Find u(x, t) for the string of length :=1 and c²=1 when the initial velocity is zero and the inital deflection with small k(say, 0.01) is as follows. Sketch or graph u(x, t). kx²(1-x)

A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket $|R\rangle(|L\rangle)$, where we have neglected spatial variations within each half of the box. The most general state vector can then be written as $|\alpha\rangle=|R\rangle\langle R | \alpha\rangle+|L\rangle\langle L | \alpha\rangle$, where $\langle R | \alpha\rangle$ and $\langle L | \alpha\rangle$ can be regarded as “wave functions.” The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian $H=\Delta(|L\rangle\langle R|+| R\rangle\langle L|)$, where $\Delta$ is a real number with the dimension of energy. (a) Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?(b) In the Schrödinger picture the base kets $|R\rangle$ and $|L\rangle$ are fixed, and the state vector moves with time. Suppose the system is represented by $|\alpha\rangle$ as given above at t = 0. Find the state vector $\left|\alpha, t_{0}=0 ; t\right\rangle$ for t > 0 by applying the appropriate time-evolution operator to $|\alpha\rangle$. (c) Suppose that at t = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (d) Write down the coupled Schrödinger equations for the wave functions $\langle R| \alpha, t_{0}= 0 ; t\rangle$ and $\left\langle L | \alpha, t_{0}=0 ; t\right\rangle$. Show that the solutions to the coupled Schrodinger equations are just what you expect from (b). (e) Suppose the printer made an error and wrote H as $H=\Delta|L\rangle\langle R|$. By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated.

Write down the derivation in this section for length L=π, to see the very substantial simplification of formulas in this case that may show ideas more clearly.

The faces of the thin square plate with side a=24 are perfectly insulated. The upper side is kept at 25°C and the other sides are kept at 0°C. Find the steady-state temperature u(x, y) in the plate.

The smooth block B, having a mass of 0.2 kg, is attached to the vertex A of the right circular cone using a light cord. If the block has a speed of 0.5 m s around the cone, determine the tension in the cord and the reaction which the cone exerts on the block. Neglect the size of the block.

Find the temperature u(x, t) in a rod of length L if the initial temperature is f (x) throughout and if the ends x = 0 and x = L are insulated.

A rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature $u(x, t)$. The left end is held at temperature $100^{\circ}$, and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is f(x) throughout.

Find the Fourier series of f(x) as given over one period and sketch f(x) and partial sums. f(x)=|x| (-π<x<π)

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. xy-plane r(u, v)=(u, v)(thus ui+vj)

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Cone $\mathbf{r}(u, v)=[u \cos v, \quad u \sin v, \quad c u]$.

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Elliptic cylinder r(u, v)=[a cos v, b sin v, u]

A 40-m-long, 12-mm-diameter pipe with a friction factor of 0.020 is used to siphon $30^{\circ} \mathrm{C}$ water from a tank as shown . Determine the maximum value of $h$ allowed if there is to be no cavitation within the hose. Neglect minor losses

Find a parametric representation. Circle in the plane z=1 with center (3, 2) and passing through the origin.

Verifiy (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.

$$u-e^{-9t}sinωx$$