Modern Physics Part III : Quantum Mechanics Ch.# 41

1. Modern Physics Part III : Quantum Mechanics Ch.# 41 مدرس المادة الدكتور :………………………… 1431 – 1430 ادة 1

2. Lecture # 1 Ch. 41 1. An interpretation of Quantum Mechanics Quantum mechanics is a theory.Itis our current “standard model” for describing the behavior of matter and energy at the smallest scales (photons, atoms, nuclei, quarks, gluons, leptons, …). Probability: a quantity that connects wave and particles. Electromagnetic waves: 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 V at that time which its proportional to the intensity of the radiation I or the Square of the electric field amplitude E. 2

3. A particle bouncing back and forth between the walls of a box of length L. Within the box the particle is free ( no forces) : all its energy is KE 2. A particle in a Box 2.1 The quantum particle under boundary conditions Problem: A particle is confined to a one-dimensional region of space with length L (“box”). It is bouncing elastically back and forth. Potential energy: Assume that the collisions between the particle and the wall are elastic (KE conserved) Classical Physics Result: The particle can have any energy between 0 and ∞ Wave function: 1) The walls are impenetrable: ψ(x) = 0 for x < 0 and x > L. 2) The wave function is continuous: ψ(0) = 0 and ψ(L) = 0. These are boundary conditions. 3) The wave function: n is the quantum number 3

4. Graphicalrepresentation: • Only certain wavelengths for the particle are allowed: • |ψ|2 is zero at the boundaries as well as some other locations depending on n. • The number of zero points increases when the quantum number increases. 4

5. Momentum and energy: The momentum is restricted to The energy is just the kinetic energy: The energy of the particle is quantized. Energy level diagram: • Ground state: The sate having the lowest allowed energy. • Excited states:En = n2E1. • E = 0 is not an allowed state since ψ(x) = 0. The particle can never be at rest. Zero energy also means an infinite wavelength. Note that the energy levels increase as n2, and that their separation increases as the quantum number increases. 5

6. Example 41.2 An electron is confined between two impenetrable walls 0.200 nm apart. Determine the energy levels for the state n=1. n=1 h=6.63x10-34 J.s m=9.11x10-31 kg l=0.2x10-9 m E=1.51x10-18 J 6

7. 3. An electron is confined to a one-dimensional region in which its ground-state (n = 1) energy is 2.00 eV. (a) What is the length of the region? (b) How much energy is required to promote the electron to its first excited state? (a) , For the ground-state, n=1 (b) For the first excited state, n=2 7

8. 8. A laser emits light of wavelength λ. Assume this light is due to a transition of an electron in a box from its n = 2 state to its n = 1 state. Find the length of the box. so 8

9. 12. A photon with wavelength λ is absorbed by an electron confined to a box. As a result, the electron moves from state n = 1 to n = 4. (a) Find the length of the box. (b) What is the wavelength of the photon emitted in the transition of that electron from the state n = 4 to the state n = 2? (a): The energies of the confined electron are Its energy gain in the quantum jump from state 1 to state 4 is and this is the photon energy: Then and . (b): Let λ’ represent the wavelength of the photon emitted: Then and 9

10. Homework Ch. 41, Problems: # 7, 11. 7. A ruby laser emits 694.3-nm light. Assume light of this wavelength is due to a transition of an electron in a box from its n = 2 state to its n = 1 state. Find the length of the box. 11. Use the particle-in-a-box model to calculate the first three energy levels of a neutron trapped in a nucleus of diameter 20.0 fm. 10

11. 3. Schrödinger Equation Building on de Broglie’s work, in 1926, Erwin Schrödinger devised a theory that could be used to explain the wave properties of electrons in atoms and molecules. 11

12. 3.1. The wavefunction Each particle is represented by a wavefunction Ψ(position, time) such that Ψ Ψ* = the probability of finding the particle at that position at that time. • Wavefunction Ψs are normally complex (real and imaginary parts) • Probability density | ψ |2 always positive. • We write • | ψ |2 = Ψ Ψ* • Where • Ψ* is the complex conjugate of Ψ • If Ψ = A+ iB then Ψ* = A-iB • and | ψ |2 = A2 + B2 12

13. 3.2. Schrödinger Equation • In 1926 Schrodinger wrote an equation that described both the particle and wave nature of the e-. The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. • Wave function (Ψ) describes : • . energy of e- with a given Ψ • . probability of finding e- in a volume of space • Schrodinger’s equation can only be solved exactly for the hydrogen atom. Must approximate its solution for multi-electron systems. The equation is of the form : • H Ψ= E Ψ • Ψ: wavefunction • E: Energy for the system • H: Hamiltonian operator 13

14. Philosophic Implications; Probability versus Determinism The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go. Quantum mechanics is very different – you can predict what masses of electrons will do, but have no idea what any individual one will. 14