Yes, I do ofc. I know how to find the energy by requiring the raising operator to terminate the circular orbit wavefunction, I was just wondering is you can do it by directly evaluating this integral.
I know these values - for inverse 1st, 2nd and 3rd powers - my concern is the first term - the one involving the differential - it is going to produce 2 integrals on its own.
Homework Statement
We know that:
E=<\psi|H|\psi>
where
H=-\frac{\hbar^2}{2m} \nabla^2 - \frac{Ze^2}{4\pi\epsilon_0 r}
and
\psi=R(r)Y(\theta, \phi)
with
R(r)=\frac{1}{\sqrt{(2n)!}}(\frac{2Z}{na_0})^{3/2}\left(\frac{2Zr}{na_0}\right)^{n-1}e^{-Zr/na_0}
If I want to find the energy, do...
In my lecture they give the phase space picture for a simple pendulum
http://mathematicalgarden.files.wordpress.com/2009/03/pendulum-portrait3.png?w=500&h=195
and then say that adjacent trajectories never diverge and therefore evolution is predictable. I wanted to ask, is the statement that...
The problem says find the eq of motion and then the period of oscillations (not small) but surely the first step is the equation of motion.
y is the coordinate of the centre of the sphere.
I found that the submerged volume is:
V=\frac{4\pi}{3}a^3-\frac{\pi(a-y)^2}{3}(3a-(a-y))=\frac{\pi}{3}(2a^3+3a^2y-y^3)
but how do I find the period of oscillations from here on?
Homework Statement
A sphere is floating in water. It is pushed just under the water level and released. I'm asked to write the equation of motion for the sphere, not assuming small oscillations.
Is it just:
my''=\rho V g - mg
?
Or do I have to include that the buoyant force is changing...
We need to integrate over the height, that is from 0 to 2, and for each y we need to account for a disk of radius
x=\sqrt[3]{(8-y^3)}
which moment of inertia is:
dI=\rho x^2 dy=\rho (8-y^3)^{2/3}dy
For the whole body this is:
I=\int_0^2\rho (8-y^3)^{2/3} dy
I=\frac{m}{V}\int_0^2 (8-y^3)^{2/3} dy...
Homework Statement
The first quadrant area bounded by the curve
x^3+y^3=8
is rotated around y-axis to give a solid of rotation. The question asks for an integral which represents the solid's moment of inertia around the axis.
My answer is:
I_y=M\frac{\int_0^2x^2ydx}{\int_0^2x^2dx}...
Homework Statement
A soap bubble of radius R_1 and surface tension \gamma is expanded at constant temperature by forcing in air by driving in fully a piston containing volume v. We have to show that the work needed to increase the bubble's radius to R_2 is:
\Delta W=P_2V_2ln\frac{P_2}{P_1} +...
Homework Statement
If a light beam is circularly polarized and then bounced off a mirror perpendicular to the optical axis, what is the new polarization.
My thought is, because circularly polarized light has basically 2 components behaving like waves, each of them is going to experience a...