Spectroscopic Transitions

Summary

This NPTEL-NOC IITM lecture on spectroscopic transitions dives deep into the interaction of light and matter through quantum mechanics. The discussion begins with a review of fundamental spectroscopy concepts, such as the dual nature of light and Einstein's light quantum hypothesis. Detailed explanations of Einstein's A and B coefficients follow, illustrating the probability of spectroscopic transitions like absorption and emission. The lecture further delves into the conditions for spontaneous and stimulated emission, the concept of population inversion necessary for laser operation, and the role of quantum mechanics in describing these interactions. A problem-solving segment at the end reinforces the lecture's key points, emphasizing the significance of understanding the units of Einstein's coefficients and their implications in two-level systems.

Highlights

• Discussed the dual nature of light and its implications for spectroscopy. 💡
• Explained the importance of Einstein's A and B coefficients in describing spectroscopic transitions. 📚
• Highlighted the conditions required for spontaneous and stimulated emission. 🌈
• Illuminated the concept of population inversion and its relevance to laser technology. 🔦
• Explored multi-photon processes and their dependency on photon density. 🔍

Key Takeaways

• Understanding the dual nature of light enhances the appreciation of spectroscopy. 🌟
• Einstein's A and B coefficients are crucial for describing light-matter interactions. 📊
• Population inversion is necessary for laser operation, highlighting the importance of stimulated emission. 🎆
• Quantum mechanics provides a framework to understand spectroscopic transitions. 🔬
• Multi-photon processes are rare and typically require high-photon-density sources like lasers. 🎇

Overview

In this comprehensive lecture, the fundamental aspects of spectroscopic transitions spearheaded by the dual nature of light and quantum mechanics are explored. The lecture revisits basic spectroscopy concepts before delving into the nature of light as both particles and waves, embodying the dual characteristics that are pivotal in understanding these transitions.

A significant portion of the lecture is dedicated to unraveling the intricacies of Einstein's A and B coefficients and their roles in the probability of absorption and emission within the context of spectroscopy. These coefficients form the bedrock for explaining why and how spectroscopic transitions occur, introducing fascinating concepts like population inversion, which is essential for the operation of lasers.

Towards the conclusion, the lecture provides insights into the limits of certain spectroscopic processes and the conditions necessary for alternative interactions. Specifically, it addresses multi-photon interactions, underlining that such processes are generally scarce unless under the influence of high-density photon sources such as lasers. The session wraps up with problem-solving exercises to cement these theoretical concepts in practical understanding.

Chapters

• 00:00 - 00:30: Introduction and Revision of Quantum Mechanics The chapter titled 'Introduction and Revision of Quantum Mechanics' serves as an introductory lecture where the basic concepts of quantum mechanics are revisited. This foundational knowledge is essential to understanding the more advanced topics such as spectroscopy and the interaction between light and matter. The transcript also mentions a continuity from the last lecture, ensuring students build on their existing understanding.
• 00:30 - 01:00: Revision of Spectroscopy Concepts The chapter discusses spectroscopy, revisiting the initial concepts taught in the course. It begins with describing light as an electromagnetic wave and introduces fundamental terms such as wavelength.
• 01:00 - 02:00: Wavelength and Frequency Relationship This chapter explores the inverse relationship between wavelength and frequency, emphasizing that wavelength is inversely proportional to frequency. Following this discussion, the chapter delves into the dual nature of light, highlighting Einstein's light quantum hypothesis which posits that light consists of photons.
• 02:00 - 03:30: Energy of Photons Photons can be thought of as the molecules of light, much like how molecules are to matter which consists of atoms and molecules.
• 03:30 - 07:00: Einstein's Coefficients for Spectroscopic Transition This chapter discusses the energy of a photon and its relation to Einstein's coefficients, which describe spectroscopic transitions. The energy of a single photon is given by the formula E = h * ν = h * c / λ. Additionally, the energy of one mole of photons is derived as N_A * E, with N_A representing Avogadro's number, as explored in an initial problem set.
• 07:00 - 11:00: Electromagnetic Spectrum and Spontaneous Emission This chapter explores the electromagnetic spectrum and the spontaneous emission of energy. It introduces the concept of energy as E proportional to V times H and sometimes referred to in terms of frequency mu (ν) or wavelength lambda (λ), emphasizing the quantized nature of electromagnetic radiation. The chapter aims to provide a foundational understanding of how energy is quantified and transferred in the form of electromagnetic waves.
• 11:00 - 15:00: Rate Equations and Population Inversion The chapter titled 'Rate Equations and Population Inversion' discusses the fundamental concept of expressing energy using Einstein or E as a unit, emphasizing its dependency on the frequency or wavelength of light being used. The lecture concludes with an exploration of the implications and applications of this concept in various scientific fields.
• 15:00 - 19:00: Absorption and Emission Processes The chapter begins with a discussion on Einstein's coefficients, specifically 'A' and 'B', which represent the probability of spectroscopic transitions such as absorption or emission. These coefficients are interconnected, leading to an exploration of their relationships. The chapter further delves into the concept of two energy levels, denoted as E1 (lower energy) and E2 (higher energy), and introduces three Einstein's coefficients relevant to these transitions.
• 19:00 - 21:00: Introduction to Lasers The chapter 'Introduction to Lasers' discusses various processes involved in laser operations, such as spontaneous emission, absorption, and stimulated emission. It introduces key concepts and terminology critical to understanding how lasers function, focusing on three primary processes represented by notation: a for spontaneous emission, b12 for absorption, and d21 for stimulated emission.
• 21:00 - 25:30: Multi-Photon Processes and Quantum Mechanics This chapter delves into multi-photon processes in quantum mechanics, highlighting that molecules in their ground state cannot transition to excited states without external light stimulation. The explanation emphasizes the importance of photon interaction to induce molecular excitation. The content is particularly focused on the theoretical underpinning of why a process with coefficient e1 is not possible in the absence of light interaction.
• 25:30 - 33:30: Electric Dipole Transitions This chapter discusses electric dipole transitions, with a focus on the relationship between coefficients in Einstein's theory. It highlights two main relationships: (1) B_12 equals B_21, often referred to as Einstein's B coefficient, and (2) another relation involving A_21, which can be expressed using constants like 8 and pi. These coefficients are crucial for understanding spontaneous transitions to excited states.
• 33:30 - 35:30: Time-Dependent Perturbation Theory and Transition Probability This chapter discusses Time-Dependent Perturbation Theory with a focus on Transition Probability. It derives an expression involving Planck's constant (H), frequency of transition (nu_1_2), and the speed of light (C). The transition is described by Delta e = H nu_1_2, and the expression leads to the calculation of transition probability through the relevant constants and variables.
• 35:30 - 37:00: Fermi's Golden Rule and Transition Moment Integral This chapter discusses Fermi's Golden Rule and the Transition Moment Integral, focusing on Einstein's coefficients. The main equation presented is Einstein's a coefficient, which is related to the Transition Moment Integral and can be expressed as: A = (8πhν₁₂³/c³) × B. This equation shows the dependence of the a coefficient on several constants including Planck's constant (h), frequency (ν₁₂), and the speed of light (c).
• 37:00 - 43:00: Selection Rules and Conclusion The chapter titled 'Selection Rules and Conclusion' discusses the probability of absorption and stimulated emission in two states, noting that both have the same probability. These phenomena occur through the same light-induced mechanism, according to a principle outlined in the chapter.
• 43:00 - 52:30: Problem Solving: Units of Einstein's Coefficients The chapter discusses the concept of detailed balance, focusing on the difference between spontaneous and stimulated emission of radiation. It highlights that the probabilities of these emissions, represented by coefficients A and B, are not equal. Additionally, it points out that the ratio of these coefficients, A/B, is influenced by the frequency.
• 52:30 - 55:00: Conclusion and End of Lecture The chapter discusses the electromagnetic spectrum, emphasizing the relationship between wavelength, frequency, and radiation types. It highlights that radiations with lower frequencies have relatively larger wavelengths, exemplified by comparing radio and microwaves to the UV-visible region, where frequency increases. The summary aims to encapsulate the key points about electromagnetic wave properties discussed in the lesson.

Spectroscopic Transitions Transcription

• 00:00 - 00:30 [Music] hello everyone welcome to the lecture in the last lecture you have learned about the basic concepts of quantum mechanics I hope these basic concepts will help you to understand and appreciate spectroscopy or light matter interaction
• 00:30 - 01:00 let us start with the revision of what you have learned about spectroscopy from the very first lecture of this course in the very first lecture we started by describing light as an electromagnetic wave we learnt what is wavelength that
• 01:00 - 01:30 is lambda and its inverse relation to frequency that is wavelength is inversely proportional to frequency from there on we discussed about the dual nature of light we have seen that according to the light quantum hypothesis of Einstein light consists of photons so light consists of photons and
• 01:30 - 02:00 as we know matter consists of atoms and molecules so we can think what is a molecule for matter photon is for light in other words we can write photons as molecules of light
• 02:00 - 02:30 so we have seen that energy of one photon is given by e equals H nu equals H C by lambda and from the problem that we solved in the first lecture we saw the energy of one mole of photon is given by n AV times e where in a V is Avogadro's number and we
• 02:30 - 03:00 can write in every times e as in a V times H mu or in a V times H C by lambda this amount of energy is called one
• 03:00 - 03:30 Einstein or in other words one can express energy using Einstein or E as unit note that this unit Einstein or E is dependent on the frequency or the wavelength of the light that is being used we ended the lecture with a
• 03:30 - 04:00 discussion on Einsteins coefficients Einsteins a and B coefficients representing probability of our spectroscopic transition for example absorption or emission are related so these are instance coefficients are related and now let us consider two energy levels the lower energy represented by e1 and the higher energy represented by e2 so in this case we have three Einsteins coefficients one is
• 04:00 - 04:30 a to one that is for spontaneous emission then b12 for absorption process and d21 for stimulated emission so we should
• 04:30 - 05:00 remember that there cannot be a process with coefficient e1 - this is because when the molecules are already at the ground state we need some stimulation by light to take the molecules to the excited state or in other words molecules in the ground state cannot
• 05:00 - 05:30 reach the excited state spontaneously so in the last class we saw these three coefficients are related and we found two relations number one is B 1 2 equals B 2 1 and we can write this as B as Einsteins B coefficient the other relation is a 2 1 we can write this 8 pi
• 05:30 - 06:00 H nu 1 2 cubed by C cubed times B 2 1 so here H is the Planck's constant new 1/2 is a frequency for the transition where Delta e equals H nu 1 2 and C is the speed of light so we can write this as a
• 06:00 - 06:30 that is Einsteins a coefficient equals 8 pi H nu 1 to Q by C cubed times B or in other words we can again write a by B equals eight pi H nu 1 2 cubed by C
• 06:30 - 07:00 cubed the first relation states that for the given two states the probability of absorption and the probability of stimulated emission are the same note that the absorption and the stimulated emission occurs via the same mechanism that is both a light induce phenomena and this is known as principle of
• 07:00 - 07:30 detailed balance the second relation indicates that the probabilities of the spontaneous emission that is a to one and that of the stimulated emission that is B to one they are not the same the ratio a by B depends on the frequency or
• 07:30 - 08:00 the wavelength of light for radiations with lower frequencies the ratio a by B small so this figure shows the entire electromagnetic spectrum we can see the frequencies of radio waves or microwave are smaller than the frequencies of the UV visible region so here frequency is increasing on the
• 08:00 - 08:30 left side and wavelength which is inversely proportional to frequency is increasing on the right side hence a by B ratio is small in the radio wave or the microwave range this means spontaneous emission in this wavelength range is less likely to occur compared to stimulated emission on the other hand for larger frequencies or
• 08:30 - 09:00 shorter wavelengths that is in the UV visible region the a by B is large and the spontaneous emission is more likely to occur in this wavelength range you may have already heard about the process called fluorescence so fluorescence is a spontaneous emission process occurring mainly in the visible range existence of these two relations
• 09:00 - 09:30 this 1 & 2 between the 3 parameters a 2 1 b 1 2 and B 2 1 indicates the determination of any one of them will give information of the other two when light falls on matter all the processes represented by a 2 1 B 1 2 and P 2 1
• 09:30 - 10:00 that is spontaneous emission absorption and stimulated emission can take place the net result can be studied by considering the overall rate equation so we can write an overall rate equation thus for the two-level system one can
• 10:00 - 10:30 write the overall rate equation as DN 1 DT equals minus D into DT it was a into plus B Rho nu nu 1 2 into minus B Rho nu nu 1 2 and 1 so now we can see this
• 10:30 - 11:00 first term comes from spontaneous emission the second term comes from stimulated emission and the third term comes from absorption so if we consider that the probability of the spontaneous emission is very small or this a term is very small we can write so we can neglect this and we can write an overall
• 11:00 - 11:30 emission rate so the overall emission rate and in emission process molecules go from higher energy levels to lower energy levels so it's how the molecules at a higher energy level e 2 are changing over time so we have can write minus D n 2 DT equals V Rho nu 1 to n 2
• 11:30 - 12:00 minus N 1 similarly we can write an overall absorption rate so overall absorption rate and because in the absorption process the molecules
• 12:00 - 12:30 go from lower energy level to higher energy level so in this case we are talking about the rate of change of the molecules from the lower energy level in that case the overall absorption rate is minus DN 1 DT and that is equals V Rho nu nu 1 to N 1 minus n 2 so we can see
• 12:30 - 13:00 if n 2 is less than N 1 that is the lower energy level is more populated than the upper level we will get a net absorption only in the event when n 2 is greater than n 1 we would observe a net emission normally for systems at thermodynamic
• 13:00 - 13:30 equilibrium the ratio in to yn 1 is given by the Boltzmann distribution formula Boltzmann distribution formula so this Boltzmann distribution formula tells us n 2 by N 1 equals e to the
• 13:30 - 14:00 power minus e2 minus e1 by K T so as we know that e 2 corresponds to the energy of the higher energy level so e 2 is greater than e 1 so under this condition we get into is less than n 1 that is the
• 14:00 - 14:30 lower level is always more populated than the upper level thus under ordinary condition of thermodynamic equilibrium one would always get the absorption of light for an overall induced emission to occur one requires the situation where in 2 is greater than n 1 that is population of the upper state is greater than the population of lower state and
• 14:30 - 15:00 this is known as the condition of population inversion further if we have in one equals in two there will be no net absorption or emission of radiation and this is known as the saturation condition we can see
• 15:00 - 15:30 that the absorption process can be written as m1 plus photon that with energy H nu 1 2 that gives a new state of the matter that is M 2 so M 1 was the
• 15:30 - 16:00 initial state of the matter with energy e 1 now when photon comes in with frequency nu 1 2 we get the matter in the new energy state that is M 2 and this is for absorption process similarly for stimulated emission
• 16:00 - 16:30 we can write M 2 plus photon that is H nu 1 2 gives M 1 plus two photon that is to H nu 1 in chemical parlance this process written can be thought of as an
• 16:30 - 17:00 auto catalytic reaction because the photon that is being created or in terms of a chemical reaction this photon is a product this photon is catalyzing the reaction so this is an example of auto catalysis again this process indicates that the number of photons get increased as the process continues for example we
• 17:00 - 17:30 can write there is a photon and M 2 this gives M 1 and two photons then each of these photons will interact with M 2 and will create two more photons so we have
• 17:30 - 18:00 four photons and this process will go on so we can see one can multiply the number of photons in this manner increase in the number of photons of a given frequency means an increase in the intensity of light of the frequency or in other words because there is an increase in the intensity of light we have light amplification thus we can
• 18:00 - 18:30 have light amplification by stimulated emission of radiation so if I take the first alphabets what I get
• 18:30 - 19:00 yes lasers but one has to keep in mind that the condition of population inversion is the necessary condition for laser another important thing to note in this context is that the representation of a spectroscopic transition by a process like m1 plus photon gives m2 implies
• 19:00 - 19:30 that one molecule interacts with only one photon at a given time thus the B coefficient determines the probability of absorption or emission of a single photon multi photon processes that is processes involving more than one photon interacting with one molecule cannot be treated by this procedure
• 19:30 - 20:00 intuitively the probability of such a process is very very small and in normal cases the rate of this process can be neglected but let's say for a in photon process the probability or the rate is proportional to n to the power n where
• 20:00 - 20:30 capital n is the number density of photon or in other words this is the number of photons per unit volume so for ordinary light source this capital n is small enough making the rate negligible but if one uses a light source with high photon
• 20:30 - 21:00 density for example if one uses a laser source multi photonic processes can take place a spectroscopic process can be represented as there is an initial state and the initial state interacts with light so we have interaction with light
• 21:00 - 21:30 and there is a final state according to quantum mechanics the states can be represented by a wave function shape so let say I and shy F be the wave functions of the initial and the final States respectively interaction in
• 21:30 - 22:00 quantum mechanics is represented by a term V in the Hamiltonian operator of the system light matter interaction is treated in quantum mechanics by considering matter having quantized its light is however treated classically as a source of field electric or magnetic field described by wave theory based on
• 22:00 - 22:30 Maxwell's electromagnetic theory electromagnetic waves are changing electric and magnetic fields the electric field as you can see is perpendicular to the magnetic field and both fields are directed at right angles to the direction of propagation of light for a light matter interaction the
• 22:30 - 23:00 involved interaction is between the charge or charge distribution in atoms and molecules and the electric field or the magnetic field of light since the interaction with magnetic field is very small compared to that with the electric field we will confine our discussion on the interaction with the electric field only spectroscopy transition involved in such a case is
• 23:00 - 23:30 called electric dipole transition now the electric field of light depends on time given by e equals e 0 cos Omega T or we can also write e equals 0 sine
• 23:30 - 24:00 Omega T where Omega is the angular frequency of light which is related to the frequency that we know nu so Omega is related to nu by Omega equals 2 pi nu and easy R is a constant thus the interaction term V in the Hamiltonian of a light matter system depends on time and we can write V as V
• 24:00 - 24:30 as a function of time for small magnitude of V one can use the time-dependent perturbation theory in quantum mechanics to give expression for the probability of transitions enabling
• 24:30 - 25:00 the B coefficient to be evaluated according to the time-dependent perturbation theory the transition probability per unit time from the initial to the final state is given by the Fermi golden rule so that transition probability from the initial to the final state per unit time is 2pi by H
• 25:00 - 25:30 cross and then we have this final state the interaction term initial state so this whole modulus squared and rule of EF where this Rho EF is the density of states so because the interaction is
• 25:30 - 26:00 between the electric field of light and the charge or the charge distribution that is the dipole of the matter so it comes out that the probability is proportional to shy F mu shy I whole square where this is is the MU is the
• 26:00 - 26:30 dipole operator does the integral shy F mu I is often called the transition moment integral
• 26:30 - 27:00 and this transition moment individual plays a central role in determining the probability of a transition in the event that this transition moment integral is zero the probability of the transition is also zero such transitions are said to be forbidden one can calculate the conditions under
• 27:00 - 27:30 which this transition moment integral is zero and one gets to something known as selection rules for these transitions to occur intuitively the wave functions in the transition moment integral in general depends on quantum numbers say n
• 27:30 - 28:00 and such selection rules are often stated in terms of change in quantum numbers or Delta n this will be discussed further in specific cases when we go into different forms of spectroscopy so this brings us to the end of this lecture again we will solve
• 28:00 - 28:30 a couple of problems and then we will come to the end so the first question we have is what are the units of Einsteins a and B coefficients so we will start with Einsteins a coefficient we can write minus D into DT equals a into for spontaneous ignition so from this we can write a equals minus DN 2 DT tends 1 by
• 28:30 - 29:00 into so n 2 or DN 2 these are numbers or change in numbers so we can if we look into the units of a in that case the unit on the left-hand side should be equal to the unit on the right-hand side in other words this number and number
• 29:00 - 29:30 will cancel so the unit we have is 1 by unit of time that is 1 by second so the unit of a is second inverse so now let's go to Einsteins B coefficient for B coefficient we can write minus DN 2 DT equals B Rho nu nu 1 2 into so we can
• 29:30 - 30:00 write now B equals minus T into DT 1 by Rho nu nu 1 2 times into again the units of n 2 and DN 2 they can cancel out so the unit of B because again the unit in the left hand side would be same as unit in the right hand side the unit here
• 30:00 - 30:30 will be 1 over seconds it is 1 by DT and then the unit of Rho nu nu 1 2 as I discussed in the first lecture is Joule second meter to the power minus 3 in other words the unit of B becomes Joule inverse second to the power minus
• 30:30 - 31:00 2 meter cubed so the second question we have is at equilibrium can a two-level system lead to population inversion so at equilibrium the rate of absorption will be equal to the rate of emission so we have two emission processes so in other words I can write minus D 1 DN 1 DT will be equal to minus D into DT so
• 31:00 - 31:30 minus D 1 n 1 DT is B Rho nu nu 1 to N 1 this will be equal to P Rho nu nu 1 to n 2 plus a into so we can write B Rho nu
• 31:30 - 32:00 nu 1 to N 1 this will be if I take n to common is B Rho nu nu 1 2 plus a so now if I compute in 1 by n 2 what I get is B Rho nu nu 1 2 plus a by B Rho nu nu 1 2
• 32:00 - 32:30 so now we see this a is not 0 and because a is not 0 so n1 by n2 is always greater than 1 in other words we'll always have more molecules in the lower level than at the higher level so this is true for a two-level system
• 32:30 - 33:00 and thus we can see we can never reach a case of population inversion in a two-level system however if you are interested you can see that for more than two level system population inversion can be reached [Music] [Applause]