Briefly describe how the and operator works.

When determining whether a number is inside a range, which logical operator is it best to use?

The __________ operator is used to dynamically allocate memory.

Briefly describe how the and operator works

(a) Confirm that the kinetic energy operator, $- \left( \hbar ^ { 2 } / 2 m \right) \mathrm { d } ^ { 2 } / \mathrm { d } x ^ { 2 }$, is hermitian. (b) The operator corresponding to the angular momentum of a particle is $( \hbar / \mathrm { i } ) \mathrm { d } / \mathrm { d } \phi ,$ where $\phi$ is an angle. Is this operator hermitian?

Briefly describe how the ! operator works.

Determine whether the statement is true or false, and justify your answer. If there is a nonzero vector in the kernel of the matrix operator

$$T_A: R^n→R^n$$

then this operator is not one-to-one.

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

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.

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 $A$ ?

$$\begin{array} { r } { \text { Find a Hamiltonian circuit in } } \\ { \quad \operatorname { Cay } \left( \{ ( a , 0 ) , ( b , 0 ) , ( e , 1 ) \} : Q _ { 4 } \oplus Z _ { 2 } \right) } \end{array}$$

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

$$\begin{aligned} \text { Show that there is no Hamiltonian path in } \\ & \operatorname { Cay } \left( \{ ( 1,0 ) , ( 0,1 ) \} : Z _ { 3 } \oplus Z _ { 2 } \right) \\ \text { from } ( 0,0 ) \text { to } ( 2,0 ) \end{aligned}$$

(a) If ||A|| is an operator norm, prove that

$$\| I \| = 1,$$

where I is an identity matrix. (b) Is there a vector norm that induces the Frobenius norm as an operator norm? Why or why not?