University Physics: Mechanics

1. University Physics: Mechanics Quantum Mechanics part 1: Wave Function Lecture 3 Iksan Bukhori, M.Phil iksan.bukhori@president.ac.id 2019

2. Textbook and Syllabus Textbook: “Physics for Scientists and Engineers”, Serway and Jewett9th Edition Syllabus: (tentative) Week 1-3: Intro to Quantum Physics Week 4-6: Quantum Mechanics Week 8-10: Atomic Physics Week 11-13: Particle Physics and Cosmology

3. Grade Policy Grade Point: 85 – 100 : A (GPA = 4) 70 – 84 : B (GPA = 3) 60 – 69 : C (GPA = 2) 55 – 59 : D (GPA = 1) 0 – 54 : E (GPA = 0) • The use of smartphone calculator in quizzes and exams is prohibited.

4. Grade Policy Grades: Final Grade = 10% Homeworks + 20% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points • Homeworks will be given in fairly regular basis. The average of homework grades contributes 10% of final grade. • Homeworks are to be written on A4 papers, otherwise they will not be graded. • Homeworks must be submitted on time, on the scheduled day of the lecture. If you submit late, the penalty will be –10·n points, where n is the total number of lateness made. • There will be 2 quizzes. The average of quiz grades contributes 20% of final grade. • Midterm and final exam schedule will be announced in time.

5. Grade Policy • Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points. • Make up of quizzes and exams will be held withinone week after the schedule of the respective quizzes and exams. • To maintain the integrity, the maximum score of a make up quiz or exam can be set to 90. Basic Physics 1Homework 6Rudi Bravo00920170000821 March 2021No.1. Answer: . . . . . . . . Heading of Homework Papers (Required)

6. Lecture Activities • Lectures will be held in the form of PowerPoint presentations. • You are expected to write a note along the lectures to record your own conclusions or materials which are not covered by the lecture slides. How to get good grades in this class? • Do the homeworks by yourself • Solve problems in front of the class • Take time to learn at home • Ask questions

7. Lecture Material • Latest lecture slides will be uploaded to my blog iksanbukhori@wordpress.com • You are responsible to read and understand the lecture slides. I am responsible to answer your questions. • Quizzes, midterm exam, and final exam will be open-cheat sheets. Be sure to have your own copy of lecture slides. You are not allowed to borrow or lend anything during quizzes or exams. • But: A homework can be submitted late without penalty if a scanned or photographed version of the homework is sent to iksan.bukhori@president.ac.idbefore the class begins.

8. Extra Assignments • There will be no remedial for any quizzes nor exams. • As replacement, you may submit a neat summary of your notes or collection of problems and solutions related to all topics covered up to that point in handwritten with A4 paper • This extra assignment will not be announced beforehand and has to be submitted one day after your quiz/exams paper is returned • The extra grade depends on the materials covered, how neat it is and how good your understanding of the materials actually is (Copying my slides or the book(s) will not do)

9. Particle as Wave: de Broglie wavelength = Momentum = Planck’s constant = Particle’s wavelength (deBroglie wavelength) deBroglie postulates that all particles obey the quantum description just like photon, including having energy described by • These equations present the dual nature of matter: • Particle nature, p and E • Wave nature, λ and ƒ

10. Complementarity • The principle of complementarity states that the wave and particle models of either matter or radiation complement each other. • Neither model can be used exclusively to describe matter or radiation adequately.

11. Quantum Particle • The quantum particle is a new model that is a result of the recognition of the dual nature of both light and material particles. • Entities have both particle and wave characteristics. • We must choose one appropriate behavior in order to understand a particular phenomenon.

12. Ideal Particle vs Ideal Wave • An ideal particle has zero size. • Therefore, it is localized in space. • An ideal wave has a single frequency and is infinitely long. • Therefore, it is unlocalized in space. • A localized entity can be built from infinitely long waves.

13. Quantum Mechanics • The theory of quantum mechanics was developed in the 1920s. • By Erwin Schrödinger, Werner Heisenberg and others • Enables us to understand various phenomena involving • Atoms, molecules, nuclei and solids • The discussion will follow from the quantum particle model and will incorporate some of the features of waves under boundary conditions.

14. Probability – A Particle Interpretation • From the particle point of view, the probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity.

15. Probability – A Wave Interpretation • From the point of view of a wave, the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E. • Combining the points of view gives

16. Probability – Interpretation Summary • For electromagnetic radiation, the probability per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave. • The particle is the photon • The amplitude of the wave associated with the particle is called the probability amplitude or the wave function. • The symbol is ψ

17. Wave Function • The complete wave function ψ for a system depends on the positions of all the particles in the system and on time. • The function can be written as • rj is the position of the jth particle in the system • ω = 2πƒ is the angular frequency

18. Wave Function • The wave function is often complex-valued. • The absolute square |ψ|2 = ψ*ψ is always real and positive. • ψ* is the complex conjugate of ψ. • It is proportional to the probability per unit volume of finding a particle at a given point at some instant. • The wave function contains within it all the information that can be known about the particle.

19. Wave Function Interpretation – Single Particle • Y cannot be measured. • |Y|2 is real and can be measured. • |Y|2 is also called the probability density. • The relative probability per unit volume that the particle will be found at any given point in the volume. • If dV is a small volume element surrounding some point, the probability of finding the particle in that volume element is P(x, y, z) dV = |Y |2dV

20. Postulates of Quantum Mechanics • Postulate 1: • State and wave functions. Born interpretation • The state of a quantum mechanical system is completely specified by a wave function ψ(r,t) that depends on the coordinates of the particles (r) and time t. These functions are calledwave functions or state functions. • For 2 particle system: • Wave function contains all the information about a system. • wave function  classical trajectory • (Quantum mechanics) (Newtonian mechanics) • Meaning of wave function: • P(r) = |ψ|2 = • => the probability that the particle can be found at a particular point x and a particular time t. (Born’s / Copenhagen interpretation)

21. Implications of Born’s Interpretation • Positivity: P(r) >= 0 The sign of a wavefunction has no direct physical significance: The positive and negative regions of this wavefunction both correspond to the same probability distribution. (2) Normalization: i.e. the probability of finding the particle in the universe is 1.

22. Physically acceptable wave function • The wave function and its first derivative must be: • Finite. The wave function must be single valued. This means that for any given values of x and t ,  Ψ(x,t) must have a unique value. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state. 

23. The wave function must be square-integrable. In other words, the integral of |Ψ|2   over all space must be finite. This is another way of saying that it must be possible to use  |Ψ|2  as a probability density, since any probability density must integrate over all space to give a value of 1, which is clearly not possible if the integral of  |Ψ|2  is infinite. One consequence of this proposal is that   must tend to 0 for infinite distances. 2. Square-integrable

24. A rapid change would mean that the derivative of the function was very large (either a very large positive or negative number). In the limit of a step function, this would imply an infinite derivative. Since the momentum of the system is found using the momentum operator, which is a first order derivative, this would imply an infinite momentum, which is not possible in a physically realistic system.  Continuous wavefunction

25. All first-order derivatives of the wave function must be continuous. Following the same reasoning as in condition 3, a discontinuous first derivative would imply an infinite second derivative, and since the energy of the system is found using the second derivative, a discontinuous first derivative would imply an infinite energy, which again is not physically realistic. Continuous First derivative

26. Acceptable or Not ??

27. Acceptable or not acceptable ??

28. Exp(x)

29. Sinx/x

30. Sin-1x

31. Homework 1 • The wave function for a particle is given as . Determine the value of and the probability of finding the particle in the interval