Introduction to quantum mechanics and schrodinger equation
- 1. Introduction to Quantum Mechanics and Schrodinger Equation SUBMITTED BY :- 1. GAURAV SINGH 2.DEEPAK MEENA CLASS :- B.Sc.(H) CHEMISTRY SEMESTER :- Vth SEMESTER ROLL NO. :- 18/76022(GAURAV)
- 2. Classical Mechanics and Quantum Mechanics Mechanics: the behavior study of matter when it is in motion is called as Mechanics. Classical Mechanics: describing the motion of macroscopic objects. Macroscopic: measurable or observable by naked eyes Quantum Mechanics: describing behavior of systems at atomic length scales and smaller i.e. microscopic objects.
- 3. Introduction to Quantum Mechanics Ques.:- Why Quantum Mechanics is introduced? OR Necessity of Quantum Mechanics? Ans. :- Quantum Mechanics is introduced after the failure of Classical mechanics. Earlier it was thought that the laws of classical mechanics could explain the motion of both macroscopic and microscopic particles. But later on experimental studies showed that classical mechanics failed when applied to microscopic objects such as electron, proton, etc.
- 4. Limitations of Classical Theory of radiation Classical Theory of radiation was not able to explain the following observations:- 1. Black body radiation 2. Photoelectric effect 3. Compton effect 4. Variation of heat capacity of solids as a function of temperature. 5. Line spectra of atoms with special reference to hydrogen.
- 5. Photo Electric Effect The emission of electrons from a metal plate when illuminated by light or any radiation of suitable wavelength or frequency is called Photo electrons.
- 6. 6 6 Photoelectric Effect T max 0 ν νo • Inconsistency with classical light theory According to the classical wave theory, maximum kinetic energy of the photoelectron is only dependent on the incident intensity of the light, and independent on the light frequency; however, experimental results show that the kinetic energy of the photoelectron is dependent on the light frequency. Metal Plate Incident light with frequency ν Emitted electron kinetic energy = T The photoelectric effect ( year1887 by Hertz) Experiment results Concept of “energy quanta”
- 7. Black-Body Radiation A black body is an ideal body which allows the whole of the incident radiation to pass into itself (without reflecting the energy) and absorbs within itself this whole incident radiation (without passing on the energy). This property is valid for radiation corresponding to all wavelengths and to all angels of incidence. Therefore, the black body is an ideal absorber of incident radiation.
- 8. Compton Effect When a monochromatic beam of X-rays is scattered from a material then both the wavelength of primary radiation (unmodified radiation ) and the radiation of higher wavelength (modified radiation) are found to be present in the scattered radiation. Presence of modified radiation in scattered X-rays is called Compton effect.
- 9. Max Planck In 1900, Max Planck initiated Quantum Physics by presenting his Quantum Theory. He was awarded Nobel Prize in Physics in 1918.
- 10. Planck’s Quantum Theory Main Points of this Theory are: a. Energy is not emitted or absorbed continuously. It is emitted or absorbed in the form of wave packets or quanta. In case of light, the quantum of energy is often called Photon. b. The amount of energy associated with quantum of radiation is directly proportional to the frequency of radiation. c. A body can emit or absorb energy only in terms of integral multiple of a quantum/ photon. where n =1,2,3,…. nh E h E E
- 11. Ques.:- Why Classical Mechanics failed for microscopic objects? Ans.:- This is mainly because Classical Mechanics did not take into account the concept of de-Broglie Dual nature of matter and Heisenberg’s Uncertainty Principle. Also, it does not put any restriction on the values of dynamical properties (such as position, magnitude, etc) calculated for microscopic objects and hence Classical Mechanics was failed. Therefore, to understand the behavior of such objects, Quantum Mechanics was introduced.
- 12. Wave-Particle Duality Particle-like wave behavior (example, photoelectric effect) Wave-like particle behavior (example, Davisson-Germer experiment) Wave-particle duality Mathematical descriptions: The momentum of a photon is: h p The wavelength of a particle is: p h λ is called the de Broglie wavelength
- 13. Proposed by French physicist, Louis-Victor de- Broglie. He suggested that, just as radiation possesses dual behavior i.e. Wave nature as well as particle nature, all microscopic and macroscopic materials also possess dual nature (i.e. wave and particle nature). De Broglie, gave the following relation between wavelength (λ) and momentum (p) of a material particle. λ= h/mv = h/p [here λ is wavelength, p is the momentum and h is de-Broglie Dual nature of matter
- 14. The Uncertainty Principle The Heisenberg Uncertainty Principle (year 1927): • It is impossible to simultaneously describe with absolute accuracy the position and momentum of a moving microscopic particle. • It is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy 4 / h x p 4 / h t E The Heisenberg uncertainty principle applies to electrons and states that we can not determine the exact position of an electron. Instead, we could determine the probability of finding an electron at a particular position.
- 15. Schrӧdinger’s Wave Equation Schrödinger equation is the fundamental equation of the science of submicroscopic phenomena known as Quantum Mechanics. The equation, developed (1926) by the Austrian physicist Ervin Schrodinger, has the same central importance to Quantum Mechanics as Newton’s laws of motion have for the large-scale phenomena of classical mechanics.
- 16. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. Essentially a wave equation, the Schrödinger equation describes the form of the probability waves (or wave functions ) that govern the motion of small particles. It specifies how these waves are altered by external influences. Schrödinger established the correctness of the equation by applying it to the hydrogen atom, predicting many of its properties with remarkable accuracy. The equation is used extensively in atomic, nuclear, and solid-state physics.
- 17. Schrodinger’s Equation Time Independent Schrodinger’s Equation Time dependent Schrodinger’s Equation If the particle is moving in 3- dimensional space then,
- 18. Erwin Rudolf Josef Alexander Schrödinger Austrian 1887 –1961 Without potential E = T With potential E = T + V Time-dependent Schrödinger Equation
- 19. The equation takes into account the dual nature of matter and is given as:- The solution of Schrodinger wave equation for an electron give the values of E and Ψ. The values of E represent the possible values of energy which the electron in the atom can occupy. The corresponding values of Ψ are called wavefunctions. The wavefunction corresponding to an energy state contains all the information about an electron which is present in that state.
- 20. Wave function The quantity with which Quantum Mechanics is concerned is the wave function of a body. Wave function, Ψ is a quantity associated with a moving particle. It is a complex quantity. |Ψ|2 is proportional to the probability of finding a particle at a particular point at a particular time. It is the probability density. |Ψ|2 = Ψ*Ψ Thus if Ψ = A+ iB then Ψ* = A− iB ⇒|Ψ |2 = Ψ*Ψ = A2 − i2 B2 = A2 + B2
- 21. Properties of wave function 1. It must be finite everywhere. If ψ is infinite for a particular point, it mean an infinite large probability of finding the particles at that point. This would violates the uncertainty principle. 2. It must be single valued. If ψ has more than one value at any point, it mean more than one value of probability of finding the particle at that point which is obviously wrong. 3. It must be continuous and have a continuous first derivative everywhere. 4. It must be normalisable. 5.It must be bounded in the space.