Question

A common solution to the wave equation is E(x,t) = A ei(kx+wt). On paper take the needed derivatives and show that it actually is a solution. Note that i is the square-root of -1. Upload a photograph of your work.

Answer #1

Show that E(x,t) = Emax. Cos (kx – wt)
And B(x,t) = Bmax. Cos (kx – wt)
Are solutions to the Wave Equations

Show that x(t) = A sin(wt) sin(kx) satisfies the wave equation,
where w and k are some constants. Find the relation between w, k,
and v so that the wave equation is satisfied.

4 Schr¨odinger Equation and Classical Wave Equation
Show that the wave function Ψ(x, t) = Ae^(i(kx−ωt)) satisfies
both the time-dependent Schr¨odinger equation and the classical
wave equation. One of these cases corresponds to massive particles,
such as an electron, and one corresponds to massless particles,
such as a photon. Which is which? How do you know?

A standing wave’s function is y(x,t) = Asin(kx)cos(?t). Prove
that this equation is indeed a solution to the wave equation.

(a) Write down expressions for the electric and magnetic fields
of a sinusoidal plane electromagnetic wave having a frequency of 3
GHz and traveling in the positive x direction. The amplitude of the
magnetic field is 1 μT. b) Verify that E(x, t) = Ae^(i(kx-wt)) or
B(x, t) = C sinkxsinwt is a solution of the one-dimensional wave
equation; A and C are constants.

Consider the following wave function:
Psi(x,t) = Asin(2piBx)e^(-iCt) for 0<x<1/2B
Psi(x,t) = 0 for all other x
where A,B and C are some real, positive constants.
a) Normalize Psi(x,t)
b) Calculate the expectation values of the position operator and
its square. Calculate the standard deviation of x.
c) Calculate the expectation value of the momentum operator and
its square. Calculate the standard deviation of p.
d) Is what you found in b) and c) consistent with the
uncertainty principle? Explain....

2). A particle moving on the x-axis has a time-dependent
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units of x are meters (m) and time t in seconds (s). (Hint: you
must get derivatives, you need graph paper)
(a) So that the position in x has units of meter which are the
units of the constants c and b?
Sic = 5yb = 1.Desdeti = 0satf = 3s.
(b) What is its displacement,...

A power plant emits a pollutant X to the atmosphere at a
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constant k (s-1).
a. Let m be the mass of X in the atmosphere resulting from the
power plant emissions. Write an equation for m(t). Plot your
results. What is the steady-state value m??
b. Show that the atmospheric lifetime of X is...

3. Consider the region R in the first quadrant enclosed by y =
x, y = x/2, and y = 5.
(a) Sketch this region, making sure to identify and label all
points of intersection.
(b) Find the area of R, using the method of your choice.
(c) Using the method of your choice, set up an integral for the
volume of the solid resulting from rotating R around the y-axis. Do
NOT evaluate the integral.
(d) Using the method...

In this exercise, you will analyze the supply-demand equilibrium
of a city under some special simplifying assumptions about land
use. The assumptions are: (i) all dwellings must contain exactly
1,500 square feet of floor space, regardless of location, and (ii)
apartment complexes must contain exactly 15,000 square feet of
floor space per square block of land area. These land-use
restrictions, which are imposed by a zoning authority, mean that
dwelling sizes and building heights do not vary with distance to...

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