Briefly describe how the or operator works.

Form the operator $\hat{A}^2$ if $\hat{A}=d^2 / d y^2+3 y(d / d y)-5$. Be sure to include an arbitrary function on which the operator acts.

Consider a two-state quantum system with a Hamiltonian

$$H \doteq\left(\begin{array}{cc} E_1 & 0 \\ 0 & E_2 \end{array}\right)$$

Another physical observable $A$ is described by the operator

$$A=\left(\begin{array}{ll} 0 & a \\ a & 0 \end{array}\right)$$

where $a$ is real and positive. Let the initial state of the system be $|\psi(0)\rangle=\left|a_1\right\rangle$, where $\left|a_1\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $A$. What is the frequency of oscillation (i.e., the Bohr frequency) of the expectation value of $A$ ?

Briefly describe how the && operator works.

The __________ operator is used to dynamically allocate memory.

Is $\cos x$ an eigenfunction of the operator $\hat{A}$ if $\hat{A} f(x)=x f(x)$ ?

Write the Hamiltonian function and find Hamilton's canonical equations for the three-dimensional motion of a projectile in a uniform gravitational field with no air resistance. Show that these equations lead to the same equations of motion as found.

Let $G$ be a graph. The line graph of $G$ is a new graph $L(G)$ whose vertices are the edges of $G$; two vertices of $L(G)$ are adjacent if, as edges of $G$, they share a common end point. In symbols:

$$V[L(G)]=E(G)\text{ and }E[L(G)]=\{e_1e_2:|e_1\cap e_2|=2\}$$

Prove or disprove the following statements about the relationship between a graph and its line graph : a. If $G$ is Eulerian, then $L(G)$ is also Eulerian. b. If $G$ has a Hamiltonian cycle, then $L(G)$ is Eulerian. (See Exercise 50.16 for the definition of a Hamiltonian cycle.) c. If $L(G)$ is Eulerian, then $G$ is also Eulerian. d. If $L(G)$ is Eulerian, then $G$ has a Hamiltonian cycle.

The _______ operator is used to dynamically allocate memory.

$$\begin{array} { l } { \text { Find a Hamiltonian circuit in } } \\ { \text { Cay } \left( \{ ( a , 0 ) , ( b , 0 ) , ( e , 1 ) \} : Q _ { 4 } \oplus Z _ { m } \right) } \\ { \text { where } m \text { is even. } } \end{array}$$

Show that the operator $\hat{p}^{\prime}$ obtained by applying a translation to the operator $\hat{p}$ is $\hat{p}^{\prime}=\hat{T}^{\dagger} \hat{p} \hat{T}=\hat{p}$.

The __________ operator is used to dynamically allocate memory.

Briefly describe how the or operator works

$$\begin{array} { c } { \text { Find a Hamiltonian circuit in } } \\ { \quad \operatorname { Cay } \left( \{ ( a , 0 ) , ( b , 0 ) , ( e , 1 ) \} : Q _ { 4 } \oplus Z _ { 3 } \right) } \end{array}$$