Physics Department Phys 3650 Quantum Mechanics – I Lecture Notes Dr. Ibrahim Elsayed

1. Physics Department Phys 3650 Quantum Mechanics – I Lecture Notes Dr. Ibrahim Elsayed Quantum Mechanics

2. Newtonian mechanics • Mechanics based on Newton’s law works fine for macro-particles • The aim of Newtonian mechanics is to find the evolution of a particle position , x(t) • From x(t), we can know everything about the particle • How can we do that, simply if we know the force acting on the particles, F • Then we apply Newton’s 2nd law F = m a • With the help of boundary condition, the velocity, v(t) and position, x(t) can be found

3. Newtonian mechanics • If the particle velocity is too high v(t), approach speed of light • If the particle mass is too small, like atoms and electrons • Newtonian mechanics Fails • In the first, we use relativistic mechanics • In the second we use quantum mechanics

4. The beginning of Story • Light behaves as wave when it undergoes interference, diffraction etc. • Light wave is completely described by Maxwell's equations • But the wave nature of electromagnetic radiation failed to describe phenomenalike blackbody radiation, photoelectric effect and such • Einstein proposed his idea of photon (quantization of light in quanta, hu) • The associated momentum with a photon of frequency u: • In this way, Einstein describe the particle-like nature of light.

5. The beginning of Story • Electrons are known as particles with certain mass • Electrons diffraction Experiment shows that electron has a wave-like nature • de Broglie made a hypothesis that just as radiation has particle-like properties, electrons and other material particles possess wave-like properties • For free particles,

6. Old Quantum Mechanics Electron in hydrogen-like atom moved in circular orbit, The centripetal force equal to the attraction force between the electron and nucleus r + The “angular momentum” is quantized.

7. Old Quantum Mechanics , Bohr radius Energy = Kinetic + Potential r + The “angular momentum” is quantized.

8. Old Quantum Mechanics nU EU nL EL • The wave length of the emitted lights

9. Old Quantum Mechanics The Bohr Theory of the atom (“Old” Quantum Mechanics) works perfectly for H (as well as He+, Li2+, etc.). The only problem with the Bohr Theory is that it fails as soon as you try to use it on an atom as “complex” as helium.

10. Postulate of Quantum Mechanics

11. 1- Wave function • Max Born extended the matter waves proposed by De Broglie, by assigning • a mathematical function, Ψ(r,t), called the wavefunctionto every “material” • particle Ψ(r,t) is what is “waving” But how a wave represents a particle? Localization is the nature of particles (where is the particle: at point (2,1) ) (2,1) Spread is the nature of wave (where is the wave: every where) What is the wave length? (It is 0.5 meter)

12. 1- Wave function ……. Where is the jerk? (It is moving over there) What is wave length of the jerk? (it is not a wave) If you want to precisely define the wavelength, the lessthe position is defined If you want to precisely define the position, the less the wavelength is defined There is an intermediate case in which: the wave is fairly well localized and wavelength is fairly well defined

13. 2- Uncertainty Principle If the particle has a momentum p, the associated wavelength is Thus the spread in wavelength corresponds to a spread in momentum • An experiment cannot simultaneously determine a component of the momentum of a particle (px) and the exact value of the corresponding coordinate, (x). • The best one can do according to Heisenberg Uncertainty principle is:

14. 2- Uncertainty Principle …… Example: Bullet with p = mv = 0.1 kg × 1000 m/s = 100 kg·m/s If Δp = 0.01% p = 0.01kg·m/s • Which is much more smaller than size of the atoms the bullet made of! • So for practical purposes we can know the position of the bullet precisely

15. 2- Uncertainty Principle …… Example: Electron (m = 9.11×10-31 kg) with energy 4.9 eV Assume Δp = 0.01% p • Which is much larger than the size of the atom! • So uncertainty plays a key role on atomic scale

16. 3-Porobability Density The probability P(r,t)dV to find a particle associated with the wavefunctionΨ(r,t) within a small volume dVaround a point in space with coordinate r at some instant t is called “Probability Density” z dv r dV x y For one dimension: where

17. 3-Porobability Density ….. • The probability of finding a particle somewhere in a volume V of space is • Since the probability to find particle anywhere in space is 1, we have condition of normalization • For one-dimensional case, the probability of finding the particle in the arbitrary interval a ≤ x ≤ b is

18. 4-Operators • If we have such equation: where an operator acting on the function f gives the same function fmultiplying by a factor a. In this case we call f as the eigen function of the operator with eigen value a. For example, the eigen function where

19. Operator Linear ? • Linear operator satisfy the condition: 4-Operators ….. x2 Yes No • In quantum mechanics, log No • all observable quantities are operators sin No • All operators are linear • Observable quantities like position x, momentum, p Yes Yes

20. 5-Expectation Value • Expectation value of an observable is its mean value • If I measure the momentum p, what will I get is the expectation value of p, • A class room has 10 students1 get 10/102 get 8/102 get 7/104 get 6/101 get 5/10 فى هذا المثال: إحتمالية 10/10 هى 1 وإحتمالية 8/10 هى 2 وإحتمالية 7/10 هى 2 وإحتمالية 6/10 هى 4 وإحتمالية 5/10 هى 1 • What is the average grade of the whole class? • The average grade of the whole class

21. 5-Expectation Value …… • In the integration form: Since Then • The average (or expectation) value of an observable with the operator  is given by

22. Quantum Mechanics • The methods of Quantum Mechanics consist in finding the wavefunctionassociated with a particle or a system • Once we know this wavefunction we know “everything” about the system!