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)

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 ( =h/p ), frequency ( f=v/ ), and energy ( E=p 2 /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 String Waves and Matter Waves Confinement of wave leads to quantization: existence of discrete states with discrete energies. Wave can only have those energies. (39-2)

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

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

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

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

To find the probability that an electron can be detected in any finite section of the well, e.g., between point x 1 and x 2, we must integrate p(x) between those points. 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. Correspondence Principle: Normalization: The probability of finding the electron somewhere (if we search the entire x-axis) is 1! (39-7)

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 E 1, 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! Wave Functions of a Trapped Electron, cont’d Fig Zero-Point Energy (39-8)

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

Nanocrystallites: Small ( L ~1nm) granule of a crystal trapping electron(s) 39-6 More Electron Traps Fig Only photons with energy above minimum threshold energy E t (wavelength below a maximum threshold wavelength t ) can be absorbed by an electron in nanocrystallite. Since E t  1/L 2, 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)

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

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

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? 39-8 The Bohr Model of the Hydrogen Atom Fig Balmer’s empirical (based only on observation) formula on absorption/emission of visible light for H: Bohr’s assumptions to explain Balmer formula: 1.Electron orbits nucleus 2.The magnitude of the electron’s angular momentum L is quantized (39-13)

Coulomb force attracting electron toward nucleus Orbital Radius Is Quantized in the Bohr Model 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)

The energy of the electron (or the entire atom if nucleus at rest) in a hydrogen atom is quantized with allowed values E n. The total mechanical energy of the electron in H is: Orbital Energy Is Quantized Solving the F = ma equation for mv 2 and substituting into the energy equation above: Substituting the quantized form for r : (39-15)

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. The energy of a hydrogen atom (equivalently its electron) changes when the atom emits or absorbs light: Energy Changes Substituting f = c/ and using the energies E n allowed for H: where the Rydberg constant (39-16)

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

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 Quantum Numbers for the Hydrogen Atom Symbol NameAllowed Values n Principal quantum number 1, 2, 3, … l Orbital quantum number 0, 1, 2, …, n-1 m l Orbital magnetic quantum number - l, -( l -1 ), …+( l -1), + l Table 39-2 For ground state, since n = 1 → l = 0 and m l = 0 (39-18)

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

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

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

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

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

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