Example Notebook for sho1d.py

Import the sho1d.py file as well as the test_sho1d.py file



In [1]:

    
%load_ext sympy.interactive.ipythonprinting 
from sympy import *
from IPython.display import display_pretty
from sympy.physics.quantum import *
from sympy.physics.quantum.sho1d import *
from sympy.physics.quantum.tests.test_sho1d import *









    



The sympy.interactive.ipythonprinting extension is already loaded. To reload it, use:
  %reload_ext sympy.interactive.ipythonprinting

Printing Of Operators

Create a raising and lowering operator and make sure they print correctly



In [2]:

    
ad = RaisingOp('a')
a = LoweringOp('a')



In [3]:

    
ad









    Out[3]:





RaisingOp(a)



In [4]:

    
a









    Out[4]:





a



In [5]:

    
print latex(ad)
print latex(a)









    



a^{\dag}
a



In [6]:

    
display_pretty(ad)
display_pretty(a)









    





RaisingOp(a)






    





a



In [7]:

    
print srepr(ad)
print srepr(a)









    



RaisingOp(Symbol('a'))
LoweringOp(Symbol('a'))



In [8]:

    
print repr(ad)
print repr(a)









    



RaisingOp(a)
a

Printing of States

Create a simple harmonic state and check its printing



In [9]:

    
k = SHOKet('k')
b = SHOBra('b')



In [10]:

    
k









    Out[10]:





|k>



In [11]:

    
b









    Out[11]:





<b|



In [12]:

    
print pretty(k)
print pretty(b)









    



❘k⟩
⟨b❘



In [13]:

    
print latex(k)
print latex(b)









    



{\left|k\right\rangle }
{\left\langle b\right|}



In [14]:

    
print srepr(k)
print srepr(b)









    



SHOKet(Symbol('k'))
SHOBra(Symbol('b'))

Properties

Take the dagger of the raising and lowering operators. They should return each other.



In [15]:

    
Dagger(ad)









    Out[15]:





a



In [16]:

    
Dagger(a)









    Out[16]:





RaisingOp(a)

Check Commutators of the raising and lowering operators



In [17]:

    
Commutator(ad,a).doit()









    Out[17]:





-1



In [18]:

    
Commutator(a,ad).doit()









    Out[18]:





1

Take a look at the dual states of the bra and ket



In [19]:

    
k.dual









    Out[19]:





<k|



In [20]:

    
b.dual









    Out[20]:





|b>

Taking the InnerProduct of the bra and ket will return the KroneckerDelta function



In [21]:

    
InnerProduct(b,k).doit()









    Out[21]:





KroneckerDelta(k, b)

Take a look at how the raising and lowering operators act on states. We use qapply to apply an operator to a state



In [22]:

    
qapply(ad*k)









    Out[22]:





sqrt(k + 1)*|k + 1>



In [23]:

    
qapply(a*k)









    Out[23]:





sqrt(k)*|k - 1>

But the states may have an explicit energy level. Let's look at the ground and first excited states



In [24]:

    
kg = SHOKet(0)
kf = SHOKet(1)



In [25]:

    
qapply(ad*kg)









    Out[25]:





|1>



In [26]:

    
qapply(ad*kf)









    Out[26]:





sqrt(2)*|2>



In [27]:

    
qapply(a*kg)









    Out[27]:





0



In [28]:

    
qapply(a*kf)









    Out[28]:





|0>

Notice that akg is 0 and akf is the |0> the ground state.

NumberOp & Hamiltonian

Let's look at the Number Operator and Hamiltonian Operator



In [29]:

    
k = SHOKet('k')
ad = RaisingOp('a')
a = LoweringOp('a')
N = NumberOp('N')
H = Hamiltonian('H')

The number operator is simply expressed as ad*a



In [30]:

    
N().rewrite('a').doit()









    Out[30]:





RaisingOp(a)*a

The number operator expressed in terms of the position and momentum operators



In [31]:

    
N().rewrite('xp').doit()









    Out[31]:





-1/2 + (m**2*omega**2*X**2 + Px**2)/(2*hbar*m*omega)

It can also be expressed in terms of the Hamiltonian operator



In [32]:

    
N().rewrite('H').doit()









    Out[32]:





-1/2 + H/(hbar*omega)

The Hamiltonian operator can be expressed in terms of the raising and lowering operators, position and momentum operators, and the number operator



In [33]:

    
H().rewrite('a').doit()









    Out[33]:





hbar*omega*(1/2 + RaisingOp(a)*a)



In [34]:

    
H().rewrite('xp').doit()









    Out[34]:





(m**2*omega**2*X**2 + Px**2)/(2*m)



In [35]:

    
H().rewrite('N').doit()









    Out[35]:





hbar*omega*(1/2 + N)

The raising and lowering operators can also be expressed in terms of the position and momentum operators



In [36]:

    
ad().rewrite('xp').doit()









    Out[36]:





sqrt(2)*(m*omega*X - I*Px)/(2*sqrt(hbar)*sqrt(m*omega))



In [37]:

    
a().rewrite('xp').doit()









    Out[37]:





sqrt(2)*(m*omega*X + I*Px)/(2*sqrt(hbar)*sqrt(m*omega))

Properties

Let's take a look at how the NumberOp and Hamiltonian act on states



In [38]:

    
qapply(N*k)









    Out[38]:





k*|k>

Apply the Number operator to a state returns the state times the ket



In [39]:

    
ks = SHOKet(2)
qapply(N*ks)









    Out[39]:





2*|2>



In [40]:

    
qapply(H*k)









    Out[40]:





hbar*k*omega*|k> + hbar*omega*|k>/2

Let's see how the operators commute with each other



In [41]:

    
Commutator(N,ad).doit()









    Out[41]:





RaisingOp(a)



In [42]:

    
Commutator(N,a).doit()









    Out[42]:





-a



In [43]:

    
Commutator(N,H).doit()









    Out[43]:





0

Representation

We can express the operators in NumberOp basis. There are different ways to create a matrix in Python, we will use 3 different ways.

Sympy



In [44]:

    
represent(ad, basis=N, ndim=4, format='sympy')









    Out[44]:





[0,       0,       0, 0]
[1,       0,       0, 0]
[0, sqrt(2),       0, 0]
[0,       0, sqrt(3), 0]

Numpy



In [45]:

    
represent(ad, basis=N, ndim=5, format='numpy')









    Out[45]:





array([[ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
       [ 1.        ,  0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  1.41421356,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  1.73205081,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  2.        ,  0.        ]])

Scipy.Sparse



In [46]:

    
represent(ad, basis=N, ndim=4, format='scipy.sparse', spmatrix='lil')









    Out[46]:





<4x4 sparse matrix of type '<type 'numpy.float64'>'
	with 3 stored elements in Compressed Sparse Row format>



In [47]:

    
print represent(ad, basis=N, ndim=4, format='scipy.sparse', spmatrix='lil')









    



  (1, 0)	1.0
  (2, 1)	1.41421356237
  (3, 2)	1.73205080757

The same can be done for the other operators



In [48]:

    
represent(a, basis=N, ndim=4, format='sympy')









    Out[48]:





[0, 1,       0,       0]
[0, 0, sqrt(2),       0]
[0, 0,       0, sqrt(3)]
[0, 0,       0,       0]



In [49]:

    
represent(N, basis=N, ndim=4, format='sympy')









    Out[49]:





[0, 0, 0, 0]
[0, 1, 0, 0]
[0, 0, 2, 0]
[0, 0, 0, 3]



In [50]:

    
represent(H, basis=N, ndim=4, format='sympy')









    Out[50]:





[hbar*omega/2,              0,              0,              0]
[           0, 3*hbar*omega/2,              0,              0]
[           0,              0, 5*hbar*omega/2,              0]
[           0,              0,              0, 7*hbar*omega/2]

Ket and Bra Representation



In [51]:

    
k0 = SHOKet(0)
k1 = SHOKet(1)
b0 = SHOBra(0)
b1 = SHOBra(1)



In [52]:

    
print represent(k0, basis=N, ndim=5, format='sympy')









    



[1]
[0]
[0]
[0]
[0]



In [53]:

    
print represent(k1, basis=N, ndim=5, format='sympy')









    



[0]
[1]
[0]
[0]
[0]



In [54]:

    
print represent(b0, basis=N, ndim=5, format='sympy')









    



[1, 0, 0, 0, 0]



In [55]:

    
print represent(b1, basis=N, ndim=5, format='sympy')









    



[0, 1, 0, 0, 0]



In [ ]: