Angle in Spherical coordinates

Matterwave · 2009-04-17T09:13:15+00:00

I have to proove something in QM but I'm stuck on a bit of math. Say I have two vectors: \vec{a} = (r_a,\theta_a,\phi_a) and \vec{b} = (r_b,\theta_b,\phi_b) What is the cosine of the angle between them? If my proof is to work the cosine of the angle between them have to be: cos(\theta)=1+sin(\theta_a)sin(\theta_b)cos(\phi_a-\phi_b) I think the 1 is erroneous and should be replaced with cos(\theta_a)cos(\theta_b) But I'm not sure and I can't figure out what I did wrong...Which one is it? Is it either?

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

View full post »

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Similar threads

Hot Threads

Prove that the integral is equal to ##\pi^2/8##

Calculating radius of gyration of plane figure about x-axis

Solve this problem that involves induction

The volume of a "spherical cap" using triple integrals

Finding the modulus and argument of ##\dfrac{a}{(b±ci)^n}##

Recent Insights

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Insights What Exactly is Dirac’s Delta Function? - Insight

Insights Relativator (Circular Slide-Rule): Simulated with Desmos - Insight

Insights Fixing Things Which Can Go Wrong With Complex Numbers

Insights Fermat's Last Theorem