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quantum physics

(4 points)4. Series 27. Year - 4. discharged pudding

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(„puding“), in which we can find an electron(„rozinka“). 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.

Jakub was making pudding.

(6 points)4. Series 27. Year - S. quantum

 

  • 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 $Epositive$ (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.

(5 points)1. Series 27. Year - P. speed of light

What would be the world like if the speed of light was only $c=1000\;\mathrm{km}\cdot h^{-1}$ while all the other fundamental constants stayed unchanged? What would be the impact on life on Earth? Would it even be possible for people to exist in such a world?

Karel came up with an unsolvable problem.

5. Series 23. Year - 1. photon fountain

Honza is not satisfied with the current bed standard. Thus, he started to test laser levitation. He bought a ball with a perfectly polished mirror surface of mass $m$, radius $r$ and put it on the ground. The ground was immediately lit by the laser with a wave length $\lambda$ and surface power $P$. What is the height of the ball at equilibrium? To get extra points, you may try to solve the problem for a ball made of glass. We suppose, in both the cases, that the laser will not fuse the ball and the experiment takes place in a homogenous gravitational field.

brought by Honza Humplík

4. Series 23. Year - 2. Fever

Returning home from an observatory, watching the sunrise, Janap discovered an easy way to calculate the temperature of the Sun. We do give away that the Earth is an absolute black body with a temperature of 0° C.

solved by Janap in one of her lectures on theoretical 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.

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