Abstract
In Chap. 10, we saw that the analytic and the algebraic approaches lead to very similar formulations.
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Notes
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- 2.
We remark again that in general, we do not specify the integration limits for integrals as in (11.8). It is tacitly assumed that one integrates over the entire domain of definition of the integrand. Contrary to the initial impression, these integrals are definite integrals - in other words, scalars (which may be time dependent).
- 3.
More precisely, semi-linear in the first and linear in the second component (also denoted as antilinear or conjugate linear in the first argument and linear in the second argument). Therefore, the form is not called bilinear, but sesquilinear. In mathematics, the form is usually defined the other way around, as antilinear in the second argument.
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As said above, this does not mean that the graphs of the functions are orthogonal to each other or something similar. The statement refers only to the (abstract) angle between two vectors in the vector space.
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The term ‘separable’ which occurs here has nothing to do with the requirement of ‘separability,’ which means that a system (function) is separable into functions of space and of time.
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The technical term is Cauchy sequences, see Appendix G, Vol. 1.
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The requirement of completeness has no straightforward physical meaning, but it occurs in many proofs of laws concerning Hilbert spaces.
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There are also non-separable Hilbert spaces (for example, in the quantization of fields). But in ‘our’ quantum mechanics, they play no role, so here ‘Hilbert space’ means in general ‘separable Hilbert space.’
- 9.
We note that approaching quantum mechanics by means of the Hilbert space is not the only possible option. As a starting point, one could for example consider a \(C^{*}\)-algebra (see Appendix G, Vol. 1), or the aforementioned replacement process \(\left\{ ,\right\} _\mathrm{Poisson}\rightarrow \frac{1}{i\hbar }\left[ ,\right] _\mathrm{commutator}\). This method is called canonical quantization, see Appendix W, Vol. 2.
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Only the dimension of the state space matters here. The physical system can take a variety of forms. The electronic spin with its two orientations, the polarization of a photon, e.g. with horizontally and vertically linearly-polarized states, the MZI with the basis states \(\left| H\right\rangle \) and \(\left| V\right\rangle \), in a certain sense the ammonia molecule (\(NH_{3}\), where the N atom can tunnel through the \(H_{3}\) plane and occupy two states with respect to it) are some examples of physically different systems which all ‘live’ in a two-dimensional Hilbert space.
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In fact, the notation \(\mathbf {a}\) is very abstract—it does not reveal anything about the dimension nor the individual components. We know nothing more than simply that it is a vector. Nevertheless, this notation is often not perceived as particularly abstract. This is probably due to the fact that one was introduced to it at the beginning of physics courses and it now seems familiar.
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This is also why we chose the symbol \(\cong \) to distinguish between an abstract ket and its representation as a column vector.
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It is in any case not a column vector (even if this idea sometimes proves to be helpful).
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To avoid misunderstandings: \(\mathbf {a}\) is an abstract or general column vector, whereas \(\left| \Psi \right\rangle \) is an abstract state which can be represented, where appropriate, as a column vector, but for which also other representations exist. Quite analogously, \( \mathbb {A}\) denotes a general matrix and \(A_{\text {abstract}}\) an abstract operator, which can, where appropriate, be represented as a matrix.
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However, there are books that distinguish them quite consistently.
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Note that here ‘adjoint’ means the Hermitian adjoint \({a^{\dagger }}\), as always in non-relativistic quantum mechanics. In relativistic quantum mechanics, one uses instead the Dirac adjoint \({a^{\dagger }}\) \({\gamma _{o}}\).
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In fact, there may be a difference between self-adjoint and Hermitian (see Chap. 13 and Appendix I, Vol. 1). Among the problems considered here, this difference is not noticeable.
- 18.
This form is called spectral representation; we discuss it in more detail in Chap. 13.
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Pade, J. (2018). Abstract Notation. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_11
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