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## quantum physics

### 6. Series 22. Year - S. atomic models and Rutherford experiment

• Decide if the stability (e.g. dimensions) of Saturn model depends on atomic number $Z$.
• Change equation (12) for probability of scatter of $α-particle$ at high angle $φ$ in such way, to get more practical equation for probability of impact per unit area on scintillator. and show, how this can be used to get material of target. Further estimate how the equation would change when not considering the central charge $Ze$ but $Z$ spread elementary charges $e$ as is for example in Lenard model.
• In 1896 astronomer E. C. Pickering found in the spectrum of star $ζ$ Puppis lines, which fulfilled condition (7) for $n=2$ and $m=2,5;$ 3; 3,5; 4; 4,5;…,

e.g. also for half integers. Explain this inconsistency of Bohr model.

• ( Bonus: Find a dependence similar to equation (11) for Thompson pudding model and comment on differences. Or try to modify it in such way, that it considers all atoms in thin aluminium foil. Just play a little bit.

Na rozloučenou od autorů seriálu.

### 3. Series 20. Year - E. Planck constant

Suggest and make adequate theoretical justification for methods suitable to measure Planck constant which can be realized at home or in school laboratory. Realize at least one of them. All physical quantities measure with highest accuracy (consider using statistical averages etc.) and estimate value of this fundamental constant including relevant experimental error.

Hint: LED diode with resistor costs approximately 5 Kč ( 0.10Eur).

Experiment navrhl Pavel Brom.

### 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.

### 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.

### 2. Series 19. Year - 3. spectral analysis

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?

Staré návrhy a Bzučo doformuloval.

### 6. Series 18. Year - 1. photoelectric effect

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 $h$.

Našel Honza Prachař v jedné sbírce.