Chapter 8 Vector Spaces in Quantum Mechanics

Chapter 8 Vector Spaces in Quantum Mechanics 97space of the system. Here we will give a list of the defining conditions of a state space,though we will not be concerning ourselves too much with the formalities. The followingdefinitions and concepts set up the state space of a quantum system.1. Every physical state of a quantum system is specified by a vector called a statevector, ket vector, or sometimes just state and written | . . .〉 where . . . is a labelspecifying the physical information known about the state. An arbitrary state iswritten |ψ〉, or |φ〉 and so on. The set of all state vectors describing a given physicalsystem forms a complex vector space (actually a Hilbert space, see Sec. 8.5.2) Halso known as the state space or ket space for the system.2. Every linear superposition of two or more state vectors |φ 1 〉, |φ 2 〉, |φ 3 〉, . . . , is also astate of the quantum system i.e. the state |ψ〉 given by|ψ〉 = c 1 |φ 1 〉 + c 2 |φ 2 〉 + c 3 |φ 3 〉 + . . .is a state of the system for all complex numbers c 1 , c 2 , c 3 , . . . .These last two points amount to saying that every physical state of a system is representedby a vector in the state space of the system, and every vector in the state space representsa possible physical state of the system. To guarantee this, the following condition is alsoimposed:3. If a state of the system is represented by a vector |ψ〉, then the same state is representedby the vector c|ψ〉 where c is any non-zero complex number.Finally, we need the concept of a set of basis states, and of the dimension of the statespace.4. A set of vectors {|ϕ 1 〉, |ϕ 2 〉, |ϕ 3 〉, . . . } is said to form a basis for the state space ifevery state of the quantum system can be represented as a linear superposition ofthe |ϕ i 〉’s i.e. for any state |ψ〉 we can write∑|ψ〉 = c i |ϕ i 〉.iThe set of vectors {|ϕ i 〉, i = 1, 2, . . . } are said to span the vector space. The vectorsare also termed the base states for the vector space. They are also said to becomplete. What this means, mathematically, is that for every state |φ〉 say, at leastone of the inner products 〈ϕ n |φ〉 will be non-zero, or conversely, there does not exista state |ξ〉 for which 〈ϕ n |ξ〉 = 0 for all the basis states |ϕ n 〉. Completeness clearlymeans that no more basis states are needed to describe any possible physical stateof a system.For example, returning to the spin half system, the two states |±〉 are all that is needed todescribe any state of the system, i.e. there are no spin states that cannot be described interms of these basis states. Thus, these states are said to be complete.