Download Quantum Mechanics: Understanding Wave Functions and Quantized Energy Levels - Prof. Daniel and more Study notes Chemistry in PDF only on Docsity! 1 Lecture 14: Intro. to Quantum Mechanics • Reading: Zumdahl 12.5, 12.6 • Outline – Basic concepts. – A model system: particle in a box. – Other confining potentials. Quantum Concepts • The Bohr model was capable of describing the discrete or “quantized” emission spectrum of H. • But the failure of the model for multielectron systems combined with other issues (the ultraviolet catastrophe, workfunctions 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. • Recall, photons and electrons readily demonstrate wave-particle duality. • 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. Quantum Concepts (cont.) • Louis de Broglie suggests that for the e- orbits envisioned by Bohr, only certain orbits are allowed since they satisfy the standing wave condition. not allowed Quantum Concepts (cont.) • Erwin Schrodinger develops a mathematical formalism that incorporates the wave nature of matter: ˆ H ψ = Eψ ˆ H The Hamiltonian: = ˆ p 2 2m + (PE) Kinetic Energy ψThe Wavefunction: x E = energy 2 Quantum Concepts (cont.) • What is a wavefunction? ψ = a probability amplitude • Consider a wave: y = Aei 2πν( )t +ϕ( ) y 2 = Aei 2πν( )t +ϕ( )( )Ae−i 2πν( )t +ϕ( )( )= A2Intensity = • Probability of finding a particle in space: ψ*ψProbability = • With the wavefunction, we can describe spatial distributions. Quantum Concepts (cont.) • 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 Quantum Concepts (cont.) • 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 Quantum Concepts (cont.) • Example (you’re quantum as well): What is the uncertainty in position for a 80 kg student walking across campus at 1.3 m/s with an uncertainty in velocity of 1%. ∆p = m ∆v = (80kg)(0.013 m/s) = 1.04 kg.m/s ∆x = h 4π∆p = 6.626x10−34 J.s( ) 4π 1.04kg.m /s( ) = 5.07x10−35 m Very small……we know where you are. Potentials and Quantization • Consider a particle free to move in 1 dimension: x p “The Free Particle” Potential E = 0 • The Schrodinger Eq. becomes: ˆ H ψ = ˆ p 2 2m + PE ψ(p) = ˆ p 2 2m ψ(p) = p 2 2m ψ(p) = 1 2 mv 2ψ(p) = Eψ(p) • Energy ranges from 0 to infinity….not quantized. 0 Potentials and Quantization (cont.) • What if the position of the particle is constrained by a potential: “Particle in a Box” Potential E x 0 inf. 0 L = 0 for 0 ≤ x ≤ L = ∞ all other x • Now, position of particle is limited to the dimension of the box.
