Difference between revisions of "Hilbert Space"
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* 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 c'''v'''. | * 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 c'''v'''. | ||
* 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. | * 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: | ||
==References== | ==References== | ||
[[Category: Physics]] | [[Category: Physics]] | ||
Revision as of 22:24, 26 November 2024
Full Title
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.[1]
- 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:
References
- ↑ Robert B. Griffiths, Hilbert Space Quantum Mechanics CMU (2014-01) https://quantum.phys.cmu.edu/QCQI/qitd114.pdf
