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Problem 1: Density matrix for partially polarized light A beam of light travels in the z-direction. Define the basis states $|x\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; |y\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, (1) (i) Construct the density matrix $\rho$ and its square $\rho^2$ in this representation; (ii) by checking whether $\rho^2 = \rho$ determine whether $\rho$ represents a pure or a mixed state in each case; and (iii) find $\rho$ in the representation in which it is diagonal, for the following situations: a) the $x$-polarized state; b) the $y$-polarized state; c) a state linearly polarized at 45°, i.e. the state $|45^\circ\rangle = (|x\rangle + |y\rangle)/\sqrt{2};$ d) a state linearly polarized at 135°, i.e. the state $|135^\circ\rangle = (|x\rangle - |y\rangle)/\sqrt{2};$ e) a right-circularly polarized beam, i.e. the state $|R\rangle = (|x\rangle + i|y\rangle)/\sqrt{2};$ f) a left-circularly polarized beam, i.e. the state $|L\rangle = (|x\rangle - i|y\rangle)/\sqrt{2};$ g) a beam that is a mixture of 50% $x$-polarized and 50% $y$-polarized; h) a beam that is a mixture of 50% 45°-polarized and 50% 135°-polarized; k) a beam that is a mixture of 50% right and 50% left circular polarization; l) a beam that is a mixture of 33% $x$-polarized, 33% 45°-polarized, and 33% right-circularly-polarized.

          Problem 1: Density matrix for partially polarized light
A beam of light travels in the z-direction. Define the basis states
$|x\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; |y\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$,
(1)
(i) Construct the density matrix $\rho$ and its square $\rho^2$ in this representation; (ii) by checking whether $\rho^2 = \rho$ determine
whether $\rho$ represents a pure or a mixed state in each case; and (iii) find $\rho$ in the representation in which it is diagonal,
for the following situations:
a) the $x$-polarized state;
b) the $y$-polarized state;
c) a state linearly polarized at 45°, i.e. the state $|45^\circ\rangle = (|x\rangle + |y\rangle)/\sqrt{2};$
d) a state linearly polarized at 135°, i.e. the state $|135^\circ\rangle = (|x\rangle - |y\rangle)/\sqrt{2};$
e) a right-circularly polarized beam, i.e. the state $|R\rangle = (|x\rangle + i|y\rangle)/\sqrt{2};$
f) a left-circularly polarized beam, i.e. the state $|L\rangle = (|x\rangle - i|y\rangle)/\sqrt{2};$
g) a beam that is a mixture of 50% $x$-polarized and 50% $y$-polarized;
h) a beam that is a mixture of 50% 45°-polarized and 50% 135°-polarized;
k) a beam that is a mixture of 50% right and 50% left circular polarization;
l) a beam that is a mixture of 33% $x$-polarized, 33% 45°-polarized, and 33% right-circularly-polarized.

Problem 1: Density matrix for partially polarized light
A beam of light travels in the z-direction. Define the basis states
|x⟩ =
< p m a t r i x >
; |y⟩ =
< p m a t r i x >,
(1)
(i) Construct the density matrix ρ and its square ρ^2 in this representation; (ii) by checking whether ρ^2 = ρ determine
whether ρ represents a pure or a mixed state in each case; and (iii) find ρ in the representation in which it is diagonal,
for the following situations:
a) the x-polarized state;
b) the y-polarized state;
c) a state linearly polarized at 45°, i.e. the state |45^∘⟩ = (|x⟩ + |y⟩)/√(2);
d) a state linearly polarized at 135°, i.e. the state |135^∘⟩ = (|x⟩ - |y⟩)/√(2);
e) a right-circularly polarized beam, i.e. the state |R⟩ = (|x⟩ + i|y⟩)/√(2);
f) a left-circularly polarized beam, i.e. the state |L⟩ = (|x⟩ - i|y⟩)/√(2);
g) a beam that is a mixture of 50% $x$-polarized and 50% $y$-polarized;
h) a beam that is a mixture of 50% 45°-polarized and 50% 135°-polarized;
k) a beam that is a mixture of 50% right and 50% left circular polarization;
l) a beam that is a mixture of 33% $x$-polarized, 33% 45°-polarized, and 33% right-circularly-polarized.

Added by Richard A.

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