Worked Problems in Physics

Nielsen and Chuang Exercise 2.20

Posted in Nielsen/Chuang by rpmuller on March 2, 2010

Exercise 2.20 Basis change: Suppose {A'} and {A''} are matrix representations of an operator {A} on a vector space {V} with respect to two different orthonormal bases, {\left|v_i\right>} and {\left|w_i\right>}. Then the elements of {A'} and {A''} are {\left<v_i|A|v_j\right>} and {\left<w_i|A|w_j\right>}. Characterize the relationship between {A'} and {A''}.

\displaystyle  \left<v_i|A|v_j\right> = \sum_{kl} \left<v_i|w_k\right>\left<w_k|A|w_l\right>\left<w_l|v_j\right> \ \ \ \ \ (39)

Nielsen and Chuang Exercise 2.19

Posted in Nielsen/Chuang by rpmuller on March 2, 2010

Exercise 2.19 Show that the Pauli matrices are Hermetian and unitary.

Hermetian:

\displaystyle  \left[ \begin{array}{rr} 0 & 1\\ 1 & 0 \end{array} \right]^\dagger = \left[ \begin{array}{rr} 0 & 1\\ 1 & 0 \end{array} \right] \ \ \ \ \ (33)

\displaystyle  \left[ \begin{array}{rr} 0 & -i\\ i & 0 \end{array} \right]^\dagger = \left[ \begin{array}{rr} 0 & -i\\ i & 0 \end{array} \right] \ \ \ \ \ (34)

\displaystyle  \left[ \begin{array}{rr} 1 & 0\\ 0 & -1 \end{array} \right]^\dagger = \left[ \begin{array}{rr} 1 & 0\\ 0 & -1 \end{array} \right] \ \ \ \ \ (35)

Since we’ve now shown that the matrices are Hermetian, it’s enough to show that the matrices square to {I}:

\displaystyle  X^2= \left[ \begin{array}{rr} 0 & 1\\ 1 & 0 \end{array} \right] \left[ \begin{array}{rr} 0 & 1\\ 1 & 0 \end{array} \right]=I \ \ \ \ \ (36)

\displaystyle  Y^2 = \left[ \begin{array}{rr} 0 & -i\\ i & 0 \end{array} \right] \left[ \begin{array}{rr} 0 & -i\\ i & 0 \end{array} \right]=I \ \ \ \ \ (37)

\displaystyle  Z^2 = \left[ \begin{array}{rr} 1 & 0\\ 0 & -1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0\\ 0 & -1 \end{array} \right]=I \ \ \ \ \ (38)

Nielsen and Chuang Exercise 2.18

Posted in Nielsen/Chuang by rpmuller on March 2, 2010

Exercise 2.18 Show that all eigenvalues of a unitary matrix have modulus 1, that is, can be written {e^{i\theta}} for some real {\theta}.

\displaystyle  Ux = \lambda x \ \ \ \ \ (33)

\displaystyle  x^\dagger U^\dagger = \lambda^\dagger x^\dagger \ \ \ \ \ (34)

Multiplying the two

\displaystyle  x^\dagger U^\dagger Ux = \lambda^\dagger x^\dagger \lambda x \ \ \ \ \ (35)

\displaystyle  ||x||^2 = |\lambda|^2||x||^2 \ \ \ \ \ (36)

or {|\lambda|^2=1}.

Nielsen and Chuang Exercise 2.17

Posted in Nielsen/Chuang by rpmuller on March 2, 2010

Exercise 2.17 Show that a normal (i.e. diagonalizable) matrix is Hermitian if and only if it has real eigenvalues.

Part one: show that Hermiticity implies real eigenvalues. Since the matrix is diagonalizable, we can write

\displaystyle  \lambda = U^\dagger AU \ \ \ \ \ (27)

Thus

\displaystyle  \lambda^\dagger = \left(U^\dagger AU\right)^\dagger = U^\dagger A^\dagger U \ \ \ \ \ (28)

Since the matrix {A} is Hermetian, {A^\dagger=A}, and thus {\lambda^\dagger=\lambda}, and thus the matrix has real eigenvalues.

Part two: Show that real eigenvalues imply Hermiticity. We may now write:

\displaystyle  A = U\Lambda U^\dagger \ \ \ \ \ (29)

and following a similar procedure show that

\displaystyle  A^\dagger = \left(U\Lambda U^\dagger\right)^\dagger = U\Lambda^\dagger U^\dagger. \ \ \ \ \ (30)

Now, since the eigenvalues are real, {\Lambda^\dagger = \Lambda}, and thus {A^\dagger=A}, and thus the matrix is Hermitian.