Notebook

SageMath para físicos¶

Rogério T. C.

Aula III¶

1. Vetores e matrizes¶

In [1]:

%display latex
reset()

Vetores e estrutura¶

Vetores são constrídos pelo comando vector, com sintaxe vector([x1,x2,x3,...,xn])

In [2]:

v = vector([x,x^8,5])

In [3]:

print(parent(v))

Vector space of dimension 3 over Symbolic Ring

In [ ]:

Soma, multiplicação escalar e produto vetorial¶

In [4]:

w=vector([5,2,1])

In [5]:

v.cross_product(w)

Out[5]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(x^{8} - 10,\,-x + 25,\,-5 \, x^{8} + 2 \, x\right)$

Produto interno¶

In [6]:

v*w

Out[6]:

$\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x^{8} + 5 \, x + 5$

In [7]:

v.dot_product(w)

Out[7]:

$\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x^{8} + 5 \, x + 5$

In [ ]:

Hermitiano¶

In [8]:

var('y', domain='real')

Out[8]:

$\newcommand{\Bold}[1]{\mathbf{#1}}y$

In [9]:

z = vector([I*2,x,y]);

In [10]:

z*w

Out[10]:

$\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x + y + 10 i$

In [11]:

z.hermitian_inner_product(w)

Out[11]:

$\newcommand{\Bold}[1]{\mathbf{#1}}y + 2 \, \overline{x} - 10 i$

Norma¶

In [12]:

w.norm()

Out[12]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{30}$

In [13]:

z.norm()

Out[13]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{y^{2} + {\left| x \right|}^{2} + 4}$

In [ ]:

Matrizes¶

In [14]:

a = matrix(SR,[[1,2,3],[5,23,8],[8,5,1]]);a

Out[14]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 1 & 2 & 3 \\ 5 & 23 & 8 \\ 8 & 5 & 1 \end{array}\right)$

In [15]:

(a).transpose()*z

Out[15]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(5 \, x + 8 \, y + 2 i,\,23 \, x + 5 \, y + 4 i,\,8 \, x + y + 6 i\right)$

In [16]:

z*a

Out[16]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(5 \, x + 8 \, y + 2 i,\,23 \, x + 5 \, y + 4 i,\,8 \, x + y + 6 i\right)$

Matrizes especiais¶

In [17]:

matrix.zero(10,15)

Out[17]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrrrrr} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$

In [18]:

matrix.identity(4)

Out[18]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [19]:

matrix.ones(3)

Out[19]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right)$

In [20]:

d = matrix.diagonal([y,y,y^2])

In [ ]:

Determinante, traço, transposta, inversa, conjugada¶

In [21]:

d.det()

Out[21]:

$\newcommand{\Bold}[1]{\mathbf{#1}}y^{4}$

In [22]:

d.trace()

Out[22]:

$\newcommand{\Bold}[1]{\mathbf{#1}}y^{2} + 2 \, y$

In [23]:

d.transpose()

Out[23]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} y & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & y^{2} \end{array}\right)$

In [24]:

d.conjugate()

Out[24]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} y & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & y^{2} \end{array}\right)$

In [25]:

d.inverse()

Out[25]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} \frac{1}{y} & 0 & 0 \\ 0 & \frac{1}{y} & 0 \\ 0 & 0 & \frac{1}{y^{2}} \end{array}\right)$

In [ ]:

Pausa - transformação linear¶

In [26]:

T = linear_transformation(a)

In [27]:

Out[27]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\text{vector space morphism from } \text{SR}^{3}\text{ to } \text{SR}^{3}\text{ represented by the matrix } \left(\begin{array}{rrr} 1 & 2 & 3 \\ 5 & 23 & 8 \\ 8 & 5 & 1 \end{array}\right)$

In [28]:

print(T)

Vector space morphism represented by the matrix:
[ 1  2  3]
[ 5 23  8]
[ 8  5  1]
Domain: Vector space of dimension 3 over Symbolic Ring
Codomain: Vector space of dimension 3 over Symbolic Ring

In [29]:

T.is_bijective()

Out[29]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Autovalores, auto vetores e diagonalização¶

In [30]:

d.charpoly()

Out[30]:

$\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + \left(-y^{2} - 2 \, y\right) x^{2} + \left(2 \, y^{3} + y^{2}\right) x - y^{4}$

In [31]:

a.eigenvalues()

Out[31]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-\frac{1}{2} \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{353 \, {\left(-i \, \sqrt{3} + 1\right)}}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}, -\frac{1}{2} \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} {\left(-i \, \sqrt{3} + 1\right)} - \frac{353 \, {\left(i \, \sqrt{3} + 1\right)}}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} + \frac{706}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}\right]$

In [32]:

a.eigenvectors_right()

Out[32]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(-\frac{1}{2} \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{353 \, {\left(-i \, \sqrt{3} + 1\right)}}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}, \left[\right], 1\right), \left(-\frac{1}{2} \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} {\left(-i \, \sqrt{3} + 1\right)} - \frac{353 \, {\left(i \, \sqrt{3} + 1\right)}}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}, \left[\right], 1\right), \left({\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}} + \frac{706}{9 \, {\left(\frac{1}{18} i \, \sqrt{24785605} \sqrt{3} + \frac{27173}{54}\right)}^{\frac{1}{3}}} + \frac{25}{3}, \left[\right], 1\right)\right]$

In [33]:

k = matrix([[1,1,-3],[0,4,0],[-3,3,1]])

In [34]:

k.eigenvectors_right()

Out[34]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(-2, \left[\left(1,\,0,\,1\right)\right], 1\right), \left(4, \left[\left(1,\,0,\,-1\right)\right], 2\right)\right]$

In [35]:

k.eigenmatrix_right()

Out[35]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & -1 & 0 \end{array}\right)\right)$

In [ ]:

Exponencial de matrizes¶

$e^A \equiv I+\sum_{n=1}^{\infty}\frac{A^n}{n!},$

onde $I\,$ é a matriz identidade

In [36]:

m1 = matrix([[1,2],[2,2]]);m1

Out[36]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 2 \\ 2 & 2 \end{array}\right)$

In [37]:

exp(m1)

Out[37]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -\frac{1}{34} \, {\left({\left(\sqrt{17} - 17\right)} e^{\sqrt{17}} - \sqrt{17} - 17\right)} e^{\left(-\frac{1}{2} \, \sqrt{17} + \frac{3}{2}\right)} & \frac{2}{17} \, {\left(\sqrt{17} e^{\sqrt{17}} - \sqrt{17}\right)} e^{\left(-\frac{1}{2} \, \sqrt{17} + \frac{3}{2}\right)} \\ \frac{2}{17} \, {\left(\sqrt{17} e^{\sqrt{17}} - \sqrt{17}\right)} e^{\left(-\frac{1}{2} \, \sqrt{17} + \frac{3}{2}\right)} & \frac{1}{34} \, {\left({\left(\sqrt{17} + 17\right)} e^{\sqrt{17}} - \sqrt{17} + 17\right)} e^{\left(-\frac{1}{2} \, \sqrt{17} + \frac{3}{2}\right)} \end{array}\right)$

In [ ]:

Exercício¶

Dada a matriz de Pauli

$\sigma _{2}={\begin{pmatrix}0&i\\-i&0\end{pmatrix}}.$

Calcule $e^{i\frac{\theta}{2}\sigma_2}$

In [38]:

var('theta')

Out[38]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\theta$

In [39]:

exp((i/2)*theta*matrix([[0,i],[-i,0]]))

Out[39]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \frac{1}{2} \, {\left(e^{\left(i \, \theta\right)} + 1\right)} e^{\left(-\frac{1}{2} i \, \theta\right)} & \frac{1}{2} \, {\left(i \, e^{\left(i \, \theta\right)} - i\right)} e^{\left(-\frac{1}{2} i \, \theta\right)} \\ -\frac{1}{2} \, {\left(i \, e^{\left(i \, \theta\right)} - i\right)} e^{\left(-\frac{1}{2} i \, \theta\right)} & \frac{1}{2} \, {\left(e^{\left(i \, \theta\right)} + 1\right)} e^{\left(-\frac{1}{2} i \, \theta\right)} \end{array}\right)$

In [40]:

x.simplify_rectform()

Out[40]:

$\newcommand{\Bold}[1]{\mathbf{#1}}x$

In [ ]:

Geradores do grupo de Lorentz (Representação de spin 1 da algébra de Lie $\mathfrak{so}(1,3)$ )¶

Rotações

In [41]:

J1 = I*matrix([[0,0,0,0],[0,0,0,0],[0,0,0,-1],[0,0,1,0]])
J2 = I*matrix([[0,0,0,0],[0,0,0,1],[0,0,0,0],[0,-1,0,0]])
J3 = I*matrix([[0,0,0,0],[0,0,-1,0],[0,1,0,0],[0,0,0,0]])

In [42]:

show(J1,J2,J3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{array}\right) \left(\begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{array}\right) \left(\begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right)$

In [43]:

th = var('theta',n=4);th

Out[43]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\theta_{0}, \theta_{1}, \theta_{2}, \theta_{3}\right)$

In [44]:

R1 = exp(I*th[1]*J1)

In [45]:

R2 = exp(I*th[2]*J2)

In [46]:

R3 = exp(I*th[3]*J3)

In [47]:

show(R1,R2,R3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{1}\right)} + 1\right)} e^{\left(-i \, \theta_{1}\right)} & -\frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{1}\right)} - i\right)} e^{\left(-i \, \theta_{1}\right)} \\ 0 & 0 & \frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{1}\right)} - i\right)} e^{\left(-i \, \theta_{1}\right)} & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{1}\right)} + 1\right)} e^{\left(-i \, \theta_{1}\right)} \end{array}\right) \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{2}\right)} + 1\right)} e^{\left(-i \, \theta_{2}\right)} & 0 & \frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{2}\right)} - i\right)} e^{\left(-i \, \theta_{2}\right)} \\ 0 & 0 & 1 & 0 \\ 0 & -\frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{2}\right)} - i\right)} e^{\left(-i \, \theta_{2}\right)} & 0 & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{2}\right)} + 1\right)} e^{\left(-i \, \theta_{2}\right)} \end{array}\right) \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{3}\right)} + 1\right)} e^{\left(-i \, \theta_{3}\right)} & -\frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{3}\right)} - i\right)} e^{\left(-i \, \theta_{3}\right)} & 0 \\ 0 & \frac{1}{2} \, {\left(i \, e^{\left(2 i \, \theta_{3}\right)} - i\right)} e^{\left(-i \, \theta_{3}\right)} & \frac{1}{2} \, {\left(e^{\left(2 i \, \theta_{3}\right)} + 1\right)} e^{\left(-i \, \theta_{3}\right)} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [48]:

R1[2,2].expand().simplify_rectform()

Out[48]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(\theta_{1}\right)$

In [49]:

R1.expand()

Out[49]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2} \, e^{\left(i \, \theta_{1}\right)} + \frac{1}{2} \, e^{\left(-i \, \theta_{1}\right)} & -\frac{1}{2} i \, e^{\left(i \, \theta_{1}\right)} + \frac{1}{2} i \, e^{\left(-i \, \theta_{1}\right)} \\ 0 & 0 & \frac{1}{2} i \, e^{\left(i \, \theta_{1}\right)} - \frac{1}{2} i \, e^{\left(-i \, \theta_{1}\right)} & \frac{1}{2} \, e^{\left(i \, \theta_{1}\right)} + \frac{1}{2} \, e^{\left(-i \, \theta_{1}\right)} \end{array}\right)$

In [ ]:

Funçãozinha em Python

In [50]:

def resolve(M):
    for lin in range(4):
        for col in range(4):
            M[lin,col] = M[lin,col].expand().simplify_rectform()
    return M

In [51]:

resolve(R1);resolve(R2);resolve(R3)

Out[51]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \cos\left(\theta_{3}\right) & \sin\left(\theta_{3}\right) & 0 \\ 0 & -\sin\left(\theta_{3}\right) & \cos\left(\theta_{3}\right) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [ ]:

In [52]:

show(R1,R2,R3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\left(\theta_{1}\right) & \sin\left(\theta_{1}\right) \\ 0 & 0 & -\sin\left(\theta_{1}\right) & \cos\left(\theta_{1}\right) \end{array}\right) \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \cos\left(\theta_{2}\right) & 0 & -\sin\left(\theta_{2}\right) \\ 0 & 0 & 1 & 0 \\ 0 & \sin\left(\theta_{2}\right) & 0 & \cos\left(\theta_{2}\right) \end{array}\right) \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \cos\left(\theta_{3}\right) & \sin\left(\theta_{3}\right) & 0 \\ 0 & -\sin\left(\theta_{3}\right) & \cos\left(\theta_{3}\right) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

Boosts

In [53]:

K1 = -I*matrix([[0,1,0,0],[1,0,0,0],[0,0,0,0],[0,0,0,0]])
K2 = -I*matrix([[0,0,1,0],[0,0,0,0],[1,0,0,0],[0,0,0,0]])
K3 = -I*matrix([[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]])

In [54]:

show(K1,K2,K3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & -i & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \left(\begin{array}{rrrr} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \left(\begin{array}{rrrr} 0 & 0 & 0 & -i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -i & 0 & 0 & 0 \end{array}\right)$

In [55]:

B1 = exp(I*th[1]*K1).expand()
B2 = exp(I*th[1]*K2).expand()
B3 = exp(I*th[1]*K3).expand()

In [56]:

show(B1,B2,B3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & 0 \\ -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \left(\begin{array}{rrrr} \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \left(\begin{array}{rrrr} \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & 0 & -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} & 0 & 0 & \frac{1}{2} \, e^{\left(-\theta_{1}\right)} + \frac{1}{2} \, e^{\theta_{1}} \end{array}\right)$

In [ ]:

Verificando que $R_i$ e $B_i$ são transformações de Lorentz¶

$\Lambda^T\eta\Lambda = \eta$

In [57]:

eta = diagonal_matrix([-1,1,1,1]);eta

Out[57]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [58]:

(R1.transpose()*eta*R1).simplify_trig()

Out[58]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [59]:

(B3.transpose()*eta*B3).expand()

Out[59]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

In [ ]:

Espaço de Minkowski¶

In [60]:

M = VectorSpace(SR,4, inner_product_matrix=eta)

In [61]:

w = M([3,2,1,2]);w

Out[61]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(3,\,2,\,1,\,2\right)$

In [62]:

u = M([3,0,1,2]);u

Out[62]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(3,\,0,\,1,\,2\right)$

In [63]:

w.dot_product(w) #Euclidiano

Out[63]:

$\newcommand{\Bold}[1]{\mathbf{#1}}18$

$w^\mu u_\mu = \eta_{\mu\nu}w^\mu u^\nu = \eta(w,u)$

In [64]:

w.inner_product(u) #Minkowski

Out[64]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-4$

In [65]:

w*eta*u

Out[65]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-4$

In [66]:

(w).inner_product(w)

Out[66]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

In [67]:

(u).inner_product(u)

Out[67]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-4$

In [68]:

w.norm()

Out[68]:

$\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \sqrt{2}$

In [69]:

sqrt(u*eta*u)

Out[69]:

$\newcommand{\Bold}[1]{\mathbf{#1}}2 i$

In [ ]:

SageMath para físicos¶

Aula III¶

1. Vetores e matrizes¶

Vetores e estrutura¶

Soma, multiplicação escalar e produto vetorial¶

Produto interno¶

Hermitiano¶

Norma¶

Matrizes¶

Matrizes especiais¶

Determinante, traço, transposta, inversa, conjugada¶

Pausa - transformação linear¶

Autovalores, auto vetores e diagonalização¶

Exponencial de matrizes¶

Exercício¶

Geradores do grupo de Lorentz (Representação de spin 1 da algébra de Lie so(1,3)\mathfrak{so}(1,3))¶

Verificando que RiR_i e BiB_i são transformações de Lorentz¶

Espaço de Minkowski¶

Geradores do grupo de Lorentz (Representação de spin 1 da algébra de Lie $\mathfrak{so}(1,3)$ )¶

Verificando que $R_i$ e $B_i$ são transformações de Lorentz¶