Review of Elementary Quantum Mechanics

This slide contains study material related to first and second unit of Laser and its application according to APJAKTU syllabus 2017

1. Review of Elementary Quantum Mechanics Unit I & II Dr Md Kaleem Department of Applied Sciences Jahangirabad Institute of Technology (JIT), Jahangirabad , Barabanki (UP) - 225203 1/22/2017 1 DR MD KALEEM/ ASSISTANT PROFESSOR

2. Objective Know the background for and the main features in the historical development of quantum mechanics. Explain, qualitatively and quantitatively, the role of photons in understanding phenomena such as the photoelectric effect Be able to discuss and interpret experiments displaying wavelike behavior of matter, and how this motivates the need to replace classical mechanics by a wave equation of motion for matter (the Schrödinger equation). Understand the central concepts and principles of quantum mechanics: the Schrödinger equation, the wave function and its physical interpretation. Interpret and discuss physical phenomena in light of the uncertainty relation. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

3. An idealized  body  that absorbs all incident  EM radiation incident on it, regardless of frequency or angle of incidence is called a black body Implication: 1. Zero reflection, 2. Zero transmittance It is true for : 1. All wavelengths, 2. All incident directions Corollary: A black body emits maximum amount of radiation at a given temperature Black Body Radiation 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

4. Black Body Radiation 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

5. Planck’s Radiation Law The exchange of energy by radiation with matters do not takes place in continuous manner but discretely as an integral multiple of energy E= h v , where h is the Planck’s constant. On the basis of his assumption Planck derived a relation for energy density u(λ) of resonators emitting radiation of wavelength lying between λ and λ + d λ as 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR Max Planck

6. Photoelectric Effect The emission, or ejection, of electrons (Photoelectrons) from the surface of, generally, a metal in response to incident light is called photoelectric effect . This Phenomenon was observed by Lenard but explained by Albert Einstein in 1905, by describing light as composed of discrete quanta, now called photon, rather than continuous waves . 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR Albert Einstein

7. Dual Nature of Radiation Matter mass momentum Two object can’t exist together at same time and place Wavelength Frequency Two wave co-exist together at same time and place Matter Wave 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

8. de – Broglie Hypothesis In his 1924, depending on the source doctoral dissertation, the French physicist Louis de Broglie made a bold assertion that just as light has both wave-like and particle-like properties, electrons also have wave-like properties 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

9. de – Broglie Relation λ = h/p Wave Particle By rearranging the momentum equation stated in the above section, we find a relationship between the wavelength,  λ  associated with an electron and its momentum,  p , through the Planck constant,  h : λ = h/p 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

10. Matter Wave A wave associated with the motion of a particle e of atomic or subatomic size that describes effects such as the diffraction of beams of particles by crystals. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

11. Wave Packet   wave packet is a combination of waves with about the same momentum. Combining waves into wave packets can provide localization of particles. The envelope of the wave packet shows the region where the particle is likely to be found. This region propagates with the classical particle velocity. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

12. Heisenberg's Uncertainty Principle The uncertainty principle also called the Heisenberg Uncertainty Principle, or Indeterminacy Principle, articulated (1927) by the German physicist Werner Heisenberg, that the position and the velocity of an object cannot both be measured exactly, at the same time. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

13. Heisenberg's Uncertainty Principle The product of the uncertainties in the momentum and the position of a particle equals h/(2) or more. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

14. Group Velocity The group velocity is  d ω / dk ​. It describes how fast wave packets move. It is the velocity with which the envelop of the wave packet moves 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR Group Velocity Phase Velocity

15. Phase Velocity The velocity of the component waves of a wave packet is called Phase velocity. The phase velocity is ω /k ​. It describes how fast the individual wave moves. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

16. Dependency on the medium phase velocity v p of waves are typically larger than the group velocity v g of waves. However, this really depends on the properties of the medium. The media in which v g = v p is called the non-dispersive medium. The media in which v g < v p is called normal dispersive medium. The media in which v g > v p is called anomolous dispersive media. It must be emphasized that dispersion is a property of the medium in which a wave travels. It is not the property of the waves themselves. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

17. Wave Function It is variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle’s being there at the time. By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

18. Physical interpretation of wave function The wave function, at a particular time, contains all the information that anybody at that time can have about the particle.  But the wave function itself has no physical interpretation.  It is not measurable.  However, the square of the absolute value of the wave function has a physical interpretation.  We interpret |ψ( x,t )| 2  as a probability density, a probability per unit length of finding the particle at a time t at position x. The probability of finding the particle at time t in an interval ∆x about the position x is proportional to |ψ( x,t )| 2 ∆x. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

19. Schrödinger Equation Schrödinger equation is a linear, second order partial differential equation. It is a wave equation in terms of the wave function which predicts analytically the probability of the properties the system or events or outcome with precision. The Schrödinger equation is a more general and fundamental postulate of quantum physics. It plays the same role in quantum physics which Newton's laws and the conservation of energy play in classical physics i.e., it predicts the time evolution i.e. future behavior of a quantum dynamical system. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

20. An important feature of the Schrödinger equation is that it is  linear. Hence it allows for the superposition of its solutions i.e. wave functions. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

21. The Schrodinger equation has  two forms’,  one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a, wave, and so the equation is, often referred to as time dependent Schrodinger wave equation. The other is the equation in which the time dependence has been removed and hence is known as the time independent Schrodinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

22. Schrödinger Time Independent Equation 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR Wave function Second derivative wrt x Position Energy Potential Energy Schrodinger established his equation based on de Broglie’s Hypothesis of matter wave, Classical plane wave equation & Conservation of Energy. 

23. Schrödinger Time Dependent Equation 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR The TDSE is consistent with energy conservation. The TDSE is linear and singular value, which implies that solutions can be constructed by superposition of two or more independent solutions. The free-particle solution (U(x) = 0) is consistent with a single de Broglie wave.

24. Introduction of LASER 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

25. Spontaneous and Stimulated Emission Stimulated Absorption: An atom in a lower level absorbs a photon of frequency hν and moves to an upper level. Spontaneous Emission: An atom in an upper level can decay spontaneously to the lower level and emit a photon of frequency hν if the transition between E 2 and E1 is radiative . This photon has a random direction and phase. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

26. Spontaneous and Stimulated Emission 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

27. Stimulated Emission An incident photon causes an upper level atom to decay, emitting a “stimulated” photon whose properties are identical to those of the incident photon. The term “stimulated” underlines the fact that this kind of radiation only occurs if an incident photon is present. The amplification arises due to the similarities between the incident and emitted photons. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

28. Stimulated Emission 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

29. Metasatable State 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR It is excited state of an atom, nucleus, or other system that has a longer lifetime than the ordinary excited states and that generally has a shorter lifetime than the lowest, often stable, energy state, called the ground state. It has a life time between 10 -3 to 10 -5 sec.

30. Population Inversion 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR It is the process in which the population of upper energy level is increased in comparison to lower energy level. It is used obtain optical amplification via stimulated emission needed for laser action.

31. Laser Gain It is defined as the amount of stimulated emission in which a can generate as it travels a given distance in the laser medium. It is characterised by the ability of laser medium to increase the intensity of power of laser. It is also the measure of degree of amplification. 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

32. Q M in Real Life 1/22/2017 DR MD KALEEM/ AISSISTANT PROFESSOR At bottom, the entire computer industry is built on quantum mechanics.  Modern semiconductor-based electronics rely on the band structure of solid objects. This is fundamentally a quantum phenomenon, depending on the wave nature of electrons, and because we understand that wave nature, we can manipulate the electrical properties of silicon. The fibers themselves are pretty classical, but the light sources used to send messages down the fiber optic cables are lasers, which are quantum devices.