Chapter 39 More About Matter Waves What Is Physics?

1. Chapter 39 More About Matter Waves What Is Physics? One of the long-standing goals of physics has been to understand the nature of the atom. The development of quantum mechanics provided a framework for understanding this and many other mysteries. The basic premise of quantum mechanics is that moving particles (electrons, protons, etc.) are best viewed as matter waves whose motions are governed by Schrödinger’s equation. Although this premise is also correct for massive objects (baseballs, cars, planets, etc.) where classical Newtonian mechanics still predicts behavior correctly, it is more convenient to use classical mechanics in that regime. However, when particle masses are small, quantum mechanics provides the only framework for describing their motion. Before applying quantum mechanics to the atomic structure, we will first explore some simpler situations. Some of these oversimplified examples, which previously were only seen in introductory textbooks, are now realized in real devices developed by the rapidly growing field of nanotechnology. (39-1)

2. 39-2 String Waves and Matter Waves Confinement of wave leads to quantization: existence of discrete states with discrete energies. Wave can only have those energies. • In Ch. 16 we saw that two kinds of waves can be set up on a stretched string: traveling waves and standing waves. • infinitely long string → traveling waves → frequency or wavelength can have any value. • finite length string (e.g., clamped both ends) → standing waves → only discrete frequencies or wavelengths • → confining a wave in a finite region leads to the quantization of its motion with discrete states, each defined by a quantized frequency. • This observation also applies to matter waves. • • electron moving +x-direction and subject to no force (free particle) → wavelength (l=h/p), frequency (f=v/l), and energy (E=p2/2m) can have any reasonable value. • • atomic electron (e.g., valence): Coulomb attraction to nucleus →spatial confinement → electron can exist only in discrete states, each with a discrete energy. (39-2)

3. 39-3 Energies of a Trapped Electron Fig. 39-1 Fig. 16-23 Fig. 39-2 One-Dimensional Trap: n is a quantum number, identifying each state (mode). (39-3)

4. Finding the Quantized Energies Fig. 39-2 Fig. 39-3 Infinitely deep potential well (infinte potential well) (39-4)

5. Energy Changes Confined electron can absorb photon only if photon energy hf=DE, the energy difference between initial energy level and a higher final energy level. Fig. 39-4 Confined electron can emit photon of energy hf=DE, the energy difference between initial energy level and a lower final energy level. (39-5)

6. 39-4 Wave Functions of a Trapped Electron Fig. 39-6 Probability of detection: (39-6)

7. Wave Functions of a Trapped Electron, cont’d At large enough quantum numbers (n), the predictions of quantum mechanics merge smoothly with those of classical physics. To find the probability that an electron can be detected in any finite section of the well, e.g., between point x1 and x2, we must integrate p(x) between those points. Correspondence Principle: Normalization: The probability of finding the electron somewhere (if we search the entire x-axis) is 1! (39-7)

8. Wave Functions of a Trapped Electron, cont’d Fig. 39-3 Zero-Point Energy: In a quantum well, the lowest quantum number is 1 (n= 0 means there is no electron in well), so the lowest energy (ground state) is E1, which is also nonzero. →confined particles must always have at least a certain minimum nonzero energy! →since the potential energy inside the well is zero, the zero-point energy must come from the kinetic energy. →a confined particle is never at rest! Zero-Point Energy (39-8)

9. 39-5 An Electron in a Finite Well Fig. 39-7 Fig. 39-9 Fig. 39-8 barrier barrier well Leakage into barriers →longer wavelengths →lower energies than infinite well (39-9)

10. 39-6 More Electron Traps Fig. 39-11 Nanocrystallites: Small (L~1nm) granule of a crystal trapping electron(s) Only photons with energy above minimum threshold energy Et (wavelength below a maximum threshold wavelength lt) can be absorbed by an electron in nanocrystallite. Since Eta 1/L2, the threshold energy can be increased by decreasing the size of the nanocrystallite. Quantum Dots: Electrons sandwiched in semiconductor layer →artificial atom with controllable number of electrons trapped →new electronics, new computing capabilities, new data storage capacity… Quantum Corral: Electrons “fenced in” by a corral of surrounding atoms. (39-10)

11. 39-7 Two- and Three-Dimensional Electron Traps Fig. 39-13 Rectangular Corral: Infinite potential wells in the x and y directions If Lx= Ly Unlike a 1D well, in 2D certain energies may not be uniquely associated with a single state (nx, ny) since different combinations of nx,and nycan produce the same energy. Different states with the same energy are called degenerate. Fig. 39-15 (39-11)

12. Fig. 39-14 Two- and Three-Dimensional Electron Traps Rectangular Box: Infinite potential wells in the x, y, and z directions As in 2D, certain energies may not be uniquely associated with a single state (nx, ny, nz) since different combinations of nx, ny, and nzcan produce the same (degenerate) energy. (39-12)

13. 39-8 The Bohr Model of the Hydrogen Atom Fig. 39-16 Hydrogen (H) is the simplest “natural” atom and contains +e charge at center surrounded by –e charge (electron). Why doesn’t the electrical attraction between the two charges cause them to collapse together? Balmer’s empirical (based only on observation) formula on absorption/emission of visible light for H: • Bohr’s assumptions to explain Balmer formula: • Electron orbits nucleus • The magnitude of the electron’s angular momentum L is quantized (39-13)

14. Orbital Radius Is Quantized in the Bohr Model Coulomb force attracting electron toward nucleus Quantize angular momentum l: Substitute v into force equation: where the smallest possible orbital radius (n = 1) is called the Bohr radius a: Orbital radius r is quantized and r = 0 is not allowed (H cannot collapse). (39-14)

15. Orbital Energy Is Quantized The total mechanical energy of the electron in H is: Solving the F = ma equation for mv2 and substituting into the energy equation above: Substituting the quantized form for r: The energy of the electron (or the entire atom if nucleus at rest) in a hydrogen atom is quantized with allowed values En. (39-15)

16. Energy Changes The energy of a hydrogen atom (equivalently its electron) changes when the atom emits or absorbs light: Substituting f = c/l and using the energies En allowed for H: where the Rydberg constant This is precisely the formula Balmer used to model experimental emission and absorption measurements in hydrogen! However, the premise that the electron orbits the nucleus is incorrect! Must treat electron as matter wave. (39-16)

17. 39-9 Schrödinger’s Equation and the Hydrogen Atom Fig. 39-17 Fig. 39-18 The potential well that traps an electron in a hydrogen atom is: Energy Levels and Spectra of the Hydrogen Atom: Can plug U(r) into Schrödinger’s equation to solve for En. (39-17)

18. Quantum Numbers for the Hydrogen Atom Table 39-2 Quantum Numbers for the Hydrogen Atom Symbol Name Allowed Values n Principal quantum number 1, 2, 3, … l Orbital quantum number 0, 1, 2, …, n-1 mlOrbital magnetic quantum number -l, -(l-1), …+(l-1), +l Principal quantum number n → energy of state Orbital quantum number l→ angular momentum of state Orbital magnetic quantum number m→ orientation of angular momentum of state For ground state, since n = 1→ l= 0 and ml= 0 (39-18)

19. Wave Function of the Hydrogen Atom’s Ground State Solving the three-dimensional Schrödinger equation and normalizing the result: (39-19)

20. Wave Function of the Hydrogen Atom’s Ground State, cont’d Radial probability density P(r): The probability of finding the electron somewhere (if we search all space) is 1! (39-20)

21. Wave Function of the Hydrogen Atom’s Ground State, cont’d Fig. 39-20 Fig. 39-21 Probability of finding electron within a small distance from a given radius Probability of finding electron within a small volume at a given position (39-21)

22. Hydrogen Atom States with n = 2 Table 39-3 Quantum Numbers for Hydrogen Atom States with n = 2 nlml 2 0 0 2 1 +1 2 1 0 2 1 -1 Solving the three-dimensional Schrödinger equation. (39-22)

23. Hydrogen Atom States with n = 2, cont’d Fig. 39-22 Fig. 39-24 Fig. 39-23 Direction of z-axis completely arbitrary (39-23)

24. Hydrogen Atom States with n >> 1 Fig. 39-25 As the principal quantum number increases, electronic states appear more like classical orbits. (39-24)