Serial of year 20

You can find the serial also in the yearbook.

We are sorry, this serial has not been translated.


1. Series 20. Year - S. Bohr hypothesis

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.

Zadal autor seriálu Jarda Trnka.

2. Series 20. Year - S. particle with 1/2 spin

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
⊗〉 = ( ↑〉 + ↓〉 ) ⁄ √2 ⊕〉 = ( ↑〉 −

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.

Zadal autor seriálu Jarda Trnka.

This website uses cookies for visitor traffic analysis. By using the website, you agree with storing the cookies on your computer.More information



Media partner

Created with <love/> by ©FYKOS –