Examples of using BosonOperator and FermionOperator¶

J. R. Johansson, http://jrjohansson.github.com, robert@riken.jp

Note: This notebook requires and experimental branch of sympy available at: https://github.com/jrjohansson/sympy

In [1]:

from sympy import *
init_printing()

In [2]:

from sympy.physics.quantum import *
from sympy.physics.quantum.boson import *
from sympy.physics.quantum.fermion import *
from sympy.physics.quantum.operatorordering import *

Docstrings¶

In [3]:

#help(BosonOp)

In [4]:

#help(FermionOp)

In [5]:

help(normal_order)

Help on function normal_order in module sympy.physics.quantum.operatorordering:

normal_order(expr, recursive_limit=10, _recursive_depth=0)
    Normal order an expression with bosonic or fermionic operators. Note
    that this normal order is not equivalent to the original expression, but
    the creation and annihilation operators in each term in expr is reordered
    so that the expression becomes normal ordered.
    
    Parameters
    ==========
    
    expr : expression
        The expression to normal order.
    
    recursive_limit : int (default 10)
        The number of allowed recursive applications of the function.
    
    Examples
    ========
    
    >>> from sympy.physics.quantum import Dagger
    >>> from sympy.physics.quantum.boson import BosonOp
    >>> from sympy.physics.quantum.operatorordering import normal_order
    >>> a = BosonOp("a")
    >>> normal_order(a * Dagger(a))
    Dagger(a)*a

In [6]:

help(normal_ordered_form)

Help on function normal_ordered_form in module sympy.physics.quantum.operatorordering:

normal_ordered_form(expr, independent=False, recursive_limit=10, _recursive_depth=0)
    Write an expression with bosonic or fermionic operators on normal
    ordered form, where each term is normally ordered. Note that this
    normal ordered form is equivalent to the original expression.
    
    Parameters
    ==========
    
    expr : expression
        The expression write on normal ordered form.
    
    recursive_limit : int (default 10)
        The number of allowed recursive applications of the function.
    
    Examples
    ========
    
    >>> from sympy.physics.quantum import Dagger
    >>> from sympy.physics.quantum.boson import BosonOp
    >>> from sympy.physics.quantum.operatorordering import normal_ordered_form
    >>> a = BosonOp("a")
    >>> normal_ordered_form(a * Dagger(a))
    1 + Dagger(a)*a

Test Boson operators¶

In [7]:

a = BosonOp('a')
b = BosonOp('b')

In [8]:

a.is_annihilation, Dagger(a).is_annihilation

Out[8]:

(True, False)

In [9]:

Commutator(a, Dagger(a)).doit()

Out[9]:

$$1$$

In [10]:

Commutator(Dagger(a), a).doit()

Out[10]:

$$-1$$

In [11]:

Commutator(a, a).doit()

Out[11]:

$$0$$

In [12]:

Commutator(Dagger(a), Dagger(a)).doit()

Out[12]:

$$0$$

In [13]:

Dagger(Dagger(Dagger(a)))

Out[13]:

$${{a}^\dagger}$$

In [14]:

Commutator(a, Dagger(Dagger(a))) #.doit()

Out[14]:

$$0$$

In [15]:

Commutator(a, b).doit()

Out[15]:

$${a} {b} - {b} {a}$$

In [16]:

Commutator(a, b).doit(independent=True)

Out[16]:

$$0$$

Test Fermion operators¶

In [17]:

c = FermionOp('c')
d = FermionOp('d')

In [18]:

Commutator(c,Dagger(c))

Out[18]:

$$- \left[{{c}^\dagger},{c}\right]$$

In [19]:

AntiCommutator(c,Dagger(c))

Out[19]:

$$\left\{{{c}^\dagger},{c}\right\}$$

In [20]:

AntiCommutator(c, Dagger(c)).doit()

Out[20]:

$$1$$

In [21]:

AntiCommutator(Dagger(c), c).doit()

Out[21]:

$$1$$

In [22]:

AntiCommutator(c,d).doit()

Out[22]:

$${c} {d} + {d} {c}$$

Commutators between fermions and bosons¶

In [23]:

Commutator(a, c).doit()

Out[23]:

$$0$$

Simplifying operator expressions¶

Test operator ordering of bosons¶

In [24]:

normal_ordered_form(Dagger(a) * a)

Out[24]:

$${{a}^\dagger} {a}$$

In [25]:

normal_ordered_form(a * Dagger(a))

Out[25]:

$$1 + {{a}^\dagger} {a}$$

In [26]:

normal_ordered_form(a * Dagger(a) * a)

Out[26]:

$${{a}^\dagger} \left({a}\right)^{2} + {a}$$

In [27]:

normal_ordered_form(a * a * Dagger(a))

Out[27]:

$${{a}^\dagger} \left({a}\right)^{2} + 2 {a}$$

In [28]:

normal_ordered_form(a * a * a * Dagger(a))

Out[28]:

$${{a}^\dagger} \left({a}\right)^{3} + 3 \left({a}\right)^{2}$$

In [29]:

normal_ordered_form(a * Dagger(a) * a * Dagger(a))

Out[29]:

$$1 + 3 {{a}^\dagger} {a} + \left({{a}^\dagger}\right)^{2} \left({a}\right)^{2}$$

In [30]:

normal_ordered_form(a * Dagger(a) * Dagger(a) * Dagger(a))

Out[30]:

$$3 \left({{a}^\dagger}\right)^{2} + \left({{a}^\dagger}\right)^{3} {a}$$

In [31]:

normal_ordered_form(a * Dagger(b))

Out[31]:

$${{b}^\dagger} {a} + \left[{a},{{b}^\dagger}\right]$$

In [32]:

normal_ordered_form(a * Dagger(b)).doit()

Out[32]:

$${a} {{b}^\dagger}$$

In [33]:

normal_ordered_form(a * Dagger(b), independent=True)

Out[33]:

$${{b}^\dagger} {a}$$

In [34]:

normal_ordered_form(a * Dagger(b) * Dagger(a))

Out[34]:

$${{b}^\dagger} + {{b}^\dagger} {{a}^\dagger} {a} + \left[{a},{{b}^\dagger}\right] {{a}^\dagger}$$

In [35]:

normal_ordered_form(a * Dagger(b) * Dagger(a), independent=True)

Out[35]:

$${{a}^\dagger} {{b}^\dagger} {a} + {{b}^\dagger}$$

Normal ordering¶

In [36]:

normal_order(Dagger(a) * a)

Out[36]:

$${{a}^\dagger} {a}$$

In [37]:

normal_order(a * Dagger(a))

Out[37]:

$${{a}^\dagger} {a}$$

In [38]:

normal_order(a * a * a * Dagger(a))

Out[38]:

$${{a}^\dagger} \left({a}\right)^{3}$$

In [39]:

normal_order(a * Dagger(a) * a * Dagger(a) * a)

Out[39]:

$$\left({{a}^\dagger}\right)^{2} \left({a}\right)^{3}$$

Wick's contraction:

In [40]:

a * Dagger(a) - normal_order(a * Dagger(a))

Out[40]:

$$- {{a}^\dagger} {a} + {a} {{a}^\dagger}$$

In [41]:

normal_ordered_form(a * Dagger(a) - normal_order(a * Dagger(a)))

Out[41]:

$$1$$

Test operator ordering of fermions¶

In [42]:

normal_ordered_form(Dagger(c) * c)

Out[42]:

$${{c}^\dagger} {c}$$

In [43]:

normal_ordered_form(c * Dagger(c))

Out[43]:

$$1 - {{c}^\dagger} {c}$$

In [44]:

normal_ordered_form(c * c * Dagger(c))

Out[44]:

$${{c}^\dagger} \left({c}\right)^{2}$$

In [45]:

normal_ordered_form(c * c * c * Dagger(c))

Out[45]:

$$- {{c}^\dagger} \left({c}\right)^{3} + \left({c}\right)^{2}$$

In [46]:

normal_ordered_form(c * Dagger(c) * c * Dagger(c))

Out[46]:

$$1 - {{c}^\dagger} {c} - \left({{c}^\dagger}\right)^{2} \left({c}\right)^{2}$$

In [47]:

normal_ordered_form(c * Dagger(d))

Out[47]:

$$\left\{{c},{{d}^\dagger}\right\} - {{d}^\dagger} {c}$$

In [48]:

normal_ordered_form(c * Dagger(d), independent=True)

Out[48]:

$$- {{d}^\dagger} {c}$$

Normal ordering¶

In [49]:

normal_order(Dagger(c) * c)

Out[49]:

$${{c}^\dagger} {c}$$

In [50]:

normal_order(c * Dagger(c))

Out[50]:

$$- {{c}^\dagger} {c}$$

In [51]:

normal_order(c * Dagger(c) * c * Dagger(c) * c)

Out[51]:

$$- \left({{c}^\dagger}\right)^{2} \left({c}\right)^{3}$$

Wick's contraction

In [52]:

c * Dagger(c) - normal_order(c * Dagger(c))

Out[52]:

$${{c}^\dagger} {c} + {c} {{c}^\dagger}$$

In [53]:

normal_ordered_form(c * Dagger(c) - normal_order(c * Dagger(c)))

Out[53]:

$$1$$

States¶

Boson Fock states¶

In [54]:

ket = BosonFockKet(2)

ket

Out[54]:

$${\left|2\right\rangle }$$

In [55]:

bra = BosonFockBra(2)

bra

Out[55]:

$${\left\langle 2\right|}$$

In [56]:

type(bra), type(ket)

Out[56]:

(sympy.physics.quantum.boson.BosonFockBra,
 sympy.physics.quantum.boson.BosonFockKet)

In [57]:

a * ket, qapply(a * ket)

Out[57]:

$$\begin{pmatrix}{a} {\left|2\right\rangle }, & \sqrt{2} {\left|1\right\rangle }\end{pmatrix}$$

In [58]:

bra * a, qapply(bra * a, dagger=True)

Out[58]:

$$\begin{pmatrix}{\left\langle 2\right|} {a}, & \sqrt{3} {\left\langle 3\right|}\end{pmatrix}$$

In [59]:

qapply(Dagger(ket) * ket)

Out[59]:

$$\left\langle 2 \right. {\left|2\right\rangle }$$

In [60]:

(Dagger(ket) * ket).doit()

Out[60]:

$$1$$

In [61]:

(bra * Dagger(bra)).doit()

Out[61]:

$$1$$

In [62]:

Dagger(ket) * Dagger(a) * ket, qapply(Dagger(ket) * Dagger(a) * ket).doit()

Out[62]:

$$\begin{pmatrix}{\left\langle 2\right|} {{a}^\dagger} {\left|2\right\rangle }, & 0\end{pmatrix}$$

In [63]:

(bra * Dagger(a) * Dagger(a) * a * a * ket).doit()

Out[63]:

$${\left\langle 2\right|} \left({{a}^\dagger}\right)^{2} \left({a}\right)^{2} {\left|2\right\rangle }$$

In [64]:

qapply(Dagger(ket) * Dagger(a) * a * ket).doit()

Out[64]:

$$2$$

In [65]:

qapply(bra * Dagger(a) * a * Dagger(bra)).doit()

Out[65]:

$$2$$

In [66]:

ket = BosonFockKet(4)
bra = BosonFockBra(2)

bra * a * a * ket, qapply(bra * a * a * ket).doit()

Out[66]:

$$\begin{pmatrix}{\left\langle 2\right|} \left({a}\right)^{2} {\left|4\right\rangle }, & 2 \sqrt{3}\end{pmatrix}$$

Coherent states¶

In [67]:

alpha, beta = symbols("alpha, beta")

In [68]:

ket = BosonCoherentKet(alpha)

ket

Out[68]:

$${\left|\alpha\right\rangle }$$

In [69]:

bra = BosonCoherentBra(beta)

bra

Out[69]:

$${\left\langle \beta\right|}$$

In [70]:

bra * ket

Out[70]:

$$\left\langle \beta \right. {\left|\alpha\right\rangle }$$

In [71]:

(bra * ket).doit()

Out[71]:

$$e^{\alpha \overline{\beta} - \frac{\left\lvert{\alpha}\right\rvert^{2}}{2} - \frac{\left\lvert{\beta}\right\rvert^{2}}{2}}$$

In [72]:

qapply((bra * qapply(a * ket))).doit()

Out[72]:

$$\alpha e^{\alpha \overline{\beta}} e^{- \frac{\left\lvert{\alpha}\right\rvert^{2}}{2}} e^{- \frac{\left\lvert{\beta}\right\rvert^{2}}{2}}$$

In [73]:

qapply(Dagger(ket) * Dagger(a) * a * ket, dagger=True)

Out[73]:

$$\alpha \alpha^{\dagger}$$

In [74]:

normal_ordered_form(a * Dagger(a))

Out[74]:

$$1 + {{a}^\dagger} {a}$$

In [75]:

qapply(bra * a * Dagger(a) * ket, dagger=True)

Out[75]:

$${\left\langle \beta\right|} {a} {{a}^\dagger} {\left|\alpha\right\rangle }$$

In [76]:

qapply(bra * normal_ordered_form(a * Dagger(a)) * ket, dagger=True)

Out[76]:

$$\alpha e^{- \frac{\left\lvert{\alpha}\right\rvert^{2}}{2}} e^{- \frac{\left\lvert{\beta}\right\rvert^{2}}{2}} \beta^{\dagger} \operatorname{transpose}{\left (e^{\alpha \overline{\beta}} \right )} + e^{\alpha \overline{\beta}} e^{- \frac{\left\lvert{\alpha}\right\rvert^{2}}{2}} e^{- \frac{\left\lvert{\beta}\right\rvert^{2}}{2}}$$

Fermion Fock states¶

In [77]:

ket = FermionFockKet(1)

ket

Out[77]:

$${\left|1\right\rangle }$$

In [78]:

bra = FermionFockBra(0)

bra

Out[78]:

$${\left\langle 0\right|}$$

In [79]:

(bra * ket).doit()

Out[79]:

$$0$$

In [80]:

c * ket

Out[80]:

$${c} {\left|1\right\rangle }$$

In [81]:

qapply(c * ket)

Out[81]:

$${\left|0\right\rangle }$$

In [82]:

qapply(c * c * ket)

Out[82]:

$$0$$

In [83]:

qapply(Dagger(c) * ket)

Out[83]:

$$0$$

In [84]:

qapply(Dagger(c) * c * ket)

Out[84]:

$${\left|1\right\rangle }$$

In [85]:

Dagger(c) * c

Out[85]:

$${{c}^\dagger} {c}$$

In [86]:

qapply(Dagger(ket) * Dagger(c) * c * ket).doit()

Out[86]:

$$1$$

In [87]:

qapply(bra * Dagger(c) * c * Dagger(bra)).doit()

Out[87]:

$$0$$

TensorProduct spaces¶

In [88]:

I2 = TensorProduct(IdentityOperator(), IdentityOperator())

I2

Out[88]:

$${{\mathcal{I}}}\otimes {{\mathcal{I}}}$$

In [89]:

psi = TensorProduct(BosonFockKet(1),BosonFockKet(2))

psi

Out[89]:

$${{\left|1\right\rangle }}\otimes {{\left|2\right\rangle }}$$

In [90]:

I2 * psi

Out[90]:

$${{\mathcal{I}}}\otimes {{\mathcal{I}}} {{\left|1\right\rangle }}\otimes {{\left|2\right\rangle }}$$

In [91]:

qapply(I2 * psi)

Out[91]:

$${{\left|1\right\rangle }}\otimes {{\left|2\right\rangle }}$$

In [93]:

a1 = TensorProduct(BosonOp("a"), IdentityOperator())
a2 = TensorProduct(IdentityOperator(), BosonOp("a"))

a1, a2

Out[93]:

$$\begin{pmatrix}{{a}}\otimes {{\mathcal{I}}}, & {{\mathcal{I}}}\otimes {{a}}\end{pmatrix}$$

In [94]:

a1 * psi

Out[94]:

$${{a}}\otimes {{\mathcal{I}}} {{\left|1\right\rangle }}\otimes {{\left|2\right\rangle }}$$

In [95]:

psi1 = qapply(a1 * psi)

psi1

Out[95]:

$${{\left|0\right\rangle }}\otimes {{\left|2\right\rangle }}$$

In [96]:

psi2 = qapply(a2 * psi)

psi2

Out[96]:

$$\sqrt{2} {{\left|1\right\rangle }}\otimes {{\left|1\right\rangle }}$$

In [97]:

qapply(Dagger(psi1) * psi2).doit()

Out[97]:

$$0$$

In [98]:

qapply(Dagger(psi2) * psi2).doit()

Out[98]:

$$2 \cdot {1}\otimes {1}$$

In [99]:

e = TensorProduct(IdentityOperator(), BosonFockKet(1))

e

Out[99]:

$${{\mathcal{I}}}\otimes {{\left|1\right\rangle }}$$

In [100]:

qapply(Dagger(psi2) * e).doit()

Out[100]:

$$\sqrt{2} {{\left\langle 1\right|}}\otimes {1}$$

In [101]:

qapply(qapply(Dagger(psi2) * e) * TensorProduct(BosonFockKet(1), IdentityOperator())).doit()

Out[101]:

$$\sqrt{2} \cdot {1}\otimes {{\mathcal{I}}}$$

In [102]:

bra = TensorProduct(BosonFockBra(1),BosonFockBra(2))
ket = TensorProduct(BosonFockKet(1),BosonFockKet(2))

In [103]:

qapply(bra * ket).doit()

Out[103]:

$${1}\otimes {1}$$

In [104]:

e = qapply(qapply(TensorProduct(IdentityOperator(), BosonCoherentBra(2)) * 
       TensorProduct(a, normal_ordered_form(Dagger(b)*b*Dagger(b)*b)) * 
       TensorProduct(IdentityOperator(), BosonCoherentKet(2))))

e

Out[104]:

$$2 {{a}}\otimes \left({{\left\langle 2\right|} {{b}^\dagger} {\left|2\right\rangle }}\right) + 4 {{a}}\otimes \left({{\left\langle 2\right|} \left({{b}^\dagger}\right)^{2} {\left|2\right\rangle }}\right)$$

In [105]:

ket = TensorProduct(IdentityOperator(), BosonFockKet(2))
bra = Dagger(ket)

qapply(qapply(bra * TensorProduct(a, normal_ordered_form(Dagger(b)*b*Dagger(b)*b)) * ket)).doit()

Out[105]:

$$4 {{a}}\otimes {1}$$

Versions¶

In [106]:

#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py
%reload_ext version_information

%version_information sympy

Out[106]:

Software	Version
Python	3.4.1 (default, May 21 2014, 01:39:38) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]
IPython	2.1.0
OS	posix [darwin]
sympy	0.7.4.1-git
Thu Jun 05 15:32:12 2014 JST