Chapter 8 Vector Spaces in Quantum Mechanics

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Chapter 8 Vector Spaces in Quantum Mechanics 100vectors, which can be written in a different notation, (|φ〉, |ψ〉), that is more commonlyencountered in pure mathematics. But the inner product can be viewed in another way,which leads to a new interpretation of the expression 〈φ|ψ〉, and the introduction of a newclass of state vectors. If we consider the equationand ‘cancel’ the |ψ〉, we get the result〈φ|ψ〉 = (|φ〉, |ψ〉) (8.67)〈φ| • = (|φ〉, •) (8.68)where the ‘•’ is inserted, temporarily, to remind us that in order to complete the equation,a ket vector has to be inserted. By carrying out this procedure, we have introduced anew quantity 〈φ| which is known as a bra or bra vector, essentially because 〈φ|ψ〉 lookslike quantities enclosed between a pair of ‘bra(c)kets’. It is a vector because, as can bereadily shown, the collection of all possible bras form a vector space. For instance, by theproperties of the inner product, ifthen|ψ〉 = a 1 |ϕ 1 〉 + a 2 |ϕ 2 〉 (8.69)(|ψ〉, •) = 〈ψ| • = (a 1 |ϕ 1 〉 + a 2 |ϕ 2 〉, •) (8.70)i.e., dropping the ‘•’ symbols, we have= a ∗ 1 (|ϕ 1〉, •) + a ∗ 2 (|ϕ 2〉, •) = a ∗ 1 〈ϕ 1| • + a ∗ 2 〈ϕ 2| • (8.71)〈ψ| = a ∗ 1 〈ϕ 1| + a ∗ 2 〈ϕ 2| (8.72)so that a linear combination of two bras is also a bra, from which follows (after a bit morework checking that the other requirements of a vector space are also satisfied) the resultthat the set of all bras is a vector space. Incidentally, this last calculation above shows,once again, that if |ψ〉 = a 1 |ϕ 1 〉 + a 2 |ϕ 2 〉 then the corresponding bra is 〈ψ| = a ∗ 1 〈ϕ 1| + a ∗ 2 〈ϕ 2|.So, in a sense, the bra vectors are the ‘complex conjugates’ of the ket vectors.The vector space of all bra vectors is obviously closely linked to the vector space of allthe kets H, and is in fact usually referred to as the dual space, and represented by H ∗ . Toeach ket vector |ψ〉 belonging to H, there is then an associated bra vector 〈ψ| belonging tothe dual space H ∗ . However, the reverse is not necessarily true: there are bra vectors thatdo not necessarily have a corresponding ket vector, and therein lies the difference betweenbras and kets. It turns out that the difference only matters for Hilbert spaces of infinitedimension, in which case there can arise bra vectors whose corresponding ket vector is ofinfinite length, i.e. has infinite norm, and hence cannot be normalized to unity. Such ketvectors can therefore never represent a possible physical state of a system. But these issueswill not arise here, so will not be of any concern. The point to be taken away from all thisis that a bra vector is not the same kind of mathematical object as a ket vector. In fact,it has all the attributes of an operator in the sense that it acts on a ket vector to producea complex number, this complex number being given by the appropriate inner product.This is in contrast to the more usual sort of operators encountered in quantum mechanicsthat act on ket vectors to produce other ket vectors. In mathematical texts a bra vectoris usually referred to as a ‘linear functional’. Nevertheless, in spite of the mathematical
Chapter 8 Vector Spaces in Quantum Mechanics 101distinction that can be made between bra and ket vectors, the correspondence between thetwo kinds of vectors is in most circumstances so complete that a bra vector equally wellrepresents the state of a quantum system as a ket vector. Thus, we can talk of a systembeing in the state 〈ψ|.We can summarize all this in the general case as follows: The inner product (|ψ〉, |φ〉)defines, for all states |ψ〉, the set of functions (or linear functionals) (|ψ〉, ). The linearfunctional (|ψ〉, ) maps any ket vector |φ〉 into the complex number given by the innerproduct (|ψ〉, |φ〉).1. The set of all linear functionals (|ψ〉, ) forms a complex vector space H ∗ , the dualspace of H.2. The linear functional (|ψ〉, ) is written 〈ψ| and is known as a bra vector.3. To each ket vector |ψ〉 there corresponds a bra vector 〈ψ| such that if |φ 1 〉 → 〈φ 1 |and |φ 2 〉 → 〈φ 2 | thenc 1 |φ 1 〉 + c 2 |φ 2 〉 → c ∗ 1 〈φ 1| + c ∗ 2 〈φ 2|.8.6 State Spaces of Infinite DimensionSome examples of physical systems with state spaces of infinite dimension were providedin the previous Section. In these examples, we were able to proceed, at least as far asconstructing the state space was concerned, largely as was done in the case of finite dimensionalstate spaces. However, further investigation shows that there are features ofthe mathematics, and the corresponding physical interpretation in the infinite dimensionalcase that do not arise for systems with finite dimensional state spaces. Firstly, it is possibleto construct state vectors that cannot represent a state of the system and secondly, thepossibility arises of the basis states being continuously infinite. This latter state of affairsis not at all a rare and special case – it is just the situation needed to describe the motionof a particle in space, and hence gives rise to the wave function, and wave mechanics.8.6.1 States of Infinite NormTo illustrate the first of the difficulties mentioned above, consider the example of a systemof identical photons in the state |ψ〉 defined by Eq. (8.66). As the basis states areorthonormal we have for 〈ψ|ψ〉∞∑〈ψ|ψ〉 = |c n | 2 (8.73)If the probabilities |c n | 2 form a convergent infinite series, then the state |ψ〉 has a finitenorm, i.e. it can be normalized to unity. However, if this series does not converge, then itis not possible to supply a probability interpretation to the state vector as it is not normalizableto unity. For instance, if c 0 = 0 and c n = 1/ √ n, n = 1, 2, . . ., then〈ψ|ψ〉 =n=0∞∑n=11n(8.74)
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Chapter 8 Vector Spaces in Quantum Mechanics

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