Schrödinger wave equation

about Schrödinger wave equation and quantum mechanics

1. M.M.H. COLLEGE GHAZIABAD Session 2020-21 Department of chemistry M.Sc. (CHEMISTRY), I SEMESTER

2. SNO. Schrödinger wave equation Postulates of quantum mechanics 1. Who is Erwin Schrödinger? First and second postulate 2. The Schr ö dinger equation Third , fourth and fifth postulate 3. The Schr ö dinger equation in 1-D Sixth and seven postulates 4. The Schr ö dinger equation in 1-D: Wave packets Thank you 5. The Schr ödinger equation in 1-D: Stationary states Bibliography 6. Heisenberg vs Schrödinger 7. Conclusion INDEX

3. Schrödinger wave equation & Postulates of quantum mechanics

4. Schrödinger wave equation This Photo by Unknown Author is licensed under CC BY-SA

5. Who is Erwin Schrödinger? was a Nobel Prize-winning Austrian-Irish  physicist  who developed a number of fundamental results in  quantum theory : the  Schrödinger equation  provides a way to calculate the  wave function  of a system and how it changes dynamically in time In addition, he was the author of many works on various aspects of  physics :  statistical mechanics  and  thermodynamics , physics of dielectrics,  colour theory ,  electrodynamics ,  general relativity , and  cosmology , and he made several attempts to construct a  unified field theory . He is also known for his " Schrödinger's cat " thought experiment. 5

6. Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. ( Newton's dot notation  for the time derivative is used.)

7. The Schr ö dinger equation Schrodinger wave equation is a mathematical expression describing the energy and position of the electron in space and time, taking into account the matter wave nature of the electron inside an atom. It is based on three considerations. They are; Classical plane wave equation, Broglie’s Hypothesis of matter-wave, and Conservation of Energy. Schrodinger equation gives us a detailed account of the form of the  wave functions  or probability waves that control the motion of some smaller particles. The equation also describes how these waves are influenced by external factors. Moreover, the equation makes use of the energy conservation concept that offers details about the behaviour of an electron that is attached to the nucleus.

8. Besides, by calculating the Schrödinger equation we obtain Ψ and Ψ2, which helps us determine the quantum numbers as well as the orientations and the shape of orbitals where electrons are found in a molecule or an atom. There are two equations, which are time-dependent Schrödinger equation and a time-independent Schrödinger equation. Time-dependent Schrödinger equation is represented as; Where, I = imaginary unit, Ψ = time-dependent wavefunction, h 2  is h-bar, V(x) = potential and H^ = Hamiltonian operator. The Schr ö dinger equation

9. The Schr ö dinger equation in 1-D We found that the one-dimensional Schr ödinger equation for a free particle of mass m is How do we interpret the complex solution ? This represents a distribution of “something” in space and time. Any real quantity, however, must have a real solution. Recall that we interpreted the interference intensity pattern as representing the square of the electric field, and individual photons land on a screen with a probability given by the intensity pattern (more land where the intensity is high, fewer land where it is low). Likewise, the quantity is the (real) probability in space and time where the particle will be found, where the * represents the complex-conjugate found by replacing I with - I . The square of the abs. value of the wave function, | ( x , t )| 2 , is the probability distribution function. It tells us the probability of finding the particle near position x at time t .      

10. The Schr ö dinger equation in 1-D: Wave packets If is a solution to the Schrödinger equation, any superposition of such waves is also a solution. This would be written: A free-particle wave packet localized in space (see Figure 40.6 at right) is a superposition of states of definite momentum and energy. The function itself is “wavy,” but the probability distribution function is not. The more localized in space a wave packet is, the greater the range of momenta and energies it must include, in accordance with the Heisenberg uncertainty principle The Schrödinger equation discussed so far is only for a free particle (in a region where potential energy U ( x ) = 0). We will now add non-zero U ( x ).    

11. The Schr ödinger equation in 1-D: Stationary states If a particle of mass m moves in the presence of a potential energy function U ( x ), the one-dimensional Schrödinger equation for the particle is This equation can be thought of as an expression of conservation of energy, K + U = E . Inserting ,the first term is K times . Likewise, the term on the RHS is E times . For a particle in a region of space with non-zero U ( x ), we have to add the term U ( x ) Y ( x , t ) on the left to include the potential energy. Let’s write the wave function in separable form , where the lower-case y ( x ) is the time-independent wave function. Further, we can write , so that

12. The Schr ödinger equation in 1-D: Stationary states For such a stationary state the probability distribution function | ( x , t )| 2 = |  ( x )| 2 does not depend on time, which you can see by In this case, the time-independent one-dimensional Schr ödinger equation for a stationary state of energy E simplifies to where the time derivative has been explicitly taken on the RHS We will spend most of the rest of the lecture on solving this equation to find the stationary states and their energies for various situations. Note: The term stationary state does not refer to the motion of the particle it represents. The particles are not stationary, but rather their probability distribution function is stationary (does not depend on time), rather like a standing wave on a string.

13. Heisenberg Vs Heisenberg’s picture was basically statistical. According to them the behaviour of the world’s particles can not be described classically but only probabilistically. For them asking questions such as “where is the particle in between measurements” was simply meaningless, we can only talk about measurements, and we can only make probabilistic predictions for the outcome of those measurements. For them the famous wave-particle duality was a consequence of this intrinsic probabilistic nature of particles, but they never really considered matter particles as being real waves or as being smeared across space. Schrödinger Schrödinger as is well known developed his famous wave equation, which extended the original De-Broglie’s concept of real matter waves and achieved a wave formulation which he proved to be totally equivalent to the statistical formulation of Born and Heisenberg. Schrödinger himself was never too determined about the physical meaning of his wave, but mostly he believed that the mass and charge of the electron was indeed delocalized in between measurements, it was smeared out across the region of space described by the wave.

14. CONCLUSION In conclusion, the Schrodinger equation has been derived to be the (local) condition the wavefunction must satisfy at each point in order to fulfil the total (global) energy equation. In an analogous fashion, we can derive the three dimensional, time-dependent Schrodinger equation and also the other wavefunction equations from the respective total energy equations

15. Postulates of quantum mechanics This Photo by Unknown Author is licensed under CC BY-SA

16. First postulate : The physical state of a system at time t is described by the wave function The postulates of quantum mechanics for the mechanical treatment of the structure of atom rest upon a few postulates which, for a system moving in one dimension, say the - coordinate, are given below. This Photo by Unknown Author is licensed under CC BY-SA-NC Second postulate : The wave function Ψ (r,t) depends upon position coordinate r i.e. , Ψ (r) and also on time coordinate t i.e., Ψ (t) and can be written as Ψ (r.t) = Ψ (r). Ψ (t) Also

17. Third postulate: A physically observable quantity can be represented by a Hermitian operator. An operator is said to be Hermitian operator, if it satisfies the following

18. 19 Large Photo Slide “If you think you understand quantum mechanics, then you don’t.” By Richard feynman

19. THANK YOU SUBMITTED TO : Mrs. ANURADHA SINGH Dr. RATNA SHERRY SUBMITTED BY : VIDHI WALIA

20. [1] Griffiths, D., Introduction to Quantum Mechanics, 2nd ed., Prentice-Hall, New Jersey, 2004. [2] Greiner, W., Quantum Mechanics an Introduction, Springer, New York, 1994. [3] Principle of Physical chemistry by B.K. Puri Bibliography