INTRODUCTION TO WAVE PACKETS

1. S.RAJAGOPALA REDDY INTRODUCTION TO WAVE PACKETS

2. 1.Origin 2.Definition 3.Properties 4.Time evolution

3. Part 1 Origin of wave packets

4. Schrödinger Equation In one dimension the Schrödinger equation is Time dependent equation By variable separable method this can be divided into two parts On substitution of above equation in first, we will get Time independent equation Complete wave function

5. Applying Time independent Schrödinger equation for a Free particle For free particle potential V(x) = 0 The time independent equation becomes On solving above second order differential equation The complete solution is

6. Importance of coefficients A & B Put B=0 Applying momentum operator on wave function Put A=0 momentum Energy of free particle

7. The above solution poses three severe problems 1. The probability densities corresponds to either solutions are independent of x and t. Heisenberg uncertainty principle give counter for this Both momentum and energy precisely defined so we can't get information of position and time. 2.The wave function is not square integrable

8. 3.Speed of wave vs. particle wavelength Speed of wave = Time period Acc to Debrogle hypothesis Time period is reciprocal of frequency The particle travels twice as past as the wave which represents it So, the plane waves are not solutions for Schrödinger equation

9. Part 2 Definition of wave packet

10. The remedy for all these problems come from Mathematical expression called wave packet Classical Mechanics – Particle - localized Quantum Mechanics - Wave – not localized A localized wave function is called wave packet So, the function peaks at a certain value of x and disappear within a small span . Localized wave packet can be constructed by superposing waves of slightly different wavelength but phases and amplitudes chosen Constructively in desired region and destructively in other region

11. This can be done by Fourier Transformations Where Choosing t=0 Wave packet is given by

12. At x=0, exp(ikx)=1,so Peaks at x=0 As x 0, exp(ikx)‏ 1 In this case frequencies under go constructive interference Away from x=0 then destructive interference takes place and amplitude diminishes. Examples

13. What we can get from wave packets

14. Part 3 How wave packet give answers for previous queries

15. 1) Normalization 2) Since we are not precisely defining momentum and energy, we will not loose information about position and time . 3) Wave velocity : To be explained in few minutes

16. Uncertainty relations Gaussian wave packet Other wave packets Gaussian wave packet is called as minimum uncertainty wave packet

17. Motion of wave packets a) Non dispersive medium Which implies The wave packet travels right without any distortion Angular frequency is function of k b) Dispersive medium Expanding angular frequency using Taylor series

18. Time evolution of wave packets The motion of wave packet is Group velocity is the velocity with which the wave packet Propagates as a whole Phase velocity is the velocity of individual waves

19. The angular frequency related to phase velocity by For non dispersive medium For a free particle Group velocity nothing but classical velocity Phase velocity has no physical significance

20. We can truncate the Taylor expansion of angular frequency a) Linear approximation This can be written as travels right with phase velocity is a curve which travels with group velocity and undistorted In linear approximation the wave packet is undistorted and undergoes a uniform translation

21. 2) Quadratic approximation Apply above formalism on Gaussian wave packet Probability density Probability density

22. Width of wave packet Width of wave packet in L.A So, in Q.A wave packet broadens linearly with time

23. The Spreading of a Wave Packets Probability density Width of wave packet vg Δx0

24. References “Quantum Mechanics – concepts and applications” by Nouredine Zettili “Molecular quantum mechanics” by Peter Atkins & Ronald Friedman