AKTU - QUANTUM MECHANICS for Engineering Physics

Summary

In this lecture on Quantum Mechanics for first-year engineering physics students, the instructor covers key topics from the AKTU syllabus. The lecture begins with an introduction to Black Body Radiation, explaining the concept and its relation to Stephen's Law, which states that the energy radiated by a black body is proportional to the fourth power of its absolute temperature. The instructor then delves into the wave-particle duality of light and matter, emphasizing the importance of understanding the time-dependent and independent Schrödinger wave equations. The lecture concludes with a discussion on Compton Effect, providing the formula for the variation in wavelength when a particle undergoes this effect. The instructor also guides students on how to approach numerical problems in quantum physics, ensuring they understand the derivations and applications of these fundamental concepts.

Takeaways

• 📚 The lecture focuses on Unit 3 of the AKTU Engineering Physics syllabus, specifically on quantum mechanics.
• 🌑 Black body radiation is a central concept, with black bodies absorbing all radiation and emitting none.
• 🔢 Several important laws related to energy were discussed: Stefan's Law, Wien's Law, Rayleigh-Jeans Law, and Planck's Law, each with its own formula and principles.
• ⚛️ Wave-particle duality highlights that light behaves both as a wave and a particle under different conditions.
• 📊 Wien's Law explains how the wavelength of maximum energy emitted by a black body shifts towards shorter wavelengths as temperature increases.
• ✍️ Planck's oscillator and its average energy were derived, emphasizing the need to understand this for exams.
• ⚙️ The Schrödinger wave equation is crucial, and students should know both the time-dependent and time-independent versions.
• 💡 Compton effect and its impact on wavelength changes when a particle undergoes the effect were also covered.
• 🔑 Students are advised to memorize key laws and equations, particularly Planck's Radiation Law, which has been tested in exams.
• 📝 The lecture concluded with a review of Stefan's Law, the dual nature of light, and advice on solving numericals in quantum mechanics for exams.

Q & A

What are the five laws related to energy covered in the quantum mechanics unit of the AKTU Engineering Physics syllabus?

The five laws related to energy in the quantum mechanics unit are Black Body Radiation, Stephen's Law, Wayne's Law, Rayleigh's Law, and Planck's Law.

What is the significance of the term 'black body' in the context of quantum mechanics?

A black body is an idealized object that absorbs all incident radiation and emits no radiation. It is used as a model for understanding thermal radiation.

What is Stephen's Law and how is it expressed mathematically?

Stephen's Law states that the total radiant energy emitted by a black body is proportional to the fourth power of its absolute temperature. Mathematically, it is expressed as E = σT^4, where E is the energy, σ is Stephen's constant (5.67 x 10^-8 W/m^2K^-4), and T is the temperature in Kelvin.

What does the graph of energy versus wavelength according to Stephen's Law show?

The graph of energy versus wavelength according to Stephen's Law shows that at the peak value of energy, the wavelength is minimum, and as the energy drops, the wavelength increases.

What is the significance of Wien's Displacement Law in quantum mechanics?

Wien's Displacement Law states that the product of the wavelength corresponding to maximum energy (λm) and the temperature (T) is a constant. This law helps in understanding how the peak of the spectral distribution curve shifts towards shorter wavelengths as the temperature increases.

What is the mathematical expression for Rayleigh-Jeans Law?

Rayleigh-Jeans Law is given by Eλ = (8πkT) / (λ^4), where Eλ is the energy per unit wavelength, k is the Boltzmann constant, T is the temperature in Kelvin, and λ is the wavelength.

How is the average energy of a Planck's oscillator derived?

The average energy of a Planck's oscillator is derived by considering the total energy of the oscillators and dividing it by the total number of oscillators. The formula is given by ⟨E⟩ = (hν / (e^(hν/kT) - 1)), where h is Planck's constant, ν is the frequency, k is the Boltzmann constant, and T is the temperature.

What is Planck's radiation formula and how is it used?

Planck's radiation formula is given by u(ν)dν = (8πhν^2/c^3) dν / (e^(hν/kT) - 1), where u(ν) is the energy density per unit frequency, h is Planck's constant, ν is the frequency, c is the speed of light, and kT is the product of Boltzmann's constant and temperature. This formula is used to calculate the spectral radiance of a black body at a given temperature.

What is the Compton Effect and what is its significance in quantum mechanics?

The Compton Effect is the increase in wavelength of X-rays when they scatter off electrons. It demonstrates the particle nature of light and is significant in quantum mechanics as it provides evidence for the quantization of energy and momentum.

How does the lecturer suggest students approach numerical problems in quantum physics?

The lecturer suggests that students should approach numerical problems by showing each step of the solution, including formulating the problem, unit conversions, and applying the correct equations, to ensure they receive marks for each part of the solution.

Outlines

00:00

📚 Introduction to Quantum Mechanics

The lecture begins with an introduction to Unit 3 of the Engineering Physics syllabus for first-year BTEC students, focusing on quantum mechanics. The syllabus includes five energy laws: black body radiation, Stephen's law, Wayne's law, Rayleigh's law, and Planck's law. Students are advised to memorize these laws and understand their derivations, particularly Planck's law and the concept of energy average of a Planck's oscillator. The lecture also covers wave-particle duality, Schrödinger's wave equation, and the Compton effect. The explanation starts with the concept of a black body, which absorbs all radiation and emits none, and the construction of an approximate black body in a lab using a metallic cavity.

05:02

🌡️ Stephen's Law and Black Body Radiation

This section delves into Stephen's law, which describes the energy distribution among different wavelengths for black body radiation. It states that the total radiant energy emitted by a black body is proportional to the fourth power of its absolute temperature. The constant of proportionality, known as Stephen's constant, is introduced with a value of 5.67 x 10^-8 Wm^-2K^-4. The lecture then discusses the peculiar feature of the spectral distribution curve, which shows that at the peak energy, the wavelength is minimum, and as the energy decreases, the wavelength increases. Wayne's law is also introduced, explaining the relationship between wavelength and temperature, where the product of wavelength and temperature is a constant, indicating a shift towards shorter wavelengths as temperature increases.

10:03

🌱 Derivation of Planck's Radiation Law

The paragraph focuses on the derivation of Planck's radiation law, starting with the concept of energy levels of an oscillator and the average energy of a Planck's oscillator. The total number of oscillations and the total energy are calculated using the Boltzmann factor and the Einstein equation. The derivation involves summing geometric progressions and applying mathematical series formulas to arrive at the expression for the average energy of a Planck's oscillator. The final formula derived is E = hμ / (e^(hμ/kT) - 1), which is a key result in quantum mechanics.

15:06

🌞 Planck's Radiation Formula and Wien's Displacement Law

This part of the lecture covers Planck's radiation formula, which describes the number of oscillations per unit volume within a specific frequency range. The formula is given in terms of frequency and then converted to terms of wavelength using the relationship c = μλ, where c is the speed of light and λ is the wavelength. The lecture also discusses Wien's displacement law, which estimates the temperature of the sun based on the maximum wavelength of emitted radiation. The units of wavelength are normalized to match the units of Wien's constant to calculate the temperature.

20:08

📬 Closing Remarks and Contact Information

The lecture concludes with a summary of the key points covered, including Stephen's law, the dual character of light, Planck's radiation law, and Wien's law. The instructor invites students to subscribe to the channel for more content and provides contact information for feedback or suggestions. The instructor's email and phone number are given for direct communication.

Keywords

💡Black Body Radiation

Black body radiation refers to the theoretical concept of an object that absorbs all incident radiation and reflects none, emitting radiation based solely on its temperature. In the video, the lecturer explains how black bodies are idealized objects, and mentions real-world examples like carbon arc lamps and black platinum that approximate this behavior. It is a key concept in quantum mechanics for understanding energy distribution.

💡Stefan's Law

Stefan's Law, or the Stefan-Boltzmann Law, states that the total energy radiated per unit surface area of a black body is proportional to the fourth power of the absolute temperature. The video emphasizes its importance in understanding black body radiation, and how it helps calculate the energy emitted by such bodies, crucial for students studying thermodynamics and quantum mechanics.

💡Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics, stating that particles such as photons exhibit both wave-like and particle-like properties. The lecturer briefly touches on how light can behave as both a wave and a particle depending on the context, providing a foundation for understanding other quantum phenomena like the Compton effect.

💡Schrodinger Wave Equation

The Schrodinger wave equation describes how the quantum state of a physical system changes with time. In the lecture, both the time-dependent and time-independent versions of this equation are mentioned, stressing their significance in understanding the behavior of particles at the quantum level. This equation is crucial for analyzing systems like a particle in a one-dimensional box, a common example in quantum mechanics.

💡Planck's Law

Planck's Law describes the distribution of electromagnetic radiation emitted by a black body in thermal equilibrium. It marked a major breakthrough in physics, introducing the concept of quantized energy levels. The lecturer focuses on deriving the formula for the average energy of a Planck oscillator, explaining its importance in the broader context of quantum mechanics and black body radiation.

💡Compton Effect

The Compton effect refers to the increase in wavelength of X-rays or gamma rays when they are scattered by electrons. This phenomenon provides evidence for the particle nature of light, as discussed in the video. The lecturer mentions the Compton effect in the context of quantum mechanics, emphasizing its relevance to understanding light's dual nature and the behavior of photons.

💡Wien's Law

Wien's Law states that the wavelength corresponding to the maximum energy emitted by a black body is inversely proportional to its temperature. The lecture explains this law using a spectral distribution curve, illustrating how the peak of the radiation shifts toward shorter wavelengths as temperature increases. This law is essential for explaining the color of stars and other hot objects.

💡Rayleigh-Jeans Law

The Rayleigh-Jeans Law provides a classical approximation for the energy distribution in the thermal radiation of a black body. It works well at longer wavelengths but fails at shorter ones, leading to the ultraviolet catastrophe, which was later resolved by Planck's quantum theory. The lecturer briefly mentions this law as one of the fundamental equations students need to remember for exams.

💡Boltzmann Factor

The Boltzmann Factor is an exponential term that describes the probability of a system being in a certain energy state at a given temperature. In the video, the lecturer explains its role in determining the energy distribution among Planck oscillators, which are fundamental in understanding black body radiation and thermodynamic systems.

💡Geometric Progression

Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant. The lecturer uses this mathematical concept to explain how the number of oscillations in a system can be calculated, specifically applying it to the sum of oscillations in Planck’s radiation law.

Highlights

Introduction to Quantum Mechanics unit in the AKTU Engineering Physics BTEC first year syllabus.

Explanation of the five laws related to energy: Black body radiation, Stephen's law, Wayne's law, Rayleigh-Jeans law, and Planck's law.

Emphasis on memorizing the statements and deriving expressions for energy, specifically for a Planck's oscillator.

Discussion on wave-particle duality and the concept of matter waves.

Importance of understanding both time-dependent and time-independent Schrödinger wave equations.

Detailed case study on a particle trapped in a one-dimensional box.

Definition and formula for Compton effect and its significance.

Description of a black body and its properties, including examples like carbon arc lamp and platinum black.

Stephen's law and its mathematical expression relating total radiant energy to the fourth power of absolute temperature.

Wayne's law and its displacement law, explaining the shift of maximum energy wavelength with temperature.

Rayleigh-Jeans law and its formula for energy distribution in the thermal spectrum.

Planck's theory of black body radiation and the derivation of the average energy of an oscillator.

Derivation of Planck's radiation law and its significance in quantum physics.

Explanation of the derivation process for the average energy of Planck's oscillator.

Application of Planck's radiation formula in deriving Einstein's relationship in laser physics.

Numerical example using Wien's displacement law to estimate the temperature of the sun.

Final summary of the lecture's key points, including Stephen's law, Planck's radiation law, and Wien's law.

Invitation for feedback and suggestions to improve the lecture series.

Transcripts

good afternoon dear students in

continuation with my lecture series on

aktu engineering physics btec first year

syllabus today i am going to brief you

about unit 3 quantum mechanics okay now

first let us have a look at the syllabus

of this unit

quantum mechanics unit the first line

consists of the five laws related to

energy that is black body radiation

stephen's law wayne's law ray league

gene's law and planck's law you have to

memorize the statement of these four

laws

and derive an expression for

energy average energy of a planck's

oscillator

next wave particle duality you know

light possesses dual character at times

it behaves as a wave and at times it

behaves as a particle

matter waves which consist of

both the wave component and the particle

component the most important time

dependent and independent schrodinger

wave equation

one year if the time independent part is

asked the very next year you are asked

about the time independent schrodinger

wave equation you should be knowing

their derivation and one study case in

detail related to particle trapped in

one dimensional box

only one case and last compton effect

the definition and the formula for

variation of wavelength when a particle

undergoes compton effect okay let us

begin with the syllabus so the first

part is black body radiation what is a

black body a black body is one which

absorbs all radiation and emits none a

perfect black body is very hard to find

but we can say carbon arc lamp black

platinum black they can approximately be

treated as black body if you want to

make a black body in the lab then a very

simple method is you take a metallic

cavity in the form of a double wall here

as you can see

hollow copper sphere with lamp black

coating inside

and nickel outside when radiation falls

in this cavity it will enter the

enclosure and suffer

multiple reflections inside at the

interior surface

at each reflection a part of radiation

will be absorbed finally the incident

beam will become weak and absorbed by

the inner surface of the body this is an

approximate example of black body

not hundred percent black body now black

body stephen's law gives you the law for

energy distribution among different

wavelengths according to steepen law the

total

radiant energy that is emitted by a

black body

energy emitted by a black body

is proportional to the fourth power of

absolute temperature

when we remove this proportionality sign

we introduce

a constant here which is known as

stephen's constant and the mathematical

value of this defense constant is

5.67

into 10 to the power of minus 8 vip per

meter square kelvin to the power of

minus 4. stephen's law gave the total

energy radiated by a black body one

thing which is unique about this is that

the spectral distribution curve

plotted between the energy radiated by

the black body and the wavelength

shows one peculiar feature that is here

have a look at this graph on the y-axis

you are having the energy and on the

x-axis the wavelength at the peak value

of energy

wavelength is minimum

and as the energy drops the wavelength

increases

this is one particular feature of the

plot of energy

versus wavelength given by

wayne's law which states that the

product of

lambda m into t is a constant that means

the wavelength

corresponding to

maximum energy

represented by

the peak of the curve shifts towards the

shorter wavelength as the temperature

increases this is known as vein's

displacement law that is lambda m into t

is a constant and lambda m corresponds

to

the maximum wavelength and t corresponds

to the temperature in kelvin so from

wayne's law the maximum energy point

shifts towards the shorter wavelength

side when the temperature of body is

raised

by

or it is increased so the maximum energy

emitted by a black body is proportional

also to the four fifth power of absolute

temperature

rayleigh gene's law says that the energy

distribution in the thermal spectrum is

given by

e lambda is equal to

8 pi k t divided by lambda to the power

of 4 where lambda is wavelength k is

boltzmann constant these are just the

statements of these laws please remember

in your syllabus

not much detail is required for you to

learn you just remember the laws and

their statements and their mathematical

expressions now this derivation is

important it has been asked in your 2022

paper

what is clank's theory of black body

radiation obtain an expression for the

average energy of the oscillator and

derive the planck's radiation law see

this was asked in aktu 2022 march paper

and it was a 10 mark question from

section c now we will derive this step

by step please be ready with the print

out of this page i will tell you the

step by step derivation of this

expression and along with me even you

will be able to derive it see

the frequency of radiation that is

emitted by an oscillator is the same as

the frequency of its vibration

linear oscillator will have only

discrete energy values e and discrete

means n equal to one two three four e n

is n h mu where n is one two three four

also this is comes from the einstein

equation now average energy of planck's

oscillator

let

n be the total number of oscillations

and e the total energy so how do you get

the formula for average you take the

total energy and divide it by the number

of oscillations so let n naught n1 n2 be

number of oscillations

having energies

0

h mu 2 h mu where has this come from

this has come from the expression e n is

equal to n h mu

when i put n equal to 0 it becomes 0

when i put n equal to 1 it becomes h mu

when i put n equal to 2 it becomes 2 h

mu this way the series goes on and the

relative probability

that an oscillator has an energy h mu at

a temperature t is given by boltzmann

factor e to the power of minus h mu by

kt number of oscillations

having energy n h mu is given by

n n equal to n naught e to the power of

minus n h mu divided by k t where once

again n is the integer 0 1 2 3 etc and

when we keep the different values of n

we get the energy levels for those

specific values of n okay next step

total number of oscillations will be

given by

n equal to n naught plus n one plus n

two plus n three now in this expression

we will start putting the values of n

naught n one and

n two n naught

n 1 is n naught e to the power of minus

h mu by k t

n 2 is n naught e to the power of minus

2 h mu by k t

n naught n 3 is n naught e to the power

of minus 3 h mu by kt now we take n

naught common from this expression and

inside we get 1 plus e to the power of

minus h mu by kt plus e to the power of

minus 2 h mu by kt plus e to the power

of minus 3 h mu by kt and so on if you

remember from your mathematics the sum

of geometric progression is given by

1 plus x plus x square so on is equal to

1 upon 1 minus x so in this case x is

equal to e to the power of minus h mu by

kt so the n expression becomes

n

equal to here n equal to n naught into 1

upon

1 minus e minus

h mu by k t that is total number of

oscillations will be equal to this value

now let us find the value of total

energy why are we finding total n and

total e because we need the average so

we will divide e by h to get the average

so according to the famous einstein

equation

e equal to n

h mu where n is 1 2 3 4 etc

from the einstein's expression we will

take e equal to n naught into 0 for z 0

state n equal to 1 n 1 into h mu n equal

to 2 n 2 into 2 h mu n 3 into 3 h mu and

so on using the expression n n is equal

to n naught e to the power of minus n h

mu by k p putting these values we get

e is equal to h mu n naught e to the

power of minus h mu by k t plus 2 h mu n

naught e to the power of minus 2 h mu by

k t plus and so on there is one more

mathematical formula of the series 1

plus x plus 2x square plus 3x cube etc

equal to 1 upon 1 minus x whole square

where x is e to the power of minus h mu

by kt so

this series reduces to equal to

h mu

n naught e to the power of minus 2 h mu

by kt

into 1 upon 1 minus e to the power of

minus h mu by kt whole square where have

we got this from

this is the formula for sum of series

and x has been put as e to the power of

minus h mu by kt so the total energy

comes out as e equal to n naught e to

the power of minus h mu by kt

into h mu upon 1 minus e to the power of

minus h mu by kt whole square now we

have an expression for e we have an

expression for n what do we want we want

the average so we are going to divide e

by n to get the average energy of the

planck's oscillator here e bar is equal

to e upon n you put these values n

naught n naught gets cancelled cross

multiply this expression and you get e

equal to

h mu upon e to the power of h mu by kt

minus 1 this is average energy of

planck's oscillator it was a 10 mark

question of section c

okay now planck's radiation formula what

the why are we always using the word

planck because it is related to your

planck scientist so planck's radiation

formula you just only need to remember

the formula there are no derivations no

details in your aktu syllabus the number

of oscillations per unit volume lying in

the frequency range mu

and mu plus d mu from rayleigh gene's

law is given by 8 pi mu square square d

mu divided by c cube where c is the

velocity of light and u mu d mu is

number of oscillators per unit volume

into average velocity so the planck's

radiation formula is given by

u mu d mu is equal to 8 pi h upon c cube

into mu cube d mu upon e to the power of

h mu by kt minus 1 this has been used in

deriving einstein's relationship in

laser okay so from planck's radiation

formula we in terms of frequency we get

velocity is equal to frequency into

wavelength so c is equal to mu into

lambda you put this value in the above

expression and you can also write this

in terms of lambda

numerical on veins displacement law

estimates the temperature of sun

given lambda m is equal to 4900 angstrom

and veins constant is

0.292 centimeter per kelvin

what was wayne's law lambda m into t is

equal to a constant

now we cannot equate any equation as per

unit and dimension unless and until the

units or the dimensions on both sides of

the equation are not same let us have a

closer look at the values that have been

given to us in the question we note that

wavelength has been given in terms of

angstrom

and wayne's constant is in terms of

centimeter so our first target is to

normalize these units either change

angstrom to centimeter or change

centimeter to angstrom

so

for writing this formula in a tu exam i

will get one mark if this is a five mark

question

now i am changing angstrom to centimeter

we know that one angstrom is equal to 10

to the power of minus 10 meter

and 100 centimeters equal to 1 meter so

1 angstrom is 10 to the power of minus 8

centimeter so given lambda m 4900

angstrom is equal to 4900

into 10 to the power of minus 8

centimeter b

0.292 centimeter per kelvin so your

temperature of sun comes out as b by

lambda m putting the value

0.292 divided by 4900 into 10 to the

power of minus 8 we get

5.95 into 10 to the power of 3 so this

was this numerical is going to be of 5

marks you will be getting marks

as per step by step solution which you

are going to show in your exam like you

will get one mark if you write this

formula lambda m equal to uh lambda m

into t is equal to b

one mark if you convert angstrom to

centimeter another one mark for

normalizing the units one mark again if

you keep the numerical values properly

and last one mark for the correct answer

this is how you have to approach the

numericals in

quantum physics okay let me quickly

revise with you what did i teach you

today i told you about stephen's law e

equal to sigma t to the power of 4 i

told you that

light has a dual character and i told

you about planck's radiation law and

wayne's law thank you please subscribe

to my channel for any suggestions or

feedback please mail to me at divya

gildial

gmail.com or my number nine eight one

zero three two zero three zero three

your suggestions are always welcome

thank you