# 4th Round of 27th year

Series

### 1. ... Another unsharpened one(2 points)

A freshly sharpened pencil 6B has a tip in the shape of a cone and the radius of the cone's base is $r$ = 1 mm a and its height is $h$ = 5 mm. How long will the line that we are able to make with it be if the distance of two graphite layers is $d$ = 3,4 Å and the track of the pencil has on average $n$ = 100 such layers?

### 2. ... test tubes(2 points)

Test tubes of  volumes 3 ml and 5 ml are connected by a short thin tube in which we can find a porous thermally non-conductive barrier that allows an equilbirum in pressures to be achieved within the system. Both test tubes in the beginning are filled with oxygen at a pressure of 101,25 kPa and a temperature of 20 ° C. We submerge the first test tube (3 ml) into a container which has a system of water and ice in equilbrium inside it and the other one (5 ml) into a container with steam. What wil the pressure be in the system of the teo test tubes be after achieving mechanical equilibrium? What would the pressure be if it would have been nitrogen and not oxygen that was in the test tubes?(while keeping other conditions the same)/p>

### 3. ... Seagull(4 points)

Two ships are sailing against each other, the first one with a velocity $u$_{1} = 4 m·s^{ − 1} and  the second with a velocity of $u$_{2} = 6 m·s^{ − 1}. When they are seperated by $s$_{0} = 50 km, a seagull launches from the first ship and flies towards the second one. He is flying against the wind, his speed is $v$_{1} = 20 m·s^{ − 1}. When he arrives to the second ship he turns around and flies back  now with the wind behind his back with a velocity $v$_{2} = 30 m·s^{ − 1}.He keps on flying back and forth until the two ships meet. How long is the path that he has undertaken?

### 4. ... discharged pudding(4 points)

There are a lot of models of hydrogens and many of these have been overcome but we like pudding and so we shall return to the pudding model of hydrogen. The atom is made of a sphere with a radius $R$ with an equally distributed positive charge(&#8222;puding&#8220;), in which we can find an electron(&#8222;rozinka&#8220;). Obviously the electron prefers being in the place with the lowest possible energy and so he sits in the middle of the pudding. Overall the system is electrically neutral. What is the energy that we must give the electron to get it to infinity? What would radius have to be so that this energy would be equal to Rydberg's energy (the energy of excitation of an electron in an atom of hydrogen)? Express the radius in multiples of the Bohr radius.

### 5. ... Another unsharpened one(2 points)

By how much shall the temperature of two identical steel balls rise after their collision?They move in the same direction with speeds $v$_{1} = 0,7 $c$ and $v$_{2} = 0,9 $c$ where $c$  is the speed of light. Assume that the heat capacity is constant and that the balls are still solid.

### P. ... the true gravitational acceleration(5 points)

Faleš wanted to determine the gravitational acceleration from an experiment in Prague(V Holešovickách 2 in the first floor/ground floor). In the experiment he was dropping a round ball from a height of a couple of meters above the Earth. Think about what kind of corrections he had to apply when analysing the data. Then think up your own experiment to determine g and discuss its accuracy.

### E. ... some like it lukewarm(8 points)

Measure the relation between the temperature and time in a freshly made cup of tea. Conduct the measurements for a undisturbed cup of tea and a cup of tea stirred by a teaspoon. Finally determine if the time the tea takes to reach a drinkable temperature depends on the stirring.

### S. ... quantum(6 points)

* Look into the text to see how the operator of position $<img src=„https://latex.codecogs.com/gif.latex?\hat%20X“>$

and momentum $<img src=„https://latex.codecogs.com/gif.latex?\hat %20P“>$ acts on the components of the state vector in~$x$-

representation (wave function) and calculate their comutator, in other

words

<img src=„https://latex.codecogs.com/gif.latex?(\hat%20{X})_x%20\left((\hat%20

{P})_x%20{\psi}%20(x)\right)%20-%20(\hat%20{P})_x%20\left((\hat%20{X})_x%20

{\psi}%20(x)\right)%20“>

Tip   Find out what happens when you take the derivative

of two functions multiplied together

• The problem of levels of energy for a free quantum particle in other words

for $V$ ( $x$ ) = 0 has the

following form:

<img src=„https://latex.codecogs.com/gif.latex?-\frac%20{\hbar%20^2}

{2m}%20\dfrac{\partial^2%20{\psi}%20(x)}{\partial%20x^2}=%20E%20{\psi}%20

(x)\,.“>

• Try inputting $ψ$

( $x$ ) = e^{$αx$} as the solution

and find out for what $α$ (a general complex number)

is $E$ positive (only use such  $α$ from now on).

• Is this solution periodic? If yes then with what spatial period

(wavelength)?

• Is the gained wave function the eigenvector of the operator of momentum

(in the $x$-representation)? If yes find the relation between

wavelength and momentum (in other words the respective eigenvalue) of the state.

• Try to formally calculate the density of probability oof presence of the

particle in space.naší

vlnové funkci podle vzorce uvedeného v textu. Pravděpodobnost, že se

částice vyskytuje

v celém prostoru by měla být pro fyzikální hustotu pravděpodobnosti 1,

tj.

<img src="https://latex.codecogs.com/gif.latex?\int_\mathbb{R}%20\rho

(x)%20\mathrm{d}%20x=1.„> Show that our wave function can't be

$normalized$ (in other words multiply by some constant) so that its formal

density of probability according to the equation from the text was a real

physical density of probability.

• Bonus   What do you think that the limit of the

uncertainity of a position of a particle is if the wave function it has is close

to ours (In other words it approaches it in all properties but it always has a

normalized probability density and thus is a physical state)

Can we (using Heisenberg's relation of uncertainty) determine what is

the lowest possible imprecision while finding the momentum?

Tip   Take care when dealing with complex numbers. For

example the square of a complex number is different than that of its magnitude.

• In the second part of the series we derivated the energy levels of an

electron in hydrogen using reduced action. Due to a random happenstance the

solution of the spectrum of the hamiltonian in  a coulombic potential of a

proton would lead to thecompletely same energy,in other words

<img src=„https://latex.codecogs.com/gif.latex?E_n%20=%20-{\mathrm{Ry}}%20\frac

%20{1}{n^2}“>

where Ty = 13,6 eV is an ernergy constant that is known

as the Rydberg constant. An electron which falls from a random energy

level to $n$ = 2 shall emit energy in the form of a proton

and the magnitude of the energy shall be equal to the diference of the energies

of the two states. Which are the states that an electron can fall from so that

the light will be in the visible spectrum? What will the color of the spectral

lines be?

Tip   Remember the photoelectric

effect and the relation between the frequency of light and its

wavelength.

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