Hilbert Space¶
Hilbert spaces for quantum mechanics.
Authors: * Brian Granger * Matt Curry
- class sympy.physics.quantum.hilbert.ComplexSpace(dimension)[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
- class sympy.physics.quantum.hilbert.DirectSumHilbertSpace(*args)[source]¶
A direct sum of Hilbert spaces [R629].
This class uses the
+
operator to represent direct sums between different Hilbert spaces.A
DirectSumHilbertSpace
object takes in an arbitrary number ofHilbertSpace
objects as its arguments. Also, addition ofHilbertSpace
objects will automatically return a direct sum object.Examples
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
>>> c = ComplexSpace(2) >>> f = FockSpace() >>> hs = c+f >>> hs C(2)+F >>> hs.dimension oo >>> list(hs.spaces) [C(2), F]
References
- property spaces¶
A tuple of the Hilbert spaces in this direct sum.
- 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 [R630]. This is a mess, so we have a class to represent it directly.
Examples
>>> from sympy.physics.quantum.hilbert import FockSpace >>> hs = FockSpace() >>> hs F >>> hs.dimension oo
References
- 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 [R631].
Examples
>>> from sympy.physics.quantum.hilbert import HilbertSpace >>> hs = HilbertSpace() >>> hs H
References
- property dimension¶
Return the Hilbert dimension of the space.
- class sympy.physics.quantum.hilbert.L2(interval)[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(Interval(0, oo)) >>> hs.dimension oo >>> hs.interval Interval(0, oo)
- class sympy.physics.quantum.hilbert.TensorPowerHilbertSpace(*args)[source]¶
An exponentiated Hilbert space [R632].
Tensor powers (repeated tensor products) are represented by the operator
**
Identical Hilbert spaces that are multiplied together will be automatically combined into a single tensor power object.Any Hilbert space, product, or sum may be raised to a tensor power. The
TensorPowerHilbertSpace
takes two arguments: the Hilbert space; and the tensor power (number).Examples
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace >>> from sympy import symbols
>>> n = symbols('n') >>> c = ComplexSpace(2) >>> hs = c**n >>> hs C(2)**n >>> hs.dimension 2**n
>>> c = ComplexSpace(2) >>> c*c C(2)**2 >>> f = FockSpace() >>> c*f*f C(2)*F**2
References
- class sympy.physics.quantum.hilbert.TensorProductHilbertSpace(*args)[source]¶
A tensor product of Hilbert spaces [R633].
The tensor product between Hilbert spaces is represented by the operator
*
Products of the same Hilbert space will be combined into tensor powers.A
TensorProductHilbertSpace
object takes in an arbitrary number ofHilbertSpace
objects as its arguments. In addition, multiplication ofHilbertSpace
objects will automatically return this tensor product object.Examples
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace >>> from sympy import symbols
>>> c = ComplexSpace(2) >>> f = FockSpace() >>> hs = c*f >>> hs C(2)*F >>> hs.dimension oo >>> hs.spaces (C(2), F)
>>> c1 = ComplexSpace(2) >>> n = symbols('n') >>> c2 = ComplexSpace(n) >>> hs = c1*c2 >>> hs C(2)*C(n) >>> hs.dimension 2*n
References
- property spaces¶
A tuple of the Hilbert spaces in this tensor product.