Lecture 2

1. Lecture 2

2. Light • Classical (Wave) Description Light is an EM wave: 100 nm< l <10 microns • Quantum (Particle) Description Localized, massless quanta of energy - photons • Wave / Particle Duality Appropriate description depends on experimental device examining light

3. IIA. Classical Description of Light Properties of EM waves • Electromagnetic radiation can be considered to behave as two wave motions at right angles to each other and to the direction of propagation • One of these waves is electric (E) and the other is magnetic (B) • These waves are functions of space and time http://www.phy.ntnu.edu.tw/~hwang/emWave/emWave.html

4. IIA. Classical Description of Light

5. Classical Description of Light Wave Equation (derived from Maxwell’s equations) Any function that satisfies this eqn is a wave It describes light propagation in free space and in time (see calculus review handout)

6. IIA. Classical Description of Light Plane Wave Solution One useful solution is for plane wave E B r

7. Plane wave • A plane wave in two or three dimensions is like a sine wave in one dimension except that crests and troughs aren't points, but form lines (2-D) or planes (3-D) perpendicular to the direction of wave propagation. The Figure shows a plane sine wave in two dimensions. The large arrow is a vector called the wave vector, which defines (1) the direction of wave propagation by its orientation perpendicular to the wave fronts, and (2) the wavenumber by its length.

8. IIA. Classical Description of Light Wave number and angular frequency

9. IIB. Quantum Description of Light Historical perspective • Max Planck (1858-1947) - Introduced concept of light energy or “quanta” (blackbody radiation) and the “Planck” constant • Albert Einstein (1879-1955) - Proof for particle behavior of light came from the experiment of the photoelectric effect

10. Light as photon particles • Energy of EM wave is quantized • Light consists of localized, massless quanta of energy -photons E=hn • h=Planck’s constant=6.63x10-34 Js • n=frequency • Photon has momentum,p, associated with it • p=h/l=hn/c

11. IIC. Wave / Particle Duality Photons versus EM waves • Light is a particle and has wave like behavior • The photon concept and the wave theory of light complement each other • Depends on the specific phenomenon being observed

12. IIC. Wave / Particle Duality Photons versus EM waves (continued)

13. IIC. Wave / Particle Duality • High frequency (X-rays) Momentum and energy of photon increase Photon description dominates • Low Frequency (radio waves) Interference/diffraction easily observable Wave description dominates

14. II. Light • Classical (Wave) Description Light is an EM wave: 100 nm< l <10 microns • Quantum (Particle) Description Localized, massless quanta of energy - photons • Wave / Particle Duality Appropriate description depends on experimental device examining light

15. IV. Light-Matter Interactions • Atomic spectrum of hydrogen B. Wave mechanics C. Atomic orbitals D. Molecular orbitals

16. IVA. Atomic Spectra • Atomic spectrum of hydrogen • When hydrogen receives a high energy spark, the hydrogen atoms are excited and contain excess energy • The hydrogen will release the energy by emitting light of various wavelengths • The line spectrum (intensity vs. wavelength) is characteristic of the particular element (hydrogen) H Spectrometer

17. IVA. Atomic Spectra 2. What is the significance of the line spectrum of H? • When white light (sunlight) is passed through a prism, the spectrum is continuous (all visible wavelengths) • In contrast, when hydrogen emission spectrum is passed though a prism, only a few lines are seen corresponding to discrete wavelengths. • This suggests that only certain wavelengths (energies) are allowed for the electron in the hydrogen atom. But why?

18. IVA. Atomic Spectra 3a. Bohr quantum model of the hydrogen atom • In 1913, Bohr provided the first successful explanation of atomic spectra of hydrogen • Bohr’s model was only successful in explaining the spectral behavior of simple atoms such as hydrogen • Bohr’s model was abandoned in 1925 because it had flawed assumptions and could not be applied to more complex atomic systems.

19. m (mass) -e + v (velocity) +e r (radius) IVA. Atomic Spectra 3b. Bohr postulate (1): Planetary model • Electron has circular orbit about nucleus • Particle in motion moves in a straight line and can be made to travel in a circular orbit by the application of a coulombic force of attraction (F) between electron (-e) and nucleus (+e) k = Coulomb’s const (9 x 109 N.m2/C2) F

20. IVA. Atomic Spectra 3b. Bohr postulate (2): angular momentum quantization • Angular momentum (L) for a particle in a circular path is: • Bohr assumed that the angular momentum (L) of the electron could occur only in certain increments (quantized) to fit the experimental results of hydrogen spectrum

21. IVA. Atomic Spectra 3b. Bohr postulates (3) and (4): • Stationary states: electron can move in one of its allowed orbits without radiating energy • Energy: Atoms radiate energy when electron jumps from one stationary state to another. The frequency of radiation obeys the condition: where, Ei = energy of initial state Ef = energy of final state f = frequency h = Planck’s constant -e + hf

22. IVA. Atomic Spectra 3c. Allowed energies • Using the assumptions in Bohr’s postulates (planetary model and quantization), an expression for the allowed energies was developed.

23. IVA. Atomic Spectra 3f. Orbital and Energy level diagram E= -0.54 E= -0.85 E= -1.51 E= -3.4 E= -13.6 eV n=5 n=4 n=3 n=2 n=1 Orbital n=3 n=1 Energy Level Diagram n=2 n=4 n=5

24. IVA. Atomic Spectra 3d. Spectral wavelengths • If electron jumps from one orbit (ni) to a second orbit (nf), the energy difference is: • The corresponding frequency and wavelengths are: http://www.colorado.edu/physics/2000/quantumzone/bohr.html

25. IVA. Atomic Spectra 3f. Abandonment of the Bohr Model • Hard to describe complex atoms and assumptions lack foundation • Heisenberg’s uncertainty principle showed that it was impossible to know the exact path of the electron as it moves around the nucleus as Bohr had predicted. • De Broglie’s and Schrodinger wave description of light overcame the limitations of the Bohr model

26. IVA. Atomic Spectra 4. Wave mechanics • By mid-1920’s it was apparent that Bohr’s model did not work • Louis De Broglie, and Erwin Schrodinger developed wave mechanics • Wave mechanics is the current theory used to describe the behavior of atomic systems

27. IV. Light-Matter Interactions • Atomic spectrum of hydrogen B. Wave mechanics C. Atomic orbitals D. Molecular orbitals

28. -e + hf Properties of atoms • Atoms consist of subatomic structures. For this course, we think of atoms consisting of a nucleus (positively charged) surrounded by electrons (negatively charged) • Internal energy of matter is of discrete values (it is quantized)---line spectra of elements such as H. • It is impossible to measure simultaneously with complete precision both the position and the velocity of an electron (or a particle). (Heisenberg uncertainty principle) • Think in terms of a probability of finding a particle within a given space at a given time and discrete energy levels associated with it---wave function.

29. Wave Mechanics The wave function,  • De Broglie waves can be represented by a simple quantity Y, called a wave function, which is a complex function of time and position • A particle is completely described in quantum mechanics by the wave function • A specific wave function for an electron is called an orbital • The wave function can be be used to determine the energy levels of an atomic system

33. Wave Mechanics Time-independent Schrodinger equation Since potential energy is zero inside box, the only possible energy is kinetic energy For a particle confined to moving along the x-axis: where, V=potential energy, E = total energy

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46. Atomic Orbitals • For an atom, use Schrödinger’s equation • Find permissible energy levels for electrons around nucleus. • For each energy level, the wave function defines an orbital, a region where the probability of finding an electron is high • The orbital properties of greatest interest are size, shape (described by wave function) and energy. • Solution for multi-electron atoms is a very difficult problem, and approximations are typically used

47. Atomic Orbitals The hydrogen atom The electron of the hydrogen atom moves in three dimensions and has potential energy (attraction to positively charged nucleus) The Schrodinger equation can be solved to find the wave functions associated with the hydrogen atom In 1-D particle in a box, the wave function is a function of one quantum number; the 3-D hydrogen atom is a function of three quantum numbers

48. Atomic Orbitals Wave functions of hydrogen The solution of the Schrodinger equation for the hydrogen atom is: Rnl describes how wave function varies with distance of electron from nucleus Ylm describes the angular dependence of the wave function Subscripts n, l and m are quantum numbers of hydrogen

49. Atomic Orbitals Principal quantum number, n Has integral values of 1,2,3…… and is related to size and energy of the orbital As n increases, the orbital becomes larger and the electron is farther from the nucleus As n increases, the orbital has higher energy (less negative) and is less tightly bound to the nucleus

50. Atomic Orbitals Angular quantum number, l Can have values of 0 to n-1 for each value of n and relates to the angular momentum of the electron in an orbital The dependence of the wave function on l, determines the shape of the orbitals The value of l, for a particular orbital is commonly assigned a letter: 0 – s 1 – p 2 – d 3 – f d orbital p orbital s orbital