue problem [Music]
[Music]
now you know that quantum theory is
based on two very important constructs
operators and wave functions an operator
is a symbol that instructs
us to do is do something to whatever
follows the symbol
for example
let us consider
that d y by d x to be the d by d x
operator operating on the function y x
similarly let's take square root which
is called at sqrt
integrating operators
partial derivatives
let's say with respect to y
that is dy by dx
are other example of various kinds of operators
operators
so it is clear
that operator and operand that means the
function on which operator acts
normally an operator can be denoted
denoted
by capital letter with a current over it
which normally called a cap
thus we can write a cap f x is equal to
g x to indicate
that the operator a cap
operates on f x function to give a new
function g x
now let us take some examples for example
example
a cap 3x where
where
operator a is equal to d square by dx square
square
now performing this above operation
we can found that a cap three x
is equal to d square by d x square three
x is equal to zero
similarly we can take another example where
where
a cap is equal to d square by d x square
plus two d by d x plus 5 now when we
operate on x square
we found
that d square by d x square into x
square plus 2 d by d x into x square
plus 5 into 2 so all together
you are getting
2 plus 4 x plus 5 x square now let us understand
understand
the sum and difference of 2 operators
say a cap and e cap
by the equation known as a cap plus e
cap f x is equal to a f x plus e k f x
similarly when we subtract then it comes
like a cap minus e cap into f x is equal
to a cap f x minus e cap f x for example
if we take c cap equivalent to d by d x
then we can write c cap plus three cap
into x s cube minus five equivalent to c cap
cap
into x cube minus five
plus 3 cap into x cube minus 5 which is
equal to 3 x square plus
plus
3 x cube minus 15
so 3 x cube plus 3 x square minus 15.
now we can define the product of two
operators a cap and e cap by the equation
equation
a cap e cap f x equivalent to operator a
cap into operator e cap into f x so we
first operate on fx with the operator on
the right of the operator product and
then we take the resulting function and
operate on it with the operator on the
left of the operator product
for example
3 cap c cap f x is equal to operator 3
into c cap f x
is equal to 3 cap f prime x
is equal to 3 f prime x
for this example there is no difference
in the final result whether we first
applied one operator or the other but
not always
akp cap and ek cap have the same effect
to understand this we have to take
another example with operator c cap and
x cap where c cap is equal to d by d x
now if we write c cap
x cap f x
is equal to d by d x
x f x is equal to f x plus x f prime x
is equal to 1 cap plus x cap c cap into
f x
now if we write
x cap c cap f x
is equal to
x cap into d by d x f x
is equal to x
a prime x
thus we found that operator a cap e cap
and e cap a cap have the different
effect here a major difference between
operator algebra
and ordinary algebra
is that
numbers of the commutative law of multiplication
multiplication
but operators do not necessarily do so
now we know ae is equal to a if a and
here numbers
but operator a cap e cap and operator e
cap a cap are not necessarily equal operators
operators
now a cap e cap and e k cap are the same
then the operators a cap and operators a
are said to commute with each other
now the commutator of the two operators
a cap and e cap is defined as
a cap comma e cap is equal to
third bracket a k p cap minus e cap a f
now if a cap e k f
with the function of f x is zero
for all f x
on which the commutator acts then we
write that operator a cap and d k is
equal to zero and we say that a cap and
d cap commute with each other we have to
understand this thing very nicely
let us evaluate
the commutator of a cap e cap when a cap
is equal to
d by d x and e cap is equal to x square
now let us consider
that both these operators
act upon an arbitrary function f x
so we can write
a cap e cap f x is equal to d by d x x
square f x
now on upon differentiation we can find
2 into f x plus x square into d by d x
now e k p f f x is equal to
x square into d by d x f x
by subtracting these two results we obtain
obtain
third bracket a k p cap minus e cap a f
into f x is equal to two x f x [Music]
[Music]
now i will discuss linear operator
which is very important in quantum
mechanics the operators which are used
are all linear operators and operator a
cap is said to be linear
if the following condition holds like a
cap third bracket c 1 f 1 x plus c 2 f 2
x is equal to c 1 operator a cap f 1 x
plus c 2 operator a cap f 2 x where c 1
and c
are constants and f1 and f2 are the two functions
functions
for example the two operators
differentiate and integrates
are linear linear
because if we operate d by d x on c 1 f
1 x plus c 2 f 2 x we found c 1 d f 1 by
d x plus c 2 d f 2 by d x similarly c 1
f 1 x plus c 2 f 2 x dx
is equal to c 1 f 1 x d x plus c 2 f 2 x dx
dx
on the other hand square operator which
is normally called a square s q r is
non-linear because if you operate square
on c 1 f 1 x plus c 2 f 2 x
we found c 1 square
f 1 square x plus c 2 square f 2 square
x plus 2 c 1 c 2 f 1 x f 2 x
which is not equal to c 1 f 1 square x
plus c 2 f 2 square x
hence it is proved that square is a
non-linear operator
similarly it can be proved the square root
root
which is termed as sqrt is also a
nonlinear operator
let us say a cap fx is equal to square
root of f x
now on putting we found that a cap c 1 f
1 x plus c 2 f 2 x is equal to c 1 f 1 x
plus c 2 f 2 x to the power half so
which is
essentially not is equal to
c 1 f 1 to the half x plus c 2 f 2 to
the power half x so we can say that
square root also a non-linear operator [Music]
[Music]
it sometimes happens
that function f x when operated by
operator a
produces the same function and
multiplied by constant a
then we can write that operator a
f x is equal to small a into f x
now this equation is called an eigen
value equation which is very important
in quantum mechanics the function is
said to be an eigen function of the operator
operator
a cap and small a is called a
corresponding eigen value
eigen is a german word
meaning characteristics
eigen value
is a hybrid word basically
basically
it means
or this meaning may be characterized weird
weird
now let us solve a very simple problem
problem is that we have to show that e
to the power alpha x
is an eigen function of the operator
d n by d x that means differentiate with
respect to x
for n times
now question is what is the eigen value
we have to calculate it
now if we differentiate
e to the power alpha x n times and
obtain d n by d x e to the power alpha x
is equal to alpha to the power n e to
the alpha x and so we can say the eigen
value is alpha to the power n
now the operators can be imaginary or
complex quantities
we can see that the x component of the
momentum can be represented in quantum
mechanics by an operator of the form
operator p x
which is equal to minus i h cross del by dx
dx
now again we can solve another problem
the problem is we have to show that e to
the power i k x is an eigen function of
the momentum operator where
operator p x is equal to minus i h cross
del by dx
also we have to find the eigen value
now again if we apply the momentum
operator that means
operator p x to e to the power i k x and find
find
operator p x e to the i k x is equal to
minus i h cross del by d x e to the
power i k x
which is equal to h cross k into e to
the power i k x
so we can see that e to the i k x
is an eigen function
and h cross k is an eigen value
for the momentum operator now
now
as a home tax you can check whether the
following eigen value equation or not
that means which i am writing here
whether these are following the
eigenvaluation or not number a number
one is d by dx e to the y kx number two
you can solve whether h cross by two h
by two pi i del by d x into e to the
power i k x
a third problem you can solve it del
[Music]
now i will explain
another very important topic with
hermitian operator
in quantum mechanics the wave functions
which are allowed are always chosen from
the class of functions which are single
valued and continuous
only exception is that except at a
finite number of points where the
function may become infinite
so in the complete range of variables
you can get the value of psi and which
give a finite result
when the squares of the
of their absolute values
are integrated
over a complete range of variables
variables
so this is
very common functions of the
wave functions
now we are considering psi and phi are
two functions of the above class
now if operator a
can be such that
that integration over space
phi star a cap psi e tau is equal to
space integration
psi a cap star phi star theta
then the operator a cap is said to be an
hermitian operator that means if
any operator following the above
condition which i have just said we can
call it as a hermitian operator
now it can be proved
that eigen values of the hermitian
operator are real this is also very important
important
now the question is how we can prove it
now let us take eigen function of the
hermitian operator a cap
with the eigen value a small a
now we can write that small a f is equal
to small a cap psi is equal to s i which
you can give equation number one now
taking the complex conjugate of it we
can found
a cap star psi star is equal to a star
psi star which is equation two now
multiplying left of equation one by psi
star and integrating over space we can
get psi star a cap psi d tau is equal to
small a into psi star psi d tau which is
labeled as equation three
now by multiplying left of equation two
by psi and integrating over space we can
get integration psi a cap star psi star
d tau is equal to e star integration
sister psi theta which is labeled at
equation four now as we learnt
already that if the operator is hermitian
hermitian
then we can equate this equation three
and four
okay so the left hand side of the
equation 3
and 4 will be equal so we can write that
psi star integration psi star psi d tau
is equal to
ester into psi star psi d tau
so we can write that a minus ester
into integration psi star psi data is
equal to zero now
now
sister side theta
will be
not equal to zero so we can write a
minus ester is equal to 0 so a is equal to
to ester
ester
that means that we can say or it can be
said that the eigenvalues of hermitian
[Music]
now i will explain one very important topic
topic
of the schrodinger education so far
i have taught you about the spectroscopy
and now my subject is to connect between
the quantum mechanics which can be done
by solving that equation which is
equation for the wave function of a
particle this equation can be regarded
as fundamental axiom of quantum
mechanics as newton's law is fundamental
postulate of classical mechanics though
we cannot derive schrodinger equation at
this moment
but definitely we can trace
schrodinger's original line of thought
we can hypothesized that if matter poses
wave like properties then there must be
some wavication that will go on them
the classical wave equation can be
written as
del square u by del x square
is equal to one by f square del square u
by del t square where u is the
displacement of the string which is
called a standing wave
and is the function of two independent
variables x and
t now the equation 5 which i just mentioned
mentioned
can be solved by the method of variable
separation technique
and u x t can be written as the product
of a function of x and harmonic or
sinusoidal function of time
now we can write
u x t is equal to psi x cos omega t
which is labeled as equation six
because psi x is the special factor of amplitude
amplitude
of the mu u x t and we shall call psi x
the special amplitude of the wave
now on substituting equation 6 into
equation 5
we can write
del square psi by d x square
plus omega square by v square psi x is
equal to zero
which is enabled as equation seven
now we can introduce the idea of d
probably microwaves which i just
mentioned two classes back into equation
seven the total energy of a particle is
a sum of kinetic energy
and potential energy
we can write e is equal to p square by
two m plus u x where u x is the
potential energy now the equation 8
which i just mentioned can be rearranged as
as
p is equal to second bracket 2m e minus
ux to the power half
now according to the debugging equation
we can write lambda is equal to h by p
so by putting p we can write h by 2 m e
minus u x to the power half which is
labeled as equation 10. now the factor
omega square by v square in equation 7
can be written in terms of lambda
because we know w is equal to 2 pi nu
and nu lambda is equal to v
so putting all together
we can write
omega square by v square is equal to 4
pi square nu square by v square
is equal to 4 pi square by lambda square
now putting the value of lambda it will
be 2 m into e minus u x divided by h
cross square now substituting into
equation 7 we can write
d square psi by d x square plus 2 m by h
square into e minus u x i x is equal to 0.
0.
now this equation is known as very
famous time independent schrodinger equation
equation
importantly this is second order
differential equation for
for
psi x
and for a particle of mass m
moving in a potential field described by
capital u x
the psi x is a measure of the amplitude
of the matter wave and is called the
stationary state wave function of the particle
particle
now the equation 11 can be rewritten as
minus h cross y 2 m into del square psi
by d x square plus capital u x plus psi
x is equal to e psi x which can be
labeled equation 12. now i will explain
very important thing that how solid equation
equation
can be
evaluated or can be formulated as eigen
value problem now the left hand equation
of the equation 12 can be formulated as
minus h cos square 2 m
into del square by d x square
plus capital u x i x
is equal to e x
now we can relate the operators in
brackets by h cap and equation 13 which
i just mentioned can written as h cap
psi is equal to e psi x which is equation
equation
now the equation 14 shows that
schrodinger equation
can be formulated as an eigen value problem
problem
and the operator h cap is called a
hamiltonian operator
the wave function
is is an eigen function and the energy
is an eigen value
of the hamiltonian operator
also this shows a correlation between
the hamiltonian operator and energy the
final form will be h cap is equal to
minus h cross by two m in del square by
d x square plus u x which is labeled as
the equation 15. [Music]
[Music]
now i can summarize my talk today so in
this module we have learned
an operator is a symbol
that instructs us to do something to
whatever follows the symbol
we have learnt about various aspects of
operator algebra like how two operators
will commute with each other conditions
of an operator for being linear operator
etcetera we have discussed eigen value
eigen function and eigen equation
equation
by taking various examples
the condition of an operator forming hermitian
hermitian
is being discussed
in addition we have proved that the
eigenvalues of an hermitian operator
are real
we have
explained the condition of an operator
for being hermitian
in addition
it is proved that the eigenvalues of
hermitian operators are real
most importantly we explained
schrodinger equation from classical web equation
equation
then we have shown that how schrodinger
equation can be formulated as an
eigenvalue problem
lastly we have shown a correlation
between the hamiltonian operator and energy
energy
thank you very much for your kind attention
attention [Music]
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