Dirac notation for states.
Base class for Kets.
This class defines the dual property and the brackets for printing. This is an abstract base class and you should not instantiate it directly, instead use Ket.
Base class for Bras.
This class defines the dual property and the brackets for printing. This is an abstract base class and you should not instantiate it directly, instead use Bra.
Abstract base class for general abstract states in quantum mechanics.
All other state classes defined will need to inherit from this class. It carries the basic structure for all other states such as dual, _eval_adjoint and label.
This is an abstract base class and you should not instantiate it directly, instead use State.
General abstract quantum state used as a base class for Ket and Bra.
A general timeindependent Ket in quantum mechanics.
Inherits from State and KetBase. This class should be used as the base class for all physical, timeindependent Kets in a system. This class and its subclasses will be the main classes that users will use for expressing Kets in Dirac notation [R342].
Parameters :  args : tuple


References
[R342]  (1, 2) http://en.wikipedia.org/wiki/Braket_notation 
Examples
Create a simple Ket and looking at its properties:
>>> from sympy.physics.quantum import Ket, Bra
>>> from sympy import symbols, I
>>> k = Ket('psi')
>>> k
psi>
>>> k.hilbert_space
H
>>> k.is_commutative
False
>>> k.label
(psi,)
Ket’s know about their associated bra:
>>> k.dual
<psi
>>> k.dual_class()
<class 'sympy.physics.quantum.state.Bra'>
Take a linear combination of two kets:
>>> k0 = Ket(0)
>>> k1 = Ket(1)
>>> 2*I*k0  4*k1
2*I*0>  4*1>
Compound labels are passed as tuples:
>>> n, m = symbols('n,m')
>>> k = Ket(n,m)
>>> k
nm>
A general timeindependent Bra in quantum mechanics.
Inherits from State and BraBase. A Bra is the dual of a Ket [R343]. This class and its subclasses will be the main classes that users will use for expressing Bras in Dirac notation.
Parameters :  args : tuple


References
[R343]  (1, 2) http://en.wikipedia.org/wiki/Braket_notation 
Examples
Create a simple Bra and look at its properties:
>>> from sympy.physics.quantum import Ket, Bra
>>> from sympy import symbols, I
>>> b = Bra('psi')
>>> b
<psi
>>> b.hilbert_space
H
>>> b.is_commutative
False
Bra’s know about their dual Ket’s:
>>> b.dual
psi>
>>> b.dual_class()
<class 'sympy.physics.quantum.state.Ket'>
Like Kets, Bras can have compound labels and be manipulated in a similar manner:
>>> n, m = symbols('n,m')
>>> b = Bra(n,m)  I*Bra(m,n)
>>> b
I*<mn + <nm
Symbols in a Bra can be substituted using .subs:
>>> b.subs(n,m)
<mm  I*<mm
Base class for a general timedependent quantum state.
This class is used as a base class for any timedependent state. The main difference between this class and the timeindependent state is that this class takes a second argument that is the time in addition to the usual label argument.
Parameters :  args : tuple


General timedependent Bra in quantum mechanics.
This inherits from TimeDepState and BraBase and is the main class that should be used for Bras that vary with time. Its dual is a TimeDepBra.
Parameters :  args : tuple


Examples
>>> from sympy.physics.quantum import TimeDepBra
>>> from sympy import symbols, I
>>> b = TimeDepBra('psi', 't')
>>> b
<psi;t
>>> b.time
t
>>> b.label
(psi,)
>>> b.hilbert_space
H
>>> b.dual
psi;t>
General timedependent Ket in quantum mechanics.
This inherits from TimeDepState and KetBase and is the main class that should be used for Kets that vary with time. Its dual is a TimeDepBra.
Parameters :  args : tuple


Examples
Create a TimeDepKet and look at its attributes:
>>> from sympy.physics.quantum import TimeDepKet
>>> k = TimeDepKet('psi', 't')
>>> k
psi;t>
>>> k.time
t
>>> k.label
(psi,)
>>> k.hilbert_space
H
TimeDepKets know about their dual bra:
>>> k.dual
<psi;t
>>> k.dual_class()
<class 'sympy.physics.quantum.state.TimeDepBra'>
Class for representations in continuous bases
This class takes an expression and coordinates in its constructor. It can be used to easily calculate normalizations and probabilities.
Parameters :  expr : Expr
coords : Symbol or tuple


Examples
Particle in a box, specifying bounds in the more primitive way of using Piecewise:
>>> from sympy import Symbol, Piecewise, pi, N
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = Symbol('x', real=True)
>>> n = 1
>>> L = 1
>>> g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
>>> f = Wavefunction(g, x)
>>> f.norm
1
>>> f.is_normalized
True
>>> p = f.prob()
>>> p(0)
0
>>> p(L)
0
>>> p(0.5)
2
>>> p(0.85*L)
2*sin(0.85*pi)**2
>>> N(p(0.85*L))
0.412214747707527
Additionally, you can specify the bounds of the function and the indices in a more compact way:
>>> from sympy import symbols, pi, diff
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> f(L+1)
0
>>> f(L1)
sqrt(2)*sin(pi*n*(L  1)/L)/sqrt(L)
>>> f(1)
0
>>> f(0.85)
sqrt(2)*sin(0.85*pi*n/L)/sqrt(L)
>>> f(0.85, n=1, L=1)
sqrt(2)*sin(0.85*pi)
>>> f.is_commutative
False
All arguments are automatically sympified, so you can define the variables as strings rather than symbols:
>>> expr = x**2
>>> f = Wavefunction(expr, 'x')
>>> type(f.variables[0])
<class 'sympy.core.symbol.Symbol'>
Derivatives of Wavefunctions will return Wavefunctions:
>>> diff(f, x)
Wavefunction(2*x, x)
Return the expression which is the functional form of the Wavefunction
Examples
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, x)
>>> f.expr
x**2
Override Function’s is_commutative so that order is preserved in represented expressions
Returns true if the Wavefunction is properly normalized
Examples
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.is_normalized
True
Return the limits of the coordinates which the w.f. depends on If no limits are specified, defaults to (oo, oo).
Examples
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, (x, 0, 1))
>>> f.limits
{x: (0, 1)}
>>> f = Wavefunction(x**2, x)
>>> f.limits
{x: (oo, oo)}
>>> f = Wavefunction(x**2 + y**2, x, (y, 1, 2))
>>> f.limits
{x: (oo, oo), y: (1, 2)}
Return the normalization of the specified functional form.
This function integrates over the coordinates of the Wavefunction, with the bounds specified.
Examples
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
sqrt(2)*sqrt(L)/2
Return a normalized version of the Wavefunction
Examples
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = symbols('x', real=True)
>>> L = symbols('L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.normalize()
Wavefunction(sqrt(2)*sin(pi*n*x/L)/sqrt(L), (x, 0, L))
Return the absolute magnitude of the w.f., \(\psi(x)^2\)
Examples
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', real=True)
>>> n = symbols('n', integer=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.prob()
Wavefunction(sin(pi*n*x/L)**2, x)
Return the coordinates which the wavefunction depends on
Examples
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x,y = symbols('x,y')
>>> f = Wavefunction(x*y, x, y)
>>> f.variables
(x, y)
>>> g = Wavefunction(x*y, x)
>>> g.variables
(x,)