Lecture 17: Intro. to Quantum Mechanics

1. Lecture 17: Intro. to Quantum Mechanics • Reading: Zumdahl 12.5, 12.6 • Outline • Basic concepts. • A model system: particle in a box. • Other confining potentials.

2. 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.

3. Quantum Concepts • This new description was known as wave mechanics or quantum mechanics. • Recall that photons and electrons readily demonstrate wave-particle duality. • The idea behind wave mechanics was that the existence of the electron in a fixed energy level could be visualized as a “standing wave”.

4. Quantum Concepts Q: 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.

5. Louis de Broglie suggests that for the e- orbits envisioned by Bohr, only those orbits are allowed which satisfy the standing wave condition. not allowed

6. Erwin Schrodinger develops a mathematical formalism that incorporates the wave nature of matter: Kinetic Energy The Hamiltonian: x The Wavefunction: E = energy

7. = a probability amplitude Q: What is a wavefunction? • Consider a wave: Intensity = • Probability of finding a particle in space: Probability = • With the wavefunction, we can describe spatial distributions of probability

8. Another limitation of the Bohr model was that it assumed we could simultaneously know both the position and momentum of an electron exactly. • Werner Heisenberg’s 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 (Heisenberg Uncertainty Principle) Uncertainty in position Uncertainty in momentum

9. Example of Heisenberg Uncertainty: What is the uncertainty in velocity for an electron in a 1Å radius orbital in which the positional uncertainty is 1% of the radius. Dx = (1 Å)(0.01) = 1 x 10-12 m Huge! (Recall the speed of light is c = 3 x108 m/s)

10. Another Example (you’re a quantum wave 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%. Dp = m Dv = (80kg)(0.013 m/s) = 1.04 kg.m/s Very small uncertainty……so, we know where you are!

11. Potentials and Quantization • Consider a particle free to move in 1 dimension: Case: ‘Free’ Particle Potential E = 0 The Schrodinger Equation becomes: 0 • Energy ranges from 0 to infinity….not quantized, particle can have any arbitrary velocity

12. Case: “Particle in a Box” Q: What if the position of the particle is constrained by a potential? Potential E E = 0 for 0 ≤ x ≤ L E =  for all other x The possible position of particle is limited to the dimensions of the box, 0 to L

13. QM Solution for the particle in a box What do the wavefunctions look like? + + + n = 1, 2, …. - Like a standing wave + + y*y y

14. What does the energy look like? n = 1, 2, … Energy is quantized E y y*y

15. A real-world example: Consider the following dye molecule, the length of which can be considered the length of the “box” an electron is limited to: L = 8 Å What wavelength of light corresponds to DE from n=1 to n=2? (Observed value 680 nm, not bad!)

16. If the length L of the “box” in the particle in a box potential is reduced, what do you expect to happen to the energy levels?

17. • One effect of a “constraining potential” is that the energy of the system becomes quantized. • Back to the hydrogen atom:

18. So, for the hydrogen atom, energy becomes quantized due to the presence of a constraining potential. Solve the Wave Equation Exact q.m. solution is the Bohr formula!

19. What is the expected distribution of energy levels for the harmonic oscillator?