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 $A$ ?

Write out the Schrödinger equation as expressed in Eq. we saw earlier in matrix form for the two-state system and verify the result in Eq. we saw earlier.

Discuss the concept of polymorphism/operator overloading.

Binary operator on X: function from X x X to X

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

Which of the following is not a comparison operator? * <> = << <=

Let G be a bipartite graph with disjoint vertex sets $V_1$ and $V_2$. Show that if G has a Hamiltonian cycle, $V_1$ and $V_2$ have the same number of elements.

A particle of mass m moves under the influence of gravity along the helix $z=k \theta, r=$ constant, where k is a constant and z is vertical. Obtain the Hamiltonian equations of motion.

Briefly describe how the || operator works.

The __________ operator is used to dynamically allocate memory.

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

Briefly describe how the or operator works.

Consider the linear operator of left multiplication by an $m \times m$ matrix A on the space $F^{m \times m} \text { of all } m \times m$ matrices. Determine the trace and the determinant of this operator.

The _______ operator is used to dynamically allocate memory.