State

Dirac notation for states.

class sympy.physics.quantum.state.Bra(*args, **kwargs)[source]

A general time-independent Bra in quantum mechanics.

Inherits from State and BraBase. A Bra is the dual of a Ket [R684]. This class and its subclasses will be the main classes that users will use for expressing Bras in Dirac notation.

Parameters: args : tuple

The list of numbers or parameters that uniquely specify the ket. This will usually be its symbol or its quantum numbers. For time-dependent state, this will include the time.

Examples

Create a simple Bra and look at its properties:

>>> from sympy.physics.quantum import 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|

References

R684(1,2): https://en.wikipedia.org/wiki/Bra-ket_notation

class sympy.physics.quantum.state.BraBase(*args, **kwargs)[source]

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.

class sympy.physics.quantum.state.Ket(*args, **kwargs)[source]

A general time-independent Ket in quantum mechanics.

Inherits from State and KetBase. This class should be used as the base class for all physical, time-independent Kets in a system. This class and its subclasses will be the main classes that users will use for expressing Kets in Dirac notation [R685].

Parameters: args : tuple

The list of numbers or parameters that uniquely specify the ket. This will usually be its symbol or its quantum numbers. For time-dependent state, this will include the time.

Examples

Create a simple Ket and looking at its properties:

>>> from sympy.physics.quantum import Ket
>>> 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>

References

R685(1,2): https://en.wikipedia.org/wiki/Bra-ket_notation

class sympy.physics.quantum.state.KetBase(*args, **kwargs)[source]

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.

class sympy.physics.quantum.state.OrthogonalBra(*args, **kwargs)[source]: Orthogonal Bra in quantum mechanics.

class sympy.physics.quantum.state.OrthogonalKet(*args, **kwargs)[source]

Orthogonal Ket in quantum mechanics.

The inner product of two states with different labels will give zero, states with the same label will give one.

>>> from sympy.physics.quantum import OrthogonalBra, OrthogonalKet
>>> from sympy.abc import m, n
>>> (OrthogonalBra(n)*OrthogonalKet(n)).doit()
1
>>> (OrthogonalBra(n)*OrthogonalKet(n+1)).doit()
0
>>> (OrthogonalBra(n)*OrthogonalKet(m)).doit()
<n|m>

class sympy.physics.quantum.state.OrthogonalState(*args, **kwargs)[source]: General abstract quantum state used as a base class for Ket and Bra.

class sympy.physics.quantum.state.State(*args, **kwargs)[source]: General abstract quantum state used as a base class for Ket and Bra.

class sympy.physics.quantum.state.StateBase(*args, **kwargs)[source]

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.

property dual: Return the dual state of this one.

classmethod dual_class()[source]: Return the class used to construct the dual.

property operators: Return the operator(s) that this state is an eigenstate of

class sympy.physics.quantum.state.TimeDepBra(*args, **kwargs)[source]

General time-dependent 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

The list of numbers or parameters that uniquely specify the ket. This will usually be its symbol or its quantum numbers. For time-dependent state, this will include the time as the final argument.

Examples

>>> from sympy.physics.quantum import TimeDepBra
>>> b = TimeDepBra('psi', 't')
>>> b
<psi;t|
>>> b.time
t
>>> b.label
(psi,)
>>> b.hilbert_space
H
>>> b.dual
|psi;t>

class sympy.physics.quantum.state.TimeDepKet(*args, **kwargs)[source]

General time-dependent 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

The list of numbers or parameters that uniquely specify the ket. This will usually be its symbol or its quantum numbers. For time-dependent state, this will include the time as the final argument.

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 sympy.physics.quantum.state.TimeDepState(*args, **kwargs)[source]

Base class for a general time-dependent quantum state.

This class is used as a base class for any time-dependent state. The main difference between this class and the time-independent state is that this class takes a second argument that is the time in addition to the usual label argument.

Parameters: args : tuple

The list of numbers or parameters that uniquely specify the ket. This will usually be its symbol or its quantum numbers. For time-dependent state, this will include the time as the final argument.

property label: The label of the state.

property time: The time of the state.

class sympy.physics.quantum.state.Wavefunction(*args)[source]

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

The expression representing the functional form of the w.f.

coords : Symbol or tuple

The coordinates to be integrated over, and their bounds

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(L-1)
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)

property expr

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

property is_commutative: Override Function’s is_commutative so that order is preserved in represented expressions

property is_normalized

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

property limits

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)}

property norm

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

normalize()[source]

Return a normalized version of the Wavefunction

Examples

>>> from sympy import symbols, pi
>>> from sympy.functions import 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))

prob()[source]

Return the absolute magnitude of the w.f., \(|\psi(x)|^2\)

Examples

>>> from sympy import symbols, pi
>>> from sympy.functions import 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)

property variables

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,)