Briefly describe how the && operator works.

The Boolean operator ⊕, called the XOR operator, is defined by 1 ⊕ 1 = 0, 1 ⊕ 0 = 1, 0 ⊕ 1 = 1, and 0 ⊕ 0 = 0. Show that x ⊕ y = y ⊕ x.

Briefly describe how the and operator works

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.

Briefly describe how the ! operator works.

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

Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian L and find the generalized momentum p = ∂xL/∂ẋ. Find the Hamiltonian H as a function of x and p, and write down Hamilton's equations of motion. It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

Briefly describe how the || operator works.

A ferry operator takes tourists to an island. The operator carries an average of 500 people per day for a round-trip fare of $20. The operator estimates that for each$1 increase in fare, 20 fewer people will take the trip. What fare will maximize the number of people taking the ferry?

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 $A$ ?

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

Design the class linetype to implement a line so that the class lineType:

1. Overloads the stream insertion operator, $<<$, for easy output.
2. Overloads the stream extraction operator, $>>$, for easy intput. (The line $a x+b y=c$ is input as $(a, b, c)$.
3. Overloads the assignment operator to copy a line into another line.
4. Overloads the unary operator $+$, as a member function, so that it returns true if a line is vertical; false otherwise.
5. Overloads the unary operator -, as a member function, so that it returns true if a line is horizontal; false otherwise.
6. Overloads the operator $==$, as a member function, so that it returns true if two lines are equal; false otherwise.
7. Overloads the operator $| |$, as a member function, so that it returns true if two lines are parallel; false otherwise.
8. Overloads the operator $\& \&$, as a member function, so that it returns true if two lines are perpendicular; false otherwise. Write a program to test your class.

Recall that $K_{n}$ denotes the simple, complete graph of order n. a. For what values of n does an Euler path exist in $K_{n} ?$ b. For what values of n does a Hamiltonian circuit exist in $K_{n} ?$

The __________ operator is used to dynamically allocate memory.