I have problem to simplify the quantum system states on bloch sphere to get $\theta$, $\phi$ values, My question is not duplicated! $$ \left| \varphi \right&gt;=\frac{1+i}{\sqrt{3}} \left| 0 \right&gt; + {\sqrt{\frac{1}{3}}} \left| 1\right&gt; $$
If I have the following state:  $$ \left| \varphi \right&gt;=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right&gt; + \left| 1\right&gt;\right) $$  How can I find the $\theta$ and $\phi$ values of this qubit on the Bloch sphere?
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Additional comment: There seems to be two explanations "water adhere to the sphere and is ejected conserving horizontal momentum" and "Magnus effect" (ME). The later is not so obvious to me as the system is in equilibrium and there are two fluids, water and air, (does ME means air pushes the ball horizontally?).   Edit3: DESCRIPTION To avoid rot links and people that do not want to see the video, I'll try to explain the phenomena. You have a sphere and a water jet. The water jet is vertical (opposite to gravity), and touches the sphere almost tangentially. The ball not only rotates due to the water jet but also stays at constant height from the ground. The angular velocity can be defined in the direction of the cross product between the position of the contact point between the jet and the sphere (origin in the center of the sphere) and the direction of the jet. Some water drops fly away at the top of the sphere away from the jet.  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Basically, I wanted to solve $2^x = x^2$ to analyze algorithms with respective growth functions. Why are the laws of thermodynamics "supreme among the laws of Nature"? Eddington wrote     The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.   and Einstein wrote     [classical thermodynamics] is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.   Why did they say that? Is it a very deep insight they had, or is it something one can be convinced of quite easily? Or even, is it trivial? Shape of a string/chain/cable/rope/wire? The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall)  and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc length) .  What shape does it take?  My try so far: minimise potential energy of the whole string,  $$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$  With the constraint  $$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$ If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.  Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in  $$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$  Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.  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