006-001-energy-operators-the-hamiltonian-and-schrodinger.ipynb

In [1]:
from sympy.physics.quantum import *
from sympy.physics.quantum.cartesian import *
from sympy.physics.quantum.operator import *
from sympy.physics.quantum.state import *
from sympy import *
from sympy.core.relational import *
from sympy.physics.units import degree
from sympy.abc import a, b, x, y, z, t, alpha, n, theta, h, f, lamda, i, k
from sympy.abc import w, u, d, beta, r, psi, o, l, c, gamma, phi, j, p, epsilon
In [2]:
H = HermitianOperator('H')
ketPsi = Ket(psi)
E = symbols('E', complex = True)
E_j = symbols('E_j', complex = True)
E_i = symbols('E_i', complex = True)
m = symbols('m', constant = True)
psiOfX = Function(psi)(x)
UOfX = Function('U')(x)
blankSpace = Dummy('')
momentumOperator = HermitianOperator('P')
potential = symbols('V')
M = UnitaryOperator('M')
U = UnitaryOperator('U')
U_0 = UnitaryOperator('U_0')
U_epsilon = UnitaryOperator('U_epsilon')
identity = IdentityOperator('I')
ketPsi_TEpsilon = Ket('\psi_{t+\epsilon}')
ketPsi_T = Ket('psi_t')
ketPsi_X = Ket('psi_x')
epsilon = symbols('epsilon', constant = True)
lamda_i = symbols('lambda_i')
AverageValue = symbols('A_v')
proability = Function('P')
alpha_i = symbols('alpha_i', complex =True)
alpha_j = symbols('alpha_j', complex =True)
alpha_j_0 = Function('alpha_j')(0)
ketX_i = Ket('x_i')
braX_i = Bra('x_i')
ketX_j = Ket('x_j')
braX_j = Bra('x_j')
M = Operator('M')
ketPsi_j = Ket('psi_j')
braPsi_j = Bra('psi_j')
ketPsi_i = Ket('psi_i')
braPsi_i = Bra('psi_i')
lamda_j = symbols('lambda_j', complex = True)
In [3]:
prodOfH_KetPsi = Eq(H * ketPsi, lamda * ketPsi)
display(prodOfH_KetPsi)
$\displaystyle H {\left|\psi\right\rangle } = \lambda {\left|\psi\right\rangle }$
In [4]:
prodOfH_KetPsi = Eq(H * ketPsi, E * ketPsi)
display(prodOfH_KetPsi)
$\displaystyle H {\left|\psi\right\rangle } = E {\left|\psi\right\rangle }$
In [5]:
timeIndependantSchrodingerEq = Eq(Add(Mul(-hbar**2 / (2*m), Derivative(psiOfX, x, 2)), Mul(UOfX, psiOfX)) , E * psiOfX)
display(timeIndependantSchrodingerEq)
$\displaystyle U{\left(x \right)} \psi{\left(x \right)} - \frac{\hbar^{2} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)}}{2 m} = E \psi{\left(x \right)}$
In [6]:
solOfH = Eq(Add(Mul(-hbar**2 / (2*m), Derivative(blankSpace, x, 2)), UOfX) * ketPsi , E * ketPsi)
display(solOfH)
$\displaystyle \left(U{\left(x \right)} - \frac{\hbar^{2} \frac{d^{2}}{d x^{2}} }{2 m}\right) {\left|\psi\right\rangle } = E {\left|\psi\right\rangle }$
In [7]:
valOfmomentumOperator = Eq(momentumOperator, -I * hbar * Derivative(blankSpace, x))
display(valOfmomentumOperator)
$\displaystyle P = - \hbar i \frac{d}{d x} $
In [8]:
valOfH = Eq(H, momentumOperator**2 / (2 * m) + potential)
display(valOfH)
$\displaystyle H = V + \frac{P^{2}}{2 m}$
In [9]:
valOfH = Eq(H, Pow(-I * hbar * Derivative(blankSpace, x), 2, evaluate =False) / (2 * m) + potential)
display(valOfH)
$\displaystyle H = V + \frac{\left(-1\right) \hbar^{2} \left(\frac{d}{d x} \right)^{2}}{2 m}$
In [10]:
valOfH = Eq(H, Pow(-I * hbar, 2)* Derivative(blankSpace, x, 2) / (2 * m) + potential)
display(valOfH)
$\displaystyle H = V - \frac{\hbar^{2} \frac{d^{2}}{d x^{2}} }{2 m}$
In [11]:
prodOfM_MDagger = Eq(Mul(M, Dagger(M), evaluate = False), identity, evaluate = False)
display(prodOfM_MDagger)
$\displaystyle M M^{\dagger} = {\mathcal{I}}$
In [12]:
prodOfU_UDagger = Eq(Mul(U, Dagger(U), evaluate = False), Mul(U, Dagger(U)), evaluate = False)
display(prodOfU_UDagger)
$\displaystyle U U^{-1} = 1$
In [13]:
valOfU_0 = Eq(U_0, identity, evaluate = False)
display(valOfU_0)
$\displaystyle U_{0} = {\mathcal{I}}$
In [14]:
valOfU_epsilon = Eq(U_epsilon, Add(identity, - I * epsilon * H / hbar), evaluate = False)
display(valOfU_epsilon)
$\displaystyle U_{\epsilon} = - \frac{i \epsilon H}{\hbar} + {\mathcal{I}}$
In [15]:
prodOfU_UDagger = Eq(Mul(U, Dagger(U), evaluate = False), Mul(U, Dagger(U)), evaluate = False)
display(prodOfU_UDagger)
$\displaystyle U U^{-1} = 1$
In [16]:
prodOfU_epsilon_U_epsilonDagger = Eq(identity,
                                     Mul(Add(identity,
                                             - I * epsilon * H / hbar),
                                         Dagger(Add(identity,
                                                    - I * epsilon * H / hbar))),
                                     evaluate = False)
display(prodOfU_epsilon_U_epsilonDagger)
$\displaystyle {\mathcal{I}} = \left(- \frac{i \epsilon H}{\hbar} + {\mathcal{I}}\right) \left(\frac{i \epsilon^{\dagger} H}{\hbar} + {\mathcal{I}}\right)$
In [17]:
prodOfU_epsilon_U_epsilonDagger = Eq(identity,
                                     Add(identity,
                                             I * epsilon * Add(Dagger(H), -H, evaluate = False) / hbar),
                                     evaluate = False)
display(prodOfU_epsilon_U_epsilonDagger)
$\displaystyle {\mathcal{I}} = \frac{i \epsilon \left(- H + H\right)}{\hbar} + {\mathcal{I}}$
In [18]:
solForDerivativeOfKetPsi_T = Eq(ketPsi_TEpsilon, U_epsilon * ketPsi_T)
display(solForDerivativeOfKetPsi_T)
$\displaystyle {\left|\psi_{t+\epsilon}\right\rangle } = U_{\epsilon} {\left|\psi_{t}\right\rangle }$
In [19]:
solForDerivativeOfKetPsi_T = Eq(ketPsi_TEpsilon, Add(ketPsi_T, - I * epsilon * H * ketPsi_T / hbar))
display(solForDerivativeOfKetPsi_T)
$\displaystyle {\left|\psi_{t+\epsilon}\right\rangle } = - \frac{i \epsilon H {\left|\psi_{t}\right\rangle }}{\hbar} + {\left|\psi_{t}\right\rangle }$
In [20]:
solForDerivativeOfKetPsi_T = Eq(Add(ketPsi_TEpsilon,- ketPsi_T) / epsilon, - I * H * ketPsi_T / hbar)
display(solForDerivativeOfKetPsi_T)
$\displaystyle \frac{{\left|\psi_{t+\epsilon}\right\rangle } - {\left|\psi_{t}\right\rangle }}{\epsilon} = - \frac{i H {\left|\psi_{t}\right\rangle }}{\hbar}$
In [21]:
solForDerivativeOfKetPsi_T = Eq(Derivative(ketPsi_T, t), - I * H * ketPsi_T / hbar)
display(solForDerivativeOfKetPsi_T)
$\displaystyle \frac{d}{d t} {\left|\psi_{t}\right\rangle } = - \frac{i H {\left|\psi_{t}\right\rangle }}{\hbar}$
In [22]:
timeDependantSchrodingerEq = Eq(Derivative(ketPsi_T, t) * hbar / - I, H * ketPsi_T)
display(timeDependantSchrodingerEq)
$\displaystyle \hbar i \frac{d}{d t} {\left|\psi_{t}\right\rangle } = H {\left|\psi_{t}\right\rangle }$
In [23]:
prodOfMomentumOperator_ketPsi_X = Eq(momentumOperator * ketPsi,-I * hbar * Derivative(ketPsi, x))
display(prodOfMomentumOperator_ketPsi_X)
$\displaystyle P {\left|\psi\right\rangle } = - \hbar i \frac{d}{d x} {\left|\psi\right\rangle }$
In [24]:
averageValEq = Eq(AverageValue, Sum(lamda_i * proability(lamda_i), (i, 1, n)))
display(averageValEq)
$\displaystyle A_{v} = \sum_{i=1}^{n} \lambda_{i} P{\left(\lambda_{i} \right)}$
In [25]:
valOfKetPsi = Eq(ketPsi,
                 Sum(alpha_i * ketX_i, (i, 1, n)),
                 evaluate = False)
display(valOfKetPsi)
$\displaystyle {\left|\psi\right\rangle } = \sum_{i=1}^{n} \alpha_{i} {\left|x_{i}\right\rangle }$
In [26]:
prodOfAlpha_i_ConjugateAlpha_i = Eq(alpha_i * conjugate(alpha_i), proability( ketX_i))
display(prodOfAlpha_i_ConjugateAlpha_i)
$\displaystyle \alpha_{i} \overline{\alpha_{i}} = P{\left({\left|x_{i}\right\rangle } \right)}$
In [27]:
timeHonoredEq = Eq(M * ketPsi_j, lamda_j * ketPsi_j)
display(timeHonoredEq)
$\displaystyle M {\left|\psi_{j}\right\rangle } = \lambda_{j} {\left|\psi_{j}\right\rangle }$
In [28]:
innerprodOfBraPsi_j_KetPsi_j = Eq(Mul(braPsi_j, M, ketPsi_j), Mul(Sum(braX_i * conjugate(alpha_i), (i, 1, n)),
                                                                  M, 
                                                                  Sum(ketX_j * alpha_j, (j, 1, n))))
display(innerprodOfBraPsi_j_KetPsi_j)
$\displaystyle {\left\langle \psi_{j}\right|} M {\left|\psi_{j}\right\rangle } = \left(\sum_{i=1}^{n} \overline{\alpha_{i}} {\left\langle x_{i}\right|}\right) M \sum_{j=1}^{n} \alpha_{j} {\left|x_{j}\right\rangle }$
In [29]:
innerprodOfBraPsi_j_KetPsi_j = Eq(Mul(braPsi_j, M, ketPsi_j), Sum(Mul(conjugate(alpha_i),
                                                                      alpha_j,
                                                                      braX_i,
                                                                      M,
                                                                      ketX_j),
                                                                  (i, 1, n),
                                                                  (j, 1, n)))
display(innerprodOfBraPsi_j_KetPsi_j)
$\displaystyle {\left\langle \psi_{j}\right|} M {\left|\psi_{j}\right\rangle } = \sum_{\substack{1 \leq i \leq n\\1 \leq j \leq n}} \alpha_{j} \overline{\alpha_{i}} {\left\langle x_{i}\right|} M {\left|x_{j}\right\rangle }$
In [30]:
innerprodOfBraPsi_j_KetPsi_j = Eq(Mul(braPsi_j, M, ketPsi_j), Mul(Sum(Mul(conjugate(alpha_i),
                                                                          alpha_j,
                                                                          lamda_j,
                                                                          InnerProduct(braX_i,
                                                                                       ketX_j)),
                                                                      (i, 1, n),
                                                                      (j, 1, n))))
display(innerprodOfBraPsi_j_KetPsi_j)
$\displaystyle {\left\langle \psi_{j}\right|} M {\left|\psi_{j}\right\rangle } = \sum_{\substack{1 \leq i \leq n\\1 \leq j \leq n}} \alpha_{j} \lambda_{j} \overline{\alpha_{i}} \left\langle x_{i} \right. {\left|x_{j}\right\rangle }$
In [31]:
innerprodOfBraPsi_j_KetPsi_j = Eq(Mul(braPsi_j, M, ketPsi_j), Sum(Mul(conjugate(alpha_j),
                                                                      alpha_j,
                                                                      lamda_j),
                                                                  (j, 1, n)))
display(innerprodOfBraPsi_j_KetPsi_j)
$\displaystyle {\left\langle \psi_{j}\right|} M {\left|\psi_{j}\right\rangle } = \sum_{j=1}^{n} \alpha_{j} \lambda_{j} \overline{\alpha_{j}}$
In [32]:
innerprodOfBraPsi_j_KetPsi_j = Eq(Eq(Mul(braPsi_j, M, ketPsi_j), Sum(lamda_j * proability(lamda_j), (j, 1, n))), AverageValue)
display(innerprodOfBraPsi_j_KetPsi_j)
$\displaystyle {\left\langle \psi_{j}\right|} M {\left|\psi_{j}\right\rangle } = \sum_{j=1}^{n} \lambda_{j} P{\left(\lambda_{j} \right)} = A_{v}$
In [33]:
prodOfH_ketPsi_i = Eq(H * ketPsi_i, E_i * ketPsi_i)
display(prodOfH_ketPsi_i)
$\displaystyle H {\left|\psi_{i}\right\rangle } = E_{i} {\left|\psi_{i}\right\rangle }$
In [34]:
valOfKetPsi = Eq(ketPsi,
                 Sum(alpha_j * ketX_j, (j, 1, n)),
                 evaluate = False)
display(valOfKetPsi)
$\displaystyle {\left|\psi\right\rangle } = \sum_{j=1}^{n} \alpha_{j} {\left|x_{j}\right\rangle }$
In [35]:
diffOfKetPsi = Eq(Derivative(ketPsi, t), Sum(Derivative(alpha_j, t) * ketX_j, (j, 1, n)))
display(diffOfKetPsi)
$\displaystyle \frac{d}{d t} {\left|\psi\right\rangle } = \sum_{j=1}^{n} \frac{d}{d t} \alpha_{j} {\left|x_{j}\right\rangle }$
In [36]:
timeDependantSchrodingerEq = Eq(Derivative(ketPsi, t) * hbar / - I, H * ketPsi)
display(timeDependantSchrodingerEq)
$\displaystyle \hbar i \frac{d}{d t} {\left|\psi\right\rangle } = H {\left|\psi\right\rangle }$
In [37]:
diffOfKetPsi2 = Eq(Derivative(ketPsi, t), H * ketPsi * -I/ hbar)
display(diffOfKetPsi2)
$\displaystyle \frac{d}{d t} {\left|\psi\right\rangle } = - \frac{i H {\left|\psi\right\rangle }}{\hbar}$
In [38]:
diffOfKetPsi2 = Eq(Derivative(ketPsi, t), H * Sum(alpha_j * ketX_j, (j, 1, n)) * -I/ hbar)
display(diffOfKetPsi2)
$\displaystyle \frac{d}{d t} {\left|\psi\right\rangle } = - \frac{i H \sum_{j=1}^{n} \alpha_{j} {\left|x_{j}\right\rangle }}{\hbar}$
In [39]:
diffOfKetPsi2 = Eq(Derivative(ketPsi, t), E_j * Sum(alpha_j * ketX_j, (j, 1, n)) * -I/ hbar)
display(diffOfKetPsi2)
$\displaystyle \frac{d}{d t} {\left|\psi\right\rangle } = - \frac{i E_{j} \sum_{j=1}^{n} \alpha_{j} {\left|x_{j}\right\rangle }}{\hbar}$
In [40]:
diffOfAlpha_j = Eq(Derivative(alpha_j, t), E_j * alpha_j * -I/ hbar)
display(diffOfAlpha_j)
$\displaystyle \frac{d}{d t} \alpha_{j} = - \frac{i E_{j} \alpha_{j}}{\hbar}$
In [41]:
valOfAlpha_j = Eq(alpha_j, alpha_j_0 * exp(E_j * -I * t/ hbar))
display(valOfAlpha_j)
$\displaystyle \alpha_{j} = \alpha_{j}{\left(0 \right)} e^{- \frac{i E_{j} t}{\hbar}}$
In [42]:
waveFunc = Eq(ketPsi,
              Sum(alpha_j_0 * exp(E_j * -I * t/ hbar) * ketX_j, (j, 1, n)),
              evaluate = False)
display(waveFunc)
$\displaystyle {\left|\psi\right\rangle } = \sum_{j=1}^{n} \alpha_{j}{\left(0 \right)} e^{- \frac{i E_{j} t}{\hbar}} {\left|x_{j}\right\rangle }$
In [43]:
expansionOfExponential =  Eq(exp(-I*x), cos(x) - I * sin(x))
display(expansionOfExponential)
$\displaystyle e^{- i x} = - i \sin{\left(x \right)} + \cos{\left(x \right)}$
In [ ]:

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