Serial of year 20

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 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.