CHAPTER 28 Quantum Mechanics of Atoms Units

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CHAPTER 28 Quantum Mechanics of Atoms Units Quantum Mechanics A New Theory The Wave Function and Its Interpretation; the Double-Slit Experiment The Heisenberg Uncertainty Principle Philosophic

Elements X-Ray Spectra and Atomic Number Quantum Mechanics A New Theory Quantum mechanics incorporates wave-particle duality, and successfully explains energy states in complex atoms and molecules,

1 CHAPTER 28 Quantum Mechanics of Atoms Units Quantum Mechanics A New Theory The Wave Function and Its Interpretation; the Double-Slit Experiment The Heisenberg Uncertainty Principle Philosophic Implications; Probability versus Determinism Quantum-Mechanical View of Atoms Quantum Mechanics of the Hydrogen Atom; Quantum Numbers Complex Atoms; the Exclusion Principle The Periodic Table of Elements X-Ray Spectra and Atomic Number Quantum Mechanics A New Theory Quantum mechanics incorporates wave-particle duality, and successfully explains energy states in complex atoms and molecules, the relative brightness of spectral lines, and many other phenomena. It is widely accepted as being the fundamental theory underlying all physical processes. Quantum mechanics is essential to understanding atoms and molecules, but can also have effects on larger scales. The Wave Function and Its Interpretation; the Double-Slit Experiment An electromagnetic wave has oscillating electric and magnetic fields. What is oscillating in a matter wave? This role is played by the wave function, Ψ. The square of the wave function at any point is proportional to the number of electrons expected to be found there. For a single electron, the wave function is the probability of finding the electron at that point. For example: the interference pattern is observed after many electrons have gone through the slits. If we send the electrons through one at a time, we cannot predict the path any single electron will take, but we can predict the overall distribution. 1

Werner Heisenberg 1901 1976 Developed an abstract mathematical model to explain wavelengths of spectral lines Called matrix mechanics Other contributions Uncertainty Principle Nobel Prize in 1932

measurement not because of the limits of our instruments, but inherently. This is due to the wave-particle duality, and to interaction between the observing equipment and the object being observed.

2 Werner Heisenberg Developed an abstract mathematical model to explain wavelengths of spectral lines Called matrix mechanics Other contributions Uncertainty Principle Nobel Prize in 1932 Atomic and nuclear models Forms of molecular hydrogen The Heisenberg Uncertainty Principle Combining, we find the combination of uncertainties: Quantum mechanics tells us there are limits to measurement not because of the limits of our instruments, but inherently. This is due to the wave-particle duality, and to interaction between the observing equipment and the object being observed. Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron. The uncertainty in the momentum of the electron is taken to be the momentum of the photon it could transfer anywhere from none to all of its momentum. In addition, the position can only be measured to about one wavelength of the photon. This is called the Heisenberg uncertainty principle. It tells us that the position and momentum cannot simultaneously be measured with precision. This relation can also be written as a relation between the uncertainty in time and the uncertainty in energy: This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the time is short enough. 2

3 6 Example 1: An electron moves in a straight line with a constant speed v 1.10x10 m/ s which has been measured to a precision of 0.10%. What is the maximum precision with which its position could be simultaneously measured? p mv (9.11x10 kg)(1.10 x10 m/ s) 1.00x10 kg m/ s 27 The uncertainty is 10% - or p 1.0x10 kg m/ s 34 h 1.06x10 J s 7 x 1.1x10 m 27 p 1.0x10 kg m / s Example 2: What is the uncertainty in position, imposed by the uncertainty principle, on a 150-g baseball thrown at (42 1) m/ s? The uncertainty in speed is v 1 m / s p m v (0.150 kg)(1 m / s) 0.15 kg m / s 34 h 1.06x10 J s 34 x 7x10 m p.15 kg m / s 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. Quantum-Mechanical View of Atoms Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct. Quantum theory describes an electron probability distribution; this figure shows the distribution for the ground state of hydrogen: Quantum Mechanics of the Hydrogen Atom; Quantum Numbers There are four different quantum numbers needed to specify the state of an electron in an atom. 1. Principal quantum number n gives the total energy: 3

2. Orbital quantum number l gives the angular momentum; l can take on integer values from 0 to n - 1. 3.

This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined. 4.

4 2. Orbital quantum number l gives the angular momentum; l can take on integer values from 0 to n The magnetic quantum number, m l, gives the direction of the electron s angular momentum, and can take on integer values from l to +l. This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined. 4. The spin quantum number, m s, this for an electron can take on the values +½ and -½. The need for this quantum number was found by experiment; spin is an intrinsically quantum mechanical quantity, although it mathematically behaves as a form of angular momentum. This table summarizes the four quantum numbers. The angular momentum quantum numbers do not affect the energy level much, but they do change the spatial distribution of the electron cloud. 4

Allowed transitions between energy levels occur between states whose value of l differ by one: Other, forbidden, transitions also occur but with much lower probability.

5 Allowed transitions between energy levels occur between states whose value of l differ by one: Other, forbidden, transitions also occur but with much lower probability. Complex Atoms; the Exclusion Principle Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels. This means that the energy depends on both n and l. A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number. Wolfgang Pauli Contributions include Major review of relativity Exclusion Principle Connect between electron spin and statistics Theories of relativistic quantum electrodynamics Neutrino hypothesis Nuclear spin hypothesis Complex Atoms; the Exclusion Principle In order to understand the electron distributions in atoms, another principle is needed. This is the Pauli Exclusion Principle: No two electrons in an atom can occupy the same quantum state. The quantum state is specified by the four quantum numbers; no two electrons can have the same set. This chart shows the occupied and some unoccupied states in He, Li, and Na. 5

6 Filling Shells As a general rule, the order that electrons fill an atom s subshell is: Once one subshell is filled, the next electron goes into the vacant subshell that is lowest in energy Otherwise, the electron would radiate energy until it reached the subshell with the lowest energy A subshell is filled when it holds 2(2l+1) electrons The Periodic Table The outermost electrons are primarily responsible for the chemical properties of the atom Mendeleev arranged the elements according to their atomic masses and chemical similarities The electronic configuration of the elements explained by quantum numbers and Pauli s Exclusion Principle explains the configuration Example 3: High-energy photons are used to bombard an unknown material. The strongest peak is found for X-rays emitted with energy of 66keV. Guess what the material is. The hydrogen transition n = 2 to n = 1 would yield about 10.2eV. Energy E is proportional to 2 2 Z or ( Z 1) because the nucleus is shielded by the one electron in a 1s state. So: ( 1) Z x ev eV 6.5x10 Z , and Z 82( lead ) 3 Example 4: What is the shortest wavelength X-ray photon emitted in an X-ray tube subjected to 50kV? 34 8 hc (6.63x10 J s)(3.0x10 m / s) 11 o 2.5x10 m 19 4 ev (1.6x10 C)(5.0x10 V ) So: 0.025nm 6

Chapter 38 Quantum Mechanics

Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle