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A particle with spin 1/2 (e.g. electron) can be in two states of projection of spin to the z-axis. Either the spin is pointing up |↑〉 or down |↓〉. These two states create basis for two-dimensional Hilbert space describing particle of spin 1/2.

• Write the operator of identity in this space and language of vectors |↑〉 and |↓〉.
• Find Eigen vector and Eigen number of matrices $S_{1}$, $S_{2}$ and $S_{3}$.
• Lets have operators $S_{+}$ and $S_{-}$ in the form

$S_{+}=|↑〉〈↓|$, $S_{-}=|↓〉〈↑|$. Find its representation in basis of vector |↑〉 and |↓〉 and find how they operate on general vector |$ψ〉=a|↑〉+b|↓〉$. How do look Eigen vectors and what are the Eigen numbers?

• Lets define vectors

Show that these vectors form basis in our Hilbert space and find relation between coefficients $a$, $b$ in decomposition |$ψ〉$ into original basis and coefficients $c$, $d$ into the new basis |$ψ〉=c|⊗〉+d|⊕〉$.

• Write two spin operators $S_{1}$, $S_{2}$ a $S_{3}$ in basis of vectors |⊗〉 a |⊕〉. Find its Eigen vectors and Eigen numbers.

In this question we will deal with hydrogen atom, which consists of heavy nucleus with electric charge $e$ and light electron of mass $m$ and charge $-e$, which orbits around nucleus at circular trajectory.

• Calculate (using classical physics) the distance of electron from the nucleus depending on its total energy (kinetic and potential) $E$.
• If we accept Bohr hypothesis, that electron's momentum is quantised i.e. can have only discrete values $L=nh/2π$, where $n$ is integer number. In which distance from nucleus can electron orbit around nucleus?
• Calculate frequency of emitted photon, if the atom change its energy level from $n-th$ allowed to $m-th$ allowed distance from nucleus.

A emission spectral line of Helium was observed in spectrum of a star. The wavelength of helium line is 587,563 nm . However, the observed line in spectroscope was blurred between 587,60 nm and 587,67 nm . Estimate the temperature of the star and its speed in the space. How is this blurring caused?

The cathode of photovoltaic cell is illuminated by the light from the mercury lamp of wavelength 546,1 nm. To eliminate photoelectric voltage, the voltage of $U_{1}=1,563V$ is needed. When the light has wavelength 404,7 nm, voltage $U_{2}=2,356V$ is needed. Calculate the value of Planck constant $h$.