Hilbert Space

Context

• In Quantum Mechanics the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product.
• The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed. However, the term is often used in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness.
• We will use Dirac notation in which the vectors in the space are denoted by |v>, called a ket, where v is some symbol which identifies the vector.
• One could equally well use something like v. A multiple of a vector by a complex number c is written as c|v> -think of it as analogous to cv.
• In Dirac notation the inner product of the vectors |v> with |w> is written <v|w>. This resembles the ordinary dot product ~v · ~w except that one takes a complex conjugate of the vector on the left, thus think of ~v∗· ~w.

Problems

Quantum Theory

Unitary Gauge Transformations

• A reason to be skeptical about assigning a physical meaning directly to the Hilbert-space ingredients: quantum theory contains a little-known form of gauge invariance
• For V (t) any time-dependent, unitary operator, the following gauge transformation leaves all empirical predictions invariant:

Wave functions: 
Density matrices: 
Observables: 
Time-evolution 
operators: 
Hamiltonians: