Hilbert Space¶
Hilbert spaces for quantum mechanics.
Authors: * Brian Granger * Matt Curry
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class
sympy.physics.quantum.hilbert.
HilbertSpace
[source]¶ An abstract Hilbert space for quantum mechanics.
In short, a Hilbert space is an abstract vector space that is complete with inner products defined [R417].
References
[R417] (1, 2) http://en.wikipedia.org/wiki/Hilbert_space Examples
>>> from sympy.physics.quantum.hilbert import HilbertSpace >>> hs = HilbertSpace() >>> hs H
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dimension
¶ Return the Hilbert dimension of the space.
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class
sympy.physics.quantum.hilbert.
ComplexSpace
[source]¶ Finite dimensional Hilbert space of complex vectors.
The elements of this Hilbert space are n-dimensional complex valued vectors with the usual inner product that takes the complex conjugate of the vector on the right.
A classic example of this type of Hilbert space is spin-1/2, which is
ComplexSpace(2)
. Generalizing to spin-s, the space isComplexSpace(2*s+1)
. Quantum computing with N qubits is done with the direct product spaceComplexSpace(2)**N
.Examples
>>> from sympy import symbols >>> from sympy.physics.quantum.hilbert import ComplexSpace >>> c1 = ComplexSpace(2) >>> c1 C(2) >>> c1.dimension 2
>>> n = symbols('n') >>> c2 = ComplexSpace(n) >>> c2 C(n) >>> c2.dimension n
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class
sympy.physics.quantum.hilbert.
L2
[source]¶ The Hilbert space of square integrable functions on an interval.
An L2 object takes in a single sympy Interval argument which represents the interval its functions (vectors) are defined on.
Examples
>>> from sympy import Interval, oo >>> from sympy.physics.quantum.hilbert import L2 >>> hs = L2(Interval(0,oo)) >>> hs L2([0, oo)) >>> hs.dimension oo >>> hs.interval [0, oo)
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class
sympy.physics.quantum.hilbert.
FockSpace
[source]¶ The Hilbert space for second quantization.
Technically, this Hilbert space is a infinite direct sum of direct products of single particle Hilbert spaces [R418]. This is a mess, so we have a class to represent it directly.
References
[R418] (1, 2) http://en.wikipedia.org/wiki/Fock_space Examples
>>> from sympy.physics.quantum.hilbert import FockSpace >>> hs = FockSpace() >>> hs F >>> hs.dimension oo