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## quantum physics

### (9 points)0. Series 31. Year - P.

*We are sorry. This type of task is not translated to English.*

### (6 points)6. Series 29. Year - S. A closing one

- Find, in literature or online, the change of enthalpy and Gibbs free energy in the following reaction

$$2\,\;\mathrm{H}_2 \mathrm{O}_2\longrightarrow2\,\mathrm{H}_2\mathrm{O},$$

where both the reactants and the product are gases at standard conditions. Find the change of entropy in this reaction. Give results per mole.

- Power flux in a photon gas is given by

$j=\frac{3}{4}\frac{k_\;\mathrm{B}^4\pi^2}{45\hbar^3c^3}cT^4$.

Substitute the values of the constants and compare the result with the Stefan-Boltzmann law.

- Calculate the internal energy and the Gibbs free energy of a photon gas. Use the internal energy to write the temperature of a photon gas as a function of its volume for an adiabatic expansion (a process with $δQ=0)$.

**Hint:** The law for an adiabatic process with an ideal gas was derived in the second part of this series (Czech only).

- Considering a photon gas, show that if $δQ⁄T$ is given by

$$\delta Q / T = f_{,T} \;\mathrm{d} T f_{,V} \mathrm{d} V\,,$$

then functions $f_{,T}$ and $f_{,V}$ obey the necessary condition for the existence of entropy, that is

$$\frac{\partial f_{,T}(T, V)}{\partial V} = \frac{\partial f_{,V}(T, V)}{\partial T} $$

### (2 points)5. Series 28. Year - 1. stiffness of Mr. Planck

Maybe you have heard about the so called Planck's units ie. units expressed in the form of fundamental physics constants – speed of light $c≈3.00\cdot 10^{8}\;\mathrm{m}\cdot \mathrm{s}^{-1}$, gravitational constant $G=6.67\cdot 10^{-11}\;\mathrm{m}\cdot \mathrm{kg}^{-1}\cdot \mathrm{s}^{-2}$ and the reduced Planck's constant $h=1.05\cdot 10 ^{-34}\;\mathrm{kg}\cdot \mathrm{m}\cdot \mathrm{s}^{-1}$. This way Planck's time, Planck's length and Planck's weight are often mentioned. What if we were interested in „Planck's spring constant“? Using dimensional analysis with $c$, $G$ and $h$ the equation of the unit relating to the unit of a spring constant [ $k]=\;\mathrm{kg}\cdot \mathrm{s}^{-2}$. To determine the equation assume that the unknown and from dimensional analysis undeterminable dimensionless constant is equal to 1.

Karel was learning quantumdots

### (6 points)6. Series 27. Year - S. series

- How will the spectrum of an open string on a mass level $M=2⁄α′?$ How many possible states of the string on this level?
- If we consider the interaction of tachyons with other strings, we would find out, že ho můžeme popsat přibližně jako částici pohybující se v nějakém potenciálu. We consider a model of a string that is fastened on a unstable D-brane. The relevant potential of the tachyon is defined by

$$V(\phi)=\frac{1}{3\alpha'}\frac{1}{2\phi _0}(\phi-\phi _0)^2\left (\phi \frac{1}{2}\phi _0\right )\,,$$

where $$\alpha'$$

- The theory of superstrings enables the description of fermions. For their description one needs anticomutating variables. For those one creates an anticomutator instead of a comuator with the relation

$$\{A,B\}=AB BA$$

Find two such $$2\times 2$$

…

### (6 points)5. Series 27. Year - S. string

- We consider only open strings and we shall limit ourselves merely to three dimensions. Draw how the following things look like

- a string moving freely through timespace,
- a string fixed with both ends to a D2-brane,
- a string between a D2-brane and D1-brane.

Where can the strings end in the case of three parallel D2-branes?

- Choose one of the functions

$$\mathcal{P}_{\mu}^{\tau}$$

ot $$\mathcal{P}_{\mu}^{\sigma}$$ that was defined in the first part of the series and find its explicit

form (in other words a direct dependence on $$\dot{X}^{\mu}$$ and <img

src=„https://latex.codecogs.com/gif.latex?X'^{\mu}“>). Show that the conditions $$\vect{X}'\cdot \dot{\vect{X}}=0$$

and $$|\dot{\vect{X}}|^2=-|\vect{X}'|^2$$

- Find the spectrum of energies of a harmonic oscilator.

- The energy of the oscilator is given by the hamiltonian

$$\hat{H}=\frac{\hat{p}^2}{2m} \frac{1}{2}m\omega^2\hat{x}^2$$

The second expression is clearly the potential energy while the first gives after substituting in $$\hat{p}=m\hat{v}$$ kinetic energy. We define linear combination as

$$\hat{\alpha}=a\hat{x} \;\mathrm{i} b\hat{p}$$ . Find the real constants <img

src=„https://latex.codecogs.com/gif.latex?a“> a $b$ , such that the Hamiltonian will have the form of

<img src=„https://latex.codecogs.com/gif.latex?\hat{H}=\hbar \omega \left(\hat{\alpha} ^{\dagger}\hat{\alpha}+\frac{1}

{2}\right)\,,“> where $$\hat{\alpha} ^{\dagger}$$ is the complex conjugate <img

src=„https://latex.codecogs.com/gif.latex?\hat{\alpha}“>.

- Show from your knowledge of canoninc commutation relations for

$$\hat{x}$$

and $$\hat{p}$$ that the following is true

<img src=„https://latex.codecogs.com/gif.latex?\left[\hat{\alpha},\hat{\alpha}\right]=0\,,\quad\left[\hat{\alpha} ^{\dagger},\hat{\alpha} ^

{\dagger}\right]=0\,,\quad\left[\hat{\alpha} ,\hat{\alpha} ^{\dagger}\right]=1\,.“>

- In the spectrum of the oscilator there will surely be the state with the lowest possible energy which corresponds to the smallest possible

amount of oscilating. Lets call it $$|0\rangle$$ . This state must fulfill <img

src=„https://latex.codecogs.com/gif.latex?\alpha |0\rangle =0“>. Show that its energy is equal to $$\hbar\omega/2$$ , ie. $$\hat{H}|0\rangle=\hbar\omega/2|0\rangle$$ . Furthermore prove that if $$\alpha |0\rangle \neq 0$$ then we have a contradiction with the fact that <img

src=„https://latex.codecogs.com/gif.latex?|0\rangle“> has a minimal energy ie. <img src=„https://latex.codecogs.com/gif.latex?\hat{H}\alpha |0\rangle=E\alpha

$$E<\hbar\omega/2$$ . All the eigenstates of the Hamiltonian can be described

as $$\left(\alpha^{\dagger}\right) ^n|0\rangle$$

for $$n=0,1,2,\dots$$ Find the energy of these states, in other words find such numbers <img

src=„https://latex.codecogs.com/gif.latex?E_n“> that <img src=„https://latex.codecogs.com/gif.latex?\hat{H}\left(\alpha^{\dagger}\right) ^n|0\rangle=E_n\left

(\alpha^{\dagger}\right)^n|0\rangle“>.

*Tip* Use the commutation relation for
$$\hat{\alpha}^{\dagger}$$
a <img

src=„https://latex.codecogs.com/gif.latex?\hat{\alpha}“>.

### (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