Quantum Mechanics: Wave Nature of Matter and the Schrödinger Equation

Course Objective: The primary goal is to provide a comfortable, intuitive, and predictive understanding of how electrons form chemical bonds and determine molecular structure, moving beyond the simple Lewis dot model.
Learning Focus: Emphasize understanding why certain concepts are important rather than merely memorizing formulas or performing rote calculations. Distinguish between what is essential to know and the background justification for it.

Core Concept: The current focus is on understanding and applying the wave nature of matter, particularly how it explains chemical bonding.
Revisiting Wave Nature of Light: Building on Monday's lecture, the concept of interference, even for light, is an obvious phenomenon from everyday experience.
Particles as Waves: The lecture aims to convince students that particles also exhibit wave nature, leading to particle interference phenomena (e.g., electron diffraction).
Key Question: What vocabulary and expressions are needed to confidently discuss the wave nature of particles and its role in chemical bonding?

De Broglie Hypothesis: Postulated that particles, like photons, have wave properties, and their wavelength is inversely proportional to their momentum.
De Broglie Relation: Expected to master this relationship: $\lambda = h/p$ where $\lambda$ is the wavelength, $h$ is Planck's constant, and $p$ is the particle's momentum.
Historical Context: The speaker recalls solving De Broglie wavelength problems for objects like bowling balls without understanding their significance, highlighting the lecture's intent to provide context and purpose.
Significance: Understanding the De Broglie wavelength gives meaning to particle properties, especially for objects like electrons that play a role in atomic interactions.
Momentum of Photons vs. Particles:
- For light (photons), energy ( $E$ ) is related to frequency ( $\nu$ ) by $E = h\nu$ . Given $\nu = c/\lambda$ (where $c$ is the speed of light), then $E = hc/\lambda$ .
- Photons also have momentum ( $p$ ), where $E = pc$ . Combining these: $hc/\lambda = pc$ , which simplifies to $p = h/\lambda$ .
- This photon momentum relation ( $p = h/\lambda$ ) was postulated by De Broglie to apply to matter as well. The key difference for matter waves is that their energy-momentum relationship is not simply $pc$ .

Wavelength ( $\lambda$ ): The physical length of one complete cycle of a wave.
Wave Number ( $k$ ): A measure of the number of wave cycles per unit distance. It is defined as $\text{k} = 2\pi / \lambda$ . For an electron example, a wavelength of $2.4$ Å translates to a wave number of $2.6 \times 10^{10}$ per meter, meaning $2.6 \times 10^{10}$ cycles of the wave occur per meter.
- Purpose of $k$ : $k$ is more useful than $\lambda$ for describing the phase of a wave, which is critical for understanding interference. It describes the position within a cycle at any given distance ( $x$ ).
- Analogy: A wave traveling through space can be thought of as going around a circle; $k$ helps track the position on this circle, crucial for determining how waves interfere.
Revised De Broglie Relation: Substituting $\lambda = 2\pi/k$ into $p = h/\lambda$ gives $p = hk/(2\pi)$ .
Reduced Planck's Constant ( $\hbar$ ): To simplify expressions involving $2\pi$ , a new constant, $\hbar$ (h-bar), is introduced: $\hbar = h/(2\pi)$ .
Momentum in terms of $k$ : Using $\hbar$ , the De Broglie relation can be expressed compactly as $p = \hbar k$ .
- Units Check: The units of $\hbar$ are $kg \cdot m^2/s \cdot (1/m) = kg \cdot m/s$ , which are indeed the units of momentum (mass times velocity).

Describing a Wave: A simple sine or cosine function, e.g., $\psi(x) = A \cos(kx)$ , describes a wave but is incomplete for quantum mechanics.
Limitations of Simple Wave Function: A simple cosine function represents a