Abstract
Quantum mechanics provides the basis for the description of matter on the atomic scale. Developed in conjunction with progress in spectroscopy, it explains phenomena such as the photoelectric effect, molecular spectra, electronic structure, and the chemical bond. Challenging and at the same time fascinating, the predictions of quantum mechanics are beyond direct everyday perception. (To some extent this is in contrast to thermodynamics and changes of state (Chap. 3) where experience from everyday perception is a reasonable criterion for assessing the soundness of predictions.) The basic concepts introduce the postulates of quantum mechanics. Aside from problems dealing with black body radiation, wave-packet propagation, and the hydrogen atom, applications of operator calculus and the variational principle are highlights that show perspectives with regard to tackling advanced problems in this field.
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Notes
- 1.
The number of postulates differs from textbook to textbook, although they cover the same content.
- 2.
The Kronecker delta δ mn is defined as follows: δ mn = 1 if m = n, and δ mn = 0 if m ≠ n.
- 3.
If M ≫ m e is the mass of the nucleus, the effective mass \(\mu = \frac{Mm_{e}} {M+m_{e}}\) is close to m e . In the case of positronium consisting of an electron and a positron of the same mass, however, \(\mu = \frac{m_{e}} {2}\).
- 4.
For explicit expressions of R nl (r) and Y lm (θ, ϕ) see Sects. A.3.14 and A.3.15.
- 5.
Sample heating to high temperature, also called tempering , is, for example, needed to obtain well-defined, clean and crystalline materials.
- 6.
In Problem 7.4 we have seen that heat conduction mediated by particle collisions is largely suppressed under ultrahigh vacuum conditions.
- 7.
- 8.
XPS X-ray photoelectron spectroscopy, UPS ultraviolet photoelectron spectroscopy, ESCA electron spectroscopy for chemical analysis.
- 9.
An extensive discussion can be found in the book by David Bohm [4].
- 10.
The other function is the Lagrangian L = E kin − V
- 11.
- 12.
See also Problem 9.5, where, for the lithium atom, we estimated an effective charge of 1.26 based on the experimental value of the ionization energy.
- 13.
The justification of an effective mass in solid state physics is based on the fact that the motion of an electron in the crystal is not free. While maintaining the expression for the kinetic energy, \(E_{\text{kin}} = \frac{\hslash ^{2}\mbox{ $\mathbf{k}$}^{2}} {2m^{{\ast}}}\), the effective mass can be derived from the curvature of electron bands at special points in the crystal’s reciprocal k-space.
- 14.
We have dealt with the exact solution in Problem 9.14 on page 257. In the literature, atomic orbital functions with exponential decay in the region far from the core are also called Slater functions.
- 15.
Harmonic oscillator wave functions are illustrated in Fig. 9.1. Explicit formulas are provided in the Appendix, Sect. A.3.13.
- 16.
The most common one is LAPACK, http://www.netlib.org/lapack/.
- 17.
The probability of finding the Li+ between x and x + dx is | ψ n (x) | 2.
- 18.
See Sect. 9.1.4. Note that in atomic units the unit length is 1a 0, and the unit energy is 1 E h .
- 19.
Note that because of the mathematical form of ψ 1 and ψ 2, 〈ψ 1 | ψ 2〉 = 〈ψ 2 | ψ 1〉.
References
Gearhart CA (2002) Planck, the quantum, and the historians. Phys Perspect 4:170
Rud Nielsen J (1976) Niels Bohr collected works, vol 3. The correspondence principle. North-Holland, Amsterdam
Schrödinger E (1921) Versuch zur modellmäßigen Deutung des Terms der scharfen Nebenserien. Z Phys 4:347
Bohm D (1951) Quantum theory. Prentice Hall, New York
Grivet JP (2002) The hydrogen molecular ion revisited. J Chem Educ 79:127
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Vogt, J. (2017). Quantum Mechanics and Electronic Structure. In: Exam Survival Guide: Physical Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-319-49810-2_9
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