Question

1. 32 points - Each part below is worth 8 points a. Consider a state, $|a, b\rangle$, that is simultaneously an eigenstate of two operators $\hat{A}$ and $\hat{B}$ with eigenvalues $a$ and $b$. Show that the operators commute. That is $[\hat{A}, \hat{B}] = 0$. b. Using the definition of the Hermition Conjugate show that the Hermition Conjugate of the product of two operators is equal to the product of the Hermition Conjugate of each operator in reverse order. In other words show $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger$. c. Show that eigenstates with distinct eigenvalues are orthogonal. d. Below an operator and two states are defined in terms of matrix notation. Pick one of the states and show that it is not an eigenstate of the operator. Then find the eigenvalues and eigenvectors of the operator. $\hat{Q} = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$ $|1\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}$ $|2\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix}$ Use the following notation for the two eigenstates $|+\rangle$ and $|-\rangle$. A good way to start is to write the eigenvalue equation for $|+\rangle$ and $|-\rangle$.

          1. 32 points - Each part below is worth 8 points
a. Consider a state, $|a, b\rangle$, that is simultaneously an eigenstate of two
operators $\hat{A}$ and $\hat{B}$ with eigenvalues $a$ and $b$. Show that the operators
commute. That is $[\hat{A}, \hat{B}] = 0$.
b. Using the definition of the Hermition Conjugate show that the Hermition
Conjugate of the product of two operators is equal to the product of the
Hermition Conjugate of each operator in reverse order. In other words
show $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger$.
c. Show that eigenstates with distinct eigenvalues are orthogonal.
d. Below an operator and two states are defined in terms of matrix notation.
Pick one of the states and show that it is not an eigenstate of the operator.
Then find the eigenvalues and eigenvectors of the operator.
$\hat{Q} = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$ $|1\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}$ $|2\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix}$
Use the following notation for the two eigenstates $|+\rangle$ and $|-\rangle$. A good
way to start is to write the eigenvalue equation for $|+\rangle$ and $|-\rangle$.

1. 32 points - Each part below is worth 8 points
a. Consider a state, |a, b⟩, that is simultaneously an eigenstate of two
operators Â and B̂ with eigenvalues a and b. Show that the operators
commute. That is [Â, B̂] = 0.
b. Using the definition of the Hermition Conjugate show that the Hermition
Conjugate of the product of two operators is equal to the product of the
Hermition Conjugate of each operator in reverse order. In other words
show (ÂB̂)^† = B̂^†Â^†.
c. Show that eigenstates with distinct eigenvalues are orthogonal.
d. Below an operator and two states are defined in terms of matrix notation.
Pick one of the states and show that it is not an eigenstate of the operator.
Then find the eigenvalues and eigenvectors of the operator.
Q̂ =
< p m a t r i x > |1⟩ =
< p m a t r i x > |2⟩ =
< p m a t r i x >
Use the following notation for the two eigenstates |+⟩ and |-⟩. A good
way to start is to write the eigenvalue equation for |+⟩ and |-⟩.

Added by Kristen H.

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