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"""Known matrices related to physics""" |
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from sympy.core.numbers import I |
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from sympy.matrices.dense import MutableDenseMatrix as Matrix |
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from sympy.utilities.decorator import deprecated |
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def msigma(i): |
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r"""Returns a Pauli matrix `\sigma_i` with `i=1,2,3`. |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Pauli_matrices |
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Examples |
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======== |
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>>> from sympy.physics.matrices import msigma |
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>>> msigma(1) |
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Matrix([ |
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[0, 1], |
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[1, 0]]) |
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""" |
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if i == 1: |
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mat = ( |
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(0, 1), |
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(1, 0) |
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) |
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elif i == 2: |
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mat = ( |
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(0, -I), |
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(I, 0) |
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) |
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elif i == 3: |
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mat = ( |
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(1, 0), |
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(0, -1) |
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) |
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else: |
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raise IndexError("Invalid Pauli index") |
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return Matrix(mat) |
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def pat_matrix(m, dx, dy, dz): |
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"""Returns the Parallel Axis Theorem matrix to translate the inertia |
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matrix a distance of `(dx, dy, dz)` for a body of mass m. |
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Examples |
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======== |
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To translate a body having a mass of 2 units a distance of 1 unit along |
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the `x`-axis we get: |
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>>> from sympy.physics.matrices import pat_matrix |
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>>> pat_matrix(2, 1, 0, 0) |
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Matrix([ |
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[0, 0, 0], |
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[0, 2, 0], |
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[0, 0, 2]]) |
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""" |
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dxdy = -dx*dy |
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dydz = -dy*dz |
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dzdx = -dz*dx |
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dxdx = dx**2 |
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dydy = dy**2 |
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dzdz = dz**2 |
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mat = ((dydy + dzdz, dxdy, dzdx), |
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(dxdy, dxdx + dzdz, dydz), |
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(dzdx, dydz, dydy + dxdx)) |
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return m*Matrix(mat) |
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def mgamma(mu, lower=False): |
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r"""Returns a Dirac gamma matrix `\gamma^\mu` in the standard |
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(Dirac) representation. |
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Explanation |
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=========== |
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If you want `\gamma_\mu`, use ``gamma(mu, True)``. |
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We use a convention: |
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`\gamma^5 = i \cdot \gamma^0 \cdot \gamma^1 \cdot \gamma^2 \cdot \gamma^3` |
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`\gamma_5 = i \cdot \gamma_0 \cdot \gamma_1 \cdot \gamma_2 \cdot \gamma_3 = - \gamma^5` |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Gamma_matrices |
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Examples |
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======== |
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>>> from sympy.physics.matrices import mgamma |
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>>> mgamma(1) |
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Matrix([ |
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[ 0, 0, 0, 1], |
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[ 0, 0, 1, 0], |
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[ 0, -1, 0, 0], |
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[-1, 0, 0, 0]]) |
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""" |
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if mu not in (0, 1, 2, 3, 5): |
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raise IndexError("Invalid Dirac index") |
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if mu == 0: |
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mat = ( |
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(1, 0, 0, 0), |
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(0, 1, 0, 0), |
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(0, 0, -1, 0), |
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(0, 0, 0, -1) |
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) |
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elif mu == 1: |
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mat = ( |
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(0, 0, 0, 1), |
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(0, 0, 1, 0), |
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(0, -1, 0, 0), |
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(-1, 0, 0, 0) |
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) |
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elif mu == 2: |
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mat = ( |
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(0, 0, 0, -I), |
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(0, 0, I, 0), |
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(0, I, 0, 0), |
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(-I, 0, 0, 0) |
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) |
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elif mu == 3: |
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mat = ( |
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(0, 0, 1, 0), |
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(0, 0, 0, -1), |
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(-1, 0, 0, 0), |
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(0, 1, 0, 0) |
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) |
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elif mu == 5: |
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mat = ( |
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(0, 0, 1, 0), |
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(0, 0, 0, 1), |
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(1, 0, 0, 0), |
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(0, 1, 0, 0) |
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) |
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m = Matrix(mat) |
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if lower: |
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if mu in (1, 2, 3, 5): |
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m = -m |
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return m |
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minkowski_tensor = Matrix( ( |
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(1, 0, 0, 0), |
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(0, -1, 0, 0), |
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(0, 0, -1, 0), |
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(0, 0, 0, -1) |
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)) |
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@deprecated( |
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""" |
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The sympy.physics.matrices.mdft method is deprecated. Use |
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sympy.DFT(n).as_explicit() instead. |
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""", |
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deprecated_since_version="1.9", |
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active_deprecations_target="deprecated-physics-mdft", |
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) |
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def mdft(n): |
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r""" |
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.. deprecated:: 1.9 |
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Use DFT from sympy.matrices.expressions.fourier instead. |
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To get identical behavior to ``mdft(n)``, use ``DFT(n).as_explicit()``. |
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""" |
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from sympy.matrices.expressions.fourier import DFT |
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return DFT(n).as_mutable() |
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