Homework Statement
Determine the constant λ in the wave equation
\Psi(x) = C(2a^2 x^2 + \lambda)e^{-(a^2 x^2/2)}
where a=\sqrt{mω/\hbar}
Homework Equations
Some standard integrals I guess
The Attempt at a Solution
So I believe the wave equation just needs to be normalised...
Yeah ok, so this essentially comes down to a conversion between bases. ##\αlpha(dB/km)## should equal ##2303 \αlpha(m−1)## if you account for the difference between metres and kilometres, shouldn't it?
Where does \lambda come into it? Is that simply the wavelength of the light entering the fibre? My question sheet definitely says ##\alpha(dB/km) = 4343\alpha(m^{-1})##.
e; I see that ##P_{out} = P_{in}e^{-\lambda L}## is the same as the version we were taught where ##P_{out} = P_{in}e^{-\alpha...
Homework Statement
It is customary to express fibre loss in units of dB/km:
\alpha(dB/km) = \frac{10log(P_{in}/P_{out})}{L(km)}
where P_{in} is the power entering the fibre and P_{out} is the power leaving the fibre. Show that \alpha(dB/km) = 4343\alpha(m^{-1})
The Attempt at a Solution...
Homework Statement
Consider the function in polar coordinates
ψ(r,θ,\phi) = R(r)sinθe^{i\phi}
Show by direct calculation that ψ returns sharp values of the magnitude and z-component of the orbital angular momentum for any radial function R(r). What are these sharp values?
The Attempt at a...
Ok I believe I have that one figured out. The next problem now is to calculate the surface area of those segments using spherical coordinates. I'm told the formula S = \int^{b}_{\phi=a}\int^{d}_{\theta=c} sin\theta d\theta d\phi should be used. What are a, b, c and d here?
Homework Statement
Consider the unit sphere x^{2} + y^{2} + z^{2} = 1
Find the volume of the two pieces of the sphere when the sphere is cut by a plane at z=a.The Attempt at a Solution
My interpretation is that a is a point on the z-axis that the plane cuts at. So the height of the segment...
Homework Statement
Consider the electric field E(t,x,y,z) = Acos(ky-wt)k
1. Find a magnetic field such that \partial_tB + \nabla X E = 0
2. Show that \nabla . E = 0 and \nabla. B = 0
3. Find a relationship between k and w that enables these fields to satisfy
\nabla X B =...
Ah yes, my mistake. So this should mean that dx/dt does not exist at t=0, since \frac{1}{3}t^{-2/3}j becomes 1/0, giving i + ∞j.
Yeah. dx/ds in this case would be defined at s=0, which is what we want. Is this sort of like 'moving' the origin?
Homework Statement
Consider the parametric curve given by the equation
x(t) = ti + t^(1/3)j1. Calculate x'(t). Does the vector exist at t=0?
2. Find a new parametrisation of the curve for which the tangent vector is well
defined at all points. What is the value of the vector at the origin?The...