Abstract
The Schrödinger equation (1.14) is linear in the wave function \(\psi (\boldsymbol{x},t)\). This implies that for any set of solutions \(\psi _{1}(\boldsymbol{x},t)\), \(\psi _{2}(\boldsymbol{x},t),\ldots\), any linear combination \(\psi (\boldsymbol{x},t) = C_{1}\psi _{1}(\boldsymbol{x},t) + C_{2}\psi _{2}(\boldsymbol{x},t)+\ldots\) with complex coefficients C n is also a solution. The set of solutions of equation (1.14) for fixed potential V will therefore have the structure of a complex vector space, and we can think of the wave function \(\psi (\boldsymbol{x},t)\) as a particular vector in this vector space. Furthermore, we can map this vector bijectively into different, but equivalent representations where the wave function depends on different variables.
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Notes
- 1.
We write scalar products of vectors initially as \(\boldsymbol{u}^{\mathrm{T}} \cdot \boldsymbol{ v}\) to be consistent with proper tensor product notation used in (4.1), but we will switch soon to the shorter notations \(\boldsymbol{u} \cdot \boldsymbol{ v}\), \(\boldsymbol{u} \otimes \boldsymbol{ v}\) for scalar products and tensor products.
- 2.
For scattering off two-dimensional crystals the Laue conditions can be recast in simpler forms in special cases. E.g. for orthogonal incidence a plane grating equation can be derived from the Laue conditions, or if the momentum transfer \(\Delta \boldsymbol{k}\) is in the plane of the crystal a two-dimensional Bragg equation can be derived.
- 3.
In the case of a complex finite-dimensional vector space, the “bra vector” would actually be the transpose complex conjugate vector, \(\langle v\vert =\boldsymbol{ v}^{+} =\boldsymbol{ v}^{{\ast}\mathrm{T}}\).
- 4.
Strictly speaking, we can think of multiplication of a state | ϕ〉 with \(\langle \Psi \vert\) as projecting onto a component parallel to \(\vert \Psi \rangle\) only if \(\vert \Psi \rangle\) is normalized. It is convenient, though, to denote multiplication with \(\langle \Psi \vert\) as projection, although in the general case this will only be proportional to the coefficient of the \(\vert \Psi \rangle\) component in | ϕ〉.
- 5.
- 6.
P. Güttinger, Diplomarbeit, ETH Zürich, Z. Phys. 73, 169 (1932). Exceptionally, there is no summation convention used in equation (4.44).
- 7.
R.P. Feynman, Phys. Rev. 56, 340 (1939).
Bibliography
H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937)
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Dick, R. (2016). Notions from Linear Algebra and Bra-Ket Notation. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_4
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