Lecture 17: Introduction to Quantum Mechanics - Particle Wave Duality and Quantization, Study notes of Chemistry

Download Lecture 17: Introduction to Quantum Mechanics - Particle Wave Duality and Quantization and more Study notes Chemistry in PDF only on Docsity! Lecture 17: Intro. to Quantum Mechanics • Reading: Zumdahl 12.5, 12.6 • Outline – Particle – Wave Duality – Basic concepts. – A model system: particle in a box. – Other confining potentials. de Broglie wavelength • Drawing upon the results of Planck’s work and Einstein’s theory of relativity, Louis de Broglie proposed that a moving particle should have a wavelength associated with it. h mv λ = Where m and v are the mass and velocity of the particle. Quantum Concepts • The Bohr model was capable of describing the discrete or “quantized” emission spectrum of H. • But the failure of the model for multi-electron systems combined with other issues (the ultraviolet catastrophe, work functions of metals, etc.) suggested that a new description of atomic matter was needed. Quantum Concepts (cont.) • This new description was known as wave mechanics or quantum mechanics. • Two important experimental observations which QM had to explain were – photons and electrons show wave-particle duality. – electrons in atoms exist in states with discrete energies • The idea behind wave mechanics was that the existence of the electron in fixed energy levels could be though of as a “standing wave”. Quantum Concepts (cont.) • What is a standing wave? • A standing wave is a motion in which translation of the wave does not occur. • In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs. • Note also that integer and half- integer values of the wavelength correspond to standing waves. What is a wave function? • – Is a function of position – Is a probability amplitude • The probability of finding the particle at a given point in space is given by Probability = • The wave function describes the spatial distribution of particles. ( , , )x y xψ 2 *ψ ψ ψ= Uncertainty Principle • Another limitation of the Bohr model was that it assumed we could know both the position and momentum of an electron exactly. • Werner Heisenberg development of quantum mechanics leads him to the observation that there is a fundamental limit to how well one can know both the position and momentum of a particle. ∆x •∆p ≥ h 4π Uncertainty in position Uncertainty in momentum Uncertainty Principle • Example: What is the uncertainty in velocity for an electron in a 1Å radius orbital in which the positional uncertainty is 1% of the radius. ∆x = (1 Å)(0.01) = 1 x 10-12 m ∆p = h 4π∆x = 6.626x10−34 J.s( ) 4π 1x10−12 m( ) = 5.27x10−23 kg.m /s ∆v = ∆p m = 5.27x10−23 kg.m /s 9.11x10−31kg = 5.7x107 m s huge Potentials and Quantization • Consider a particle free to move in 1 dimension: x p “The Free Particle” Potential E = 0 •Since this corresponds to an unbound system, the wavefunction for the particle can have any wavelength. • Energy ranges from 0 to infinity….not quantized. • Free or unbound particles are not quantized. Potentials and Quantization (cont.) • What if the position of the particle is constrained by a potential: “Particle in a Box” x 0 inf. 0 L Potential E = 0 for 0 ≤ x ≤ L = ∞ all other x • Now, position of particle is limited to the dimension of the box. Potentials and Quantization (cont.) • What do the wavefunctions look like? n = 1, 2, …. Like a standing wave n is called quantum number ψ ψ∗ψ ( ) 2 sinn n xx L L πψ ⎛ ⎞= ⎜ ⎟ ⎝ ⎠ If the length of the “box” in the particle in a box potential is reduced, what do you expect to happen to the energy levels? V E V E 1 2 3 4 1 2 3 Hydrogen Atom • The electrostatic interaction between the proton and electron forms a “constraining potential” that results in the energy of the H atom becoming quantized. e- P+r V (r) = −e 2 r r0 constraining potential Hydrogen Atom • Different potential results in different spacing, but still quantized. • Note unlike particle in box, energy levels get closer together as quantum number n increases. r0 Schrodinger EquationV (r) = −e 2 r 0 ( ) 2 18 22.178 10 Jn ZE x n − ⎛ ⎞= − ⎜ ⎟⎜ ⎟ ⎝ ⎠Recovers the “Bohr” behavior