Abstract
The Schrödinger equation (1.63) is linear in the wave function ψ(x, t). This implies that for any set of solutions ψ1(x, t), ψ2(x, t), …, any linear combination ψ(x, t) = C1ψ1(x, t) + C2ψ2(x, t) + … with complex coefficients Cn is also a solution. The set of solutions of Eq. (1.63) for fixed potential V will therefore have the structure of a complex vector space, and we can think of the wave function ψ(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 uT ⋅v to be consistent with proper tensor product notation used in (4.5), but we will switch soon to the shorter notations u ⋅v, u ⊗v for scalar products and tensor products.
- 2.
It does not matter whether we write the expansion coefficients of a vector v on the right or left of the basis vectors, \(\hat {\boldsymbol {e}}_a v^a\equiv v^a\hat {\boldsymbol {e}}_a\). However, writing them on the right agrees with the conventions for bra-ket notation introduced further below, and also has the advantage that the coefficients va naturally transform with matrices \( \underline {R}^{-1}\) from the left, if the basis vectors \(\hat {\boldsymbol {e}}_a\) transform with matrix coefficients Rai from the right, see Eqs. (4.10) and (4.14) below and also Problem 4.1.
- 3.
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 Δk is in the plane of the crystal a two-dimensional Bragg equation can be derived.
- 4.
In the case of a complex finite-dimensional vector space, the “bra vector” would actually be the transpose complex conjugate vector, 〈v| = v+ = v∗T.
- 5.
We will see in a little while that these relations can also be derived using shift operators. This is explained in Problem 6.6b.
- 6.
Strictly speaking, we can think of multiplication of a state |ϕ〉 with 〈 Ψ| as projecting onto a component parallel to | Ψ〉 only if | Ψ〉 is normalized. It is convenient, though, to denote multiplication with 〈 Ψ| as projection, although in the general case this will only be proportional to the coefficient of the | Ψ〉 component in |ϕ〉.
- 7.
- 8.
References
P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford, 1958)
R.P. Feynman, Phys. Rev. 56, 340 (1939)
P. Güttinger, Diplomarbeit, ETH Zürich, Z. Phys. 73, 169 (1932)
H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937)
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Dick, R. (2020). Notions from Linear Algebra and Bra-Ket Notation. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-57870-1_4
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