# 2: Foundations of Quantum Mechanics

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- 4469

The concepts of quantum mechanics were invented to explain experimental observations that otherwise were totally inexplicable. This period of invention extended from 1900 when Max Planck introduced the revolutionary concept of quantization to 1925 when Erwin Schrödinger and Werner Heisenberg independently introduced two mathematically different but equivalent formulations of a general quantum mechanical theory. The Heisenberg method uses properties of matrices, while the Schrödinger method involves partial differential equations. We will develop and utilize Schrödinger’s approach because students usually are more familiar with elementary calculus than with matrix algebra, and because this approach provides direct insight into charge distributions in molecules, which are of prime interest in chemistry.

- 2.0: Prelude to the Foundations of Quantum Mechanics
- Heisenberg and Schrödinger were inspired by four key experimental observations: the spectral distribution of black-body radiation, the characteristics of the photoelectric effect, the Compton effect, and the luminescence spectrum of the hydrogen atom. Explanation of these phenomena required the introduction of two revolutionary concepts: physical quantities previously thought to be continuously variable, such as energy and momentum, are quantized, and momentum and wavelength are related.

- 2.1: Black-Body Radiation
- One experimental phenomenon that could not be adequately explained by classical physics was black-body radiation. Hot objects emit electromagnetic radiation. The burners on most electric stoves glow red at their highest setting. If we take a piece of metal and heat it in a flame, it begins to glow, dark red at first, then perhaps white or even blue if the temperature is high enough.

- 2.2: Photoelectric Effect
- In the photoelectric effect, light incident on the surface of a metal causes electrons to be ejected. The number of emitted electrons and their kinetic energy can be measured as a function of the intensity and frequency of the light.

- 2.3: The Compton Effect
- The Compton effect concerns the inelastic scattering of x‑rays by electrons. Scattering means dispersing in different directions, and inelastic means that energy is lost by the scattered object in the process. The intensity of the scattered x‑ray is measured as a function of the wavelength shift.

- 2.4: Hydrogen Luminescence
- The luminescence spectrum of the hydrogen atom reveals light being emitted at discrete frequencies. These spectral features appear so sharp that they are called lines. These lines, occurring in groups, are found in different regions of the spectrum; some are in the visible, some in the infrared, and some in the vacuum ultraviolet. The occurrence of these lines was very puzzling in the late 1800’s. Spectroscopists approach this type of problem by looking for patterns in the observations.

- 2.5: Early Models of the Hydrogen Atom
- Ernest Rutherford had proposed a model of atoms based on the αα -particle scattering experiments of Hans Geiger and Ernest Marsden. There are some basic problems with the Rutherford model. The Coulomb force that exists between oppositely charge particles means that a positive nucleus and negative electrons should attract each other, and the atom should collapse. Niels Bohr approached this problem by proposing that we simply must invent new physical laws.

- 2.6: Derivation of the Rydberg Equation from Bohr's Model
- Bohr postulated that electrons existed in orbits or states that had discrete energies. We therefore want to calculate the energy of these states and then take the differences in these energies to obtain the energy that is released as light when an electron makes a transition from one state to a lower energy one.

- 2.7: Summary of Bohr's Contribution
- Bohr’s proposal explained the hydrogen atom spectrum, the origin of the Rydberg formula, and the value of the Rydberg constant. Specifically it demonstrated that the integers in the Rydberg formula are a manifestation of quantization. The energy, the angular momentum, and the radius of the orbiting electron all are quantized. This quantization also parallels the concept of stable orbits in the Bohr model.

- 2.8: The Wave Properties of Matter
- De Broglie’s proposal can be applied to Bohr’s view of the hydrogen atom to show why angular momentum is quantized in units of \(ħ\). If the electron in the hydrogen atom is orbiting the nucleus in a stable orbit, then it should be described by a stable or stationary wave. Such a wave is called a standing wave. In a standing wave, the maximum and minimum amplitudes (crests and troughs) of the wave and the nodes (points where the amplitude is zero) are always found at the same position.

- 2.E: Foundations of Quantum Mechanics (Exercises)
- Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.

- 2.S: Foundations of Quantum Mechanics (Summary)
- Around 1900 several experimental observations were made that could not be explained, not even qualitatively, by existing physical laws. It therefore was necessary to invent (create) new concepts: quantization of energy and momentum, and a momentum-wavelength relation.

### Contributors

- Adapted from "Quantum States of Atoms and Molecules" by David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski