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LECTURE 21 Quantum mechanical operators

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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

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

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

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.

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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คำบรรยาย YouTube:LECTURE 21 Quantum mechanical operators

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คำบรรยาย YouTube:
LECTURE 21 Quantum mechanical operators