Notions from Linear Algebra and Bra-Ket Notation
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 s
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Notions from Linear Algebra and Bra-Ket Notation
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) = C1 ψ1 (x, t) + C2 ψ2 (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. An example of this is Fourier transformation (2.8) into a wave function which depends on a wave vector k, 1 ψ(k, t) = √ 3 2π
d 3 x exp(− ik · x) ψ(x, t).
(4.1)
We have already noticed that this is sloppy notation from the mathematical point ˜ of view. We should denote the Fourier transformed function with ψ(k, t) to make ˜ it clear that ψ(k, t) and ψ(x, t) have different dependencies on their arguments (or stated differently, to make it clear that ψ(k, t) and ψ(x, t) are really different functions). However, there is a reason for the notation in Eqs. (2.7) and (2.8). We can switch back and forth between ψ(x, t) and ψ(k, t) using Fourier transformation. This implies that any property of a particle that can be calculated from the wave function ψ(x, t) in x space can also be calculated from the wave function ψ(k, t) in k space. Therefore, following Dirac (see [42] and references there), we nowadays do not think any more of ψ(x, t) as a wave function of a particle, but we rather think more abstractly of ψ(t) as a time-dependent quantum state, with particular representations of the quantum state ψ(t) given by the wave functions ψ(x, t) or ψ(k, t). There are infinitely more possibilities to represent the quantum state ψ(t) through functions. For example, we could perform a Fourier transformation only with respect to the y variable and represent ψ(t) through the wave function
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_4
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4 Notions from Linear Algebra and Bra-Ket Notation
ψ(x, ky , z, t), or we could perform an invertible transformation to completely different independent variables. In 1939, Paul Dirac introduced a notation in quantum mechanics which emphasizes the vector space and representation aspects of quantum states in a very elegant and suggestive manner. This notation is Dirac’s bra-ket notation, and it is ubiquitous in advanced modern quantum mechanics. It is worthwhile to use bra-ket notation from the start, and it is most easily explained in the framework of linear algebra.
4.1 Notions from Linear Algebra The mathematical structure of quantum mechanics resembles linear algebra in many respects, and many notions from linear algebra are very useful in the investig
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