Hilbert Space - Revision history

Tom: /* Quantum Theory */

2024-11-27T05:41:49Z

‎Quantum Theory

Tom: /* Quantum Theory */

2024-11-27T05:29:59Z

‎Quantum Theory

Tom: /* Problems */

2024-11-27T05:26:08Z

‎Problems

Tom: /* Context */

2024-11-27T05:24:54Z

‎Context

Tom: /* Context */

2023-04-27T14:54:06Z

‎Context

Tom: /* Context */

2023-04-27T14:52:47Z

‎Context

Tom: /* Context */

2023-04-27T14:48:39Z

‎Context

Tom: /* Context */

2023-04-27T14:47:19Z

‎Context

Tom: /* Context */

2023-04-27T14:46:08Z

‎Context

Tom: Created page with "==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 produc..."

2023-04-27T14:42:36Z

Created page with "==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 produc..."

New page

==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.
◦ 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 nowadays, as in these notes, in a way
that includes finite-dimensional spaces, which automatically satisfy the condition of completeness.
1
⋆ We will use Dirac notation in which the vectors in the space are denoted by |vi, called a ket, where v
is some symbol which identifies the vector.
One could equally well use something like ~v or v. A multiple of a vector by a complex number c is written
as c|vi—think of it as analogous to c~v of cv.
⋆ In Dirac notation the inner product of the vectors |vi with |wi is written hv|wi. 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.

==References==

[[Category: Physics]]

@@ Line 12: / Line 12: @@
 * 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
 [[File:Unitary Gauge Transformations Cut.png]]
 ==References==
 [[Category: Physics]]

@@ Line 9: / Line 9: @@
 ==Problems==
 ===Quantum Theory===
-Unitary Gauge Transformations
+Unitary Gauge Transformations<ref>Jacob Barandes, ''New Foundations for Quantum Theory'' 2024-03-04 https://www.youtube.com/watch?v=dB16TzHFvj0</ref>
 * 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
 [[File:Unitary Gauge Transformations Cut.png]]

@@ Line 11: / Line 11: @@
 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:
+[[File:Unitary Gauge Transformations Cut.png]]
 ==References==
 [[Category: Physics]]

@@ Line 7: / Line 7: @@
 * 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.
 ==References==
 [[Category: Physics]]

@@ Line 3: / Line 3: @@
 ==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.<ref>Robert B. Griffiths, ''Hilbert Space Quantum Mechanics'' CMU (2014-01)  https://quantum.phys.cmu.edu/QCQI/qitd114.pdf</ref>
-* 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 nowadays, as in these notes, in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness.
+* 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  c'''v'''.