Abstract
Up to now, we have occasionally used the terms ‘vector space’ or ‘state space’. In this chapter, we will address this concept in more detail. For reasons of simplicity, we will rely heavily on the example of polarization, where the basic formulations are of course independent of the specific realization and are valid for all two-dimensional state spaces (such as polarization states, electron spin states, a double-well potential, the ammonia molecule, etc.). Moreover, the concepts introduced here retain their meaning in higher-dimensional state spaces, as well. Therefore, we can introduce and discuss many topics by using the example of the simple two-dimensional state space. From the technical point of view, this chapter is about the discussion of some of the elementary facts of complex vector spaces.
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Notes
- 1.
Of course, we treat these technical aspects not as an end in themselves, but because they are of fundamental importance for the physical description of natural phenomena in the context of quantum mechanics.
- 2.
For the notation \(\cong \), see Chap. 2.
- 3.
Later on, we will meet vector spaces where this is no longer the case; keyword ‘identical particles’ or ‘superselection rules’.
- 4.
We note that the superposition principle contains three pieces of information: (1) The multiplication of a state by a scalar is meaningful. (2) The addition of two states is meaningful. (3) Every linear combination of two states is again an element in the vector space.
- 5.
As we know, only the zero vector has length zero.
- 6.
We recall that \(^{*}\) means complex conjugation.
- 7.
Strictly speaking, there are two adjoints. The one considered here is called Hermitian adjoint; it applies so to say in non-relativistic considerations. In the relativistic case, there is another kind, called Dirac adjoint which is defined differently. The bulk of the book is devoted to non-relativistic considerations; here adjoint means always Hermitian adjoint.
- 8.
In the bra-ket notation, one cannot identify the dimension of the corresponding vector space (the same holds true for the familiar vector notations v or \(\overrightarrow{v}\), by the way). If necessary, this information must be given separately.
- 9.
In the two-dimensional vector space that we are currently addressing, such a system consists of course of two vectors; as stated above, the zero vector is excluded a priori from consideration.
- 10.
We repeat the remark that for products of numbers and vectors, it holds that \(c\cdot \left| z\right\rangle =\left| z\right\rangle \cdot c\). Because \(\left\langle h\right| \left. z\right\rangle \) is a number, we can therefore write \(\left\langle h\right| \left. z\right\rangle \left| h\right\rangle \) as \(\left| h\right\rangle \left\langle h\right| \left. z\right\rangle \).
- 11.
In equations such as (4.17), the 1 on the right side is not necessarily the number 1, but is generally something that works like a multiplication by 1, i.e. a unit operator. For instance, this is the unit matrix when working with vectors. The notation 1 for the unit operator (which implies writing simply 1 instead of \(\left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{array} \right) \), for instance) is of course quite lax. On the other hand, as said before, the effect of multiplication by the unit operator and by 1 is identical, so that the small inaccuracy is generally accepted in view of the economy of notation. If necessary, ‘one knows’ that 1 means the unit operator. But there are also special notations for it, such as \(\mathbb {E}\), \(I_{n}\) (where n indicates the dimension) and others. An analogous remark applies to the zero operator. By the way, we recall that in the case of vectors we write quite naturally \(\overrightarrow{a}=0\) and not \(\overrightarrow{a}=\overrightarrow{0}\).
- 12.
For the summation we use almost exclusively the abbreviation \( \sum \nolimits _{n}\) (instead of \(\sum \nolimits _{n=1}^{\infty }\) or \(\sum \nolimits _{n=1}^{N}\) etc.). In the shorthand notation, the range of values of n must follow from the context of the problem at hand, if necessary.
- 13.
For the connection between inner product and projection, see Appendix F, Vol. 1.
- 14.
As we shall see in Chap. 13, a projection operator in quantum mechanics must meet a further condition (self-adjointness).
- 15.
In other words, due to the process of measuring, a superposition such as \( \left| z\right\rangle =a\left| h\right\rangle +b\left| v\right\rangle \) ‘collapses’ e.g. into the state \(\left| h\right\rangle \).
- 16.
In order to make it clear once more: If, for example, we measure an arbitrarily-polarized state \(\left| z\right\rangle =a\left| h\right\rangle +b\left| v\right\rangle \) with \(\left| a\right| ^{2}+\left| b\right| ^{2}=1\) and \(ab\ne 0\), we find with probability \(\left| a\right| ^{2}\) a horizontal linearly-polarized photon. This does not permit the conclusion that the photon was in that state before the measurement. It simply makes no sense in this case to speak of a of a well-defined value of the linear polarization (\(+1\) or \(-1\)) before the measurement.
- 17.
Essentially, this operator is the x component of the orbital angular momentum operator for the angular momentum 1; see Chap. 16 Vol. 2.
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Pade, J. (2018). Complex Vector Spaces and Quantum Mechanics. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_4
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