Quantum Mechanics: the Basics II The Dreaded Quantum, 31i2017, Jeremy Butterfield (jb56) Thanks to Adam Caulton 1 Recap: Inner products and Hilbert space Vector spaces may be assigned an inner product, which is an assignment of a complex number to every pair of vectors |>>, >> >> <>|> E C. The inner product allows us to assign a length (a.k.a. norm) to each vector; || = < >>. If |o| = 1, then |>> is said to be normalized. If ($|> = 0 then |>> and |{} are said to be orthogonal. A collection of vectors are orthonormal iff they are each normalized and mutually orthogonal. It is crucial for quantum mechanics that length and orthogonality for states (represented as vectors) are definable. A state-vector's length is connected to the normalization of probabilities (see Section 3) and orthogonality represents a special sort of distinctness between states (see Section 2). The state spaces used in quantum mechanics are Hilbert spaces, usually denoted H (pro- nounce: "curly H"). A Hilbert space is a vector space over C, equipped with an inner product, that is complete in the norm | . |. (Being complete in the norm means that the limit vector of any Cauchy sequence of vectors in H is also in H, where the limit is taken in terms of distance given by | · |.) 2 Quantities (a.k.a. "observables") The abstract quantities, intended to represent physical quantities like position and momentum (a.k.a. observables), in quantum mechanics comprise a special class of linear operators on H. Linear operators are so-called because they send vectors to vectors; i.e. they preserve straight lines in H. Any linear operator A has eigenvectors, a.k.a. eigenstates, which are vectors in H that change, if anything, only their magnitude when acted on by A. Each eigenvector may be associated with an eigenvalue, which is the factor by which the length of the eigenvector changes due to having been acted on by A. The special class of linear operators that represent physical quantities are called Hermitian operators. They are formally defined by (¢| Av} = {A}}} for all |>, > E H. They are characterized by having: (i) orthogonal eigenvectors; and (ii) real eigenvalues. Any Hermitian operator may be defined by specifying: (i) an orthonormal basis of H; and (ii) a real number for each basis vector (this is the content of what is called the spectral decomposition theorem). This will be important for Section 3. Two absolutely central quantities in quantum mechanics are position and momentum. These are defined by representing the state |v) as a complex-valued wave y(x) over space. In one dimension, position Q and momentum P are defined as follows: (Q4)(x) = xv(x); (Pv)(x) = - ih- dx d?(x) . (1) Following the previous paragraph, we can instead define these quantities in terms of their normalized eigenvectors and eigenvalues: 1 -
· Q's eigenstates have wave representations that are Dirac delta "functions" 8(x - a), which, for each position a, are defined as being