(10 points)3. Series 36. Year - S. quantum of orbital
Similarly to the series, use the Hückel method to create the Hamiltonian matrix for the cyclobutadiene molecule and verify that its eigenvalues are $\alpha +2\beta $, $\alpha $, $\alpha $, $\alpha -2\beta $. Sketch the diagram of the final energies in the resulting orbitals. And show how the electrons will occupy them. $(4~b)$ Bonus: What is the main difference in the characterics of these orbitals and their occupancy compared to a benzene molecule we showed in the series? What are the consequences for the cyclobutadiene molecule? $(2~b)$
Try going back to the beta-carotene molecule and calculate again at what wavelength it should absorb using the Hückel method. What should the value of the parameter $\beta $ be equal to in order to be consistent with the experimental results Alternative: If you encounter a problem with the diagonalisation of the hamiltonian, solve the problem statement with the hexa-1,3,5-triene molecule. The experimentally determined absorption value in this case is at a wavelength of $250 \mathrm{nm}$. $(4~b)$
What happens to a molecule (a molecule with only simple bonds is sufficient) if we use UV light to excite an electron from the $\sigma $ to the $\sigma ^\ast $ orbital? $(2~b)$
Mikuláš gives presents again, this time at the right time of the year, almost.
(10 points)2. Series 36. Year - S. counting the quanta
Find a beta-carotene molecule and calculate what color should it have or rather what wavelength it absorbs. Use a simple model of an infinite potential well in which $\pi $ electrons from double bonds are „trapped“ (i.e., two electrons for each double bond). The absorption then corresponds to such a transition that an electron jumps from the highest occupied level to the first unoccupied level.
Compare the calculated value with the experimental one. Why doesn't the value obtained by our model come out the way we would expect? (5b)
Let's try to improve our model. When studying some substances, especially metals or semiconductors, we introduce the effective mass of the electron. Instead of describing the environment in which the electrons move in a complex way, we pretend that the electrons are lighter or heavier than in reality. What mass would they need to have to give us the correct experimental value? Give the result in multiples of the electron's mass. (2b)
If we produce microscopic spheres (nanoparticles) of cadmium selenide ($\ce {CdSe}$) with a size of $2{,}34 \mathrm{nm}$, they will glow bright green when irradiated by UV light with a wavelength of $536 \mathrm{nm}$. When enlarged to a size of $2{,}52 \mathrm{nm}$, the wavelength of the emitted light shifts to the yellow region with a wavelength of $570 \mathrm{nm}$. What would the size of spheres need to be to make them emit orange with a wavelength of $590 \mathrm{nm}$? (3b)
Hint: $\ce {CdSe}$ is a semiconductor, so it has a fully occupied electron band, then a (narrow!) forbidden band, and finally an empty conduction band. Thus, we must consider that the emitted photon corresponds to a jump from the conduction band (where such states are as in the infinite potential well) to the occupied band. Therefore, all the energies of the emitted photons will be shifted by an unknown constant value corresponding to the width of the forbidden band.
Finally, a bonus for those who would be disappointed if they didn't integrate – the 1s orbital of the hydrogen atom has a spherically symmetric wave function with radial progression $\psi (r) = \frac {e^{-r/a_0}}{\sqrt {\pi }a_0^{3/2}}$, where $a_0=\frac {4\pi \epsilon _0\hbar ^2}{me^2}$ is the Bohr radius. Since the orbitals as functions of three spatial variables would be hard to plot, we prefer to show the region where the electron is most likely to occur. What is the radius of the sphere centered on the nucleus in which the electron will occur with a probability of $95 \mathrm{\%}$? (+2b)
How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40 \mathrm{cm}$ and its focal length is $4 \mathrm{m}$? Our Gaussian laser beam has an input diameter $30 \mathrm{cm}$ and a wavelength $1~053 nm$. The radius of the focus (parameter $\sigma $) of the Gaussian beam can be obtained using
\[\begin{equation*}
r = \frac {2}{\pi }\lambda \frac {f}{D}
\end {equation*}\]
where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda $ is the wavelength of the laser.
The laser beam is focused on a surface of a nuclear fuel pellet of a $1 \mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14} W.cm^{-2}$? The radius of the focus is $25 \mathrm{\micro m}$ and a pulse lasts $10 \mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?
What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14} W.cm^{-2}$? Assume that we have one such beam and it shines homogenously on the pellet „from all directions“.
What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of
$0,2 \mathrm{\micro m}$,
$5 \mathrm{\micro m}$.
How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
What else can help us minimize the RT instability?
How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma1)
The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume
that the density of the plasma decreases exponentially with distance from the target,
that the density of the plasma decreases linearly with distance from the target.
What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.
What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?
Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?
What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? Note: Assume that the initial plasma is a monatomic ideal gas.
Assume a charged chord with linear density $\rho $, uniformly charged with linear charge density $\lambda $. The tension in the chord is $T$. It is placed in a magnetic field of constant magnitude $B$ pointing in the direction of the chord in equilibrium. Your task is to describe several aspects of the chord's oscillations. First, we want to write the appropriate wave equation. Neglect the effects of electromagnetic induction (assume the chord to be a perfect insulator; that also means the charge density does not change) and find the Lorentz force acting on an unit length of the chord for small oscilations in both directions perpendicular to the equilibrium position. Use this force to write the wave equation (which will also include the effects of the tension). Apply the Fourier substitution and determine the disperse relation in the approximation of a weak field $B$; more specifically, neglect the terms that are of higher than linear order in $\beta = \frac {\lambda B}{k \sqrt {\rho T}} \ll 1$, where $k$ is the wavenumber. Find two polarization vectors, this time neglect even the linear order of $\beta $. Now suppose that in a particular spot on the chord, we create a wave oscilating only in one specific direction. How far from the original spot will be the wave rotated by ninety degrees from the original direction?
Štěpán was nostalgically remembering the third serial task.
(10 points)5. Series 34. Year - S. resonance and damped oscillations
On a tense rope, waves can exist with the deflection $\f {u}{x, t}$ from the equilibrium, that satisfy the wave equation with damping
\[\begin{equation*}
\ppder {u}{t} = v^2 \ppder {u}{x} + \Gamma \pder {u}{x} ,
\end {equation*}\]
where $v$ is the phase velocity and $\Gamma $ is the coefficient of damping. Do a fourier substitution and find the dispersion relation. Solve it for the wavenumber $k$. What condition, in terms of frequency $\omega $, phase velocity $v$ and the coefficient $\Gamma $, must the waves meet in order to create nodes on the rope (i.e. points in which the rope stays in equilibrium position, but around which the rope is moving)?
Consider a jump rope attached firmly at one end to a fixed wall. At the distance $L$ from the wall, we start moving the rope up and down to create waves. The jump rope has a linear density $\lambda $ and the constant tension $T$ in the direction away from the wall. The deflection then satisfies the equation
\[\begin{equation*}
\ppder {u}{t} = \frac {T}{\lambda } \ppder {u}{x} .
\end {equation*}\]
For the deflection of the end of the rope that is moving satisfies $\f {u_0}{t} = A \f {\cos }{\omega _0 t}$. Assume the solution can be written in the form of two planar waves moving in the opposite direction to each other. Find the solution using only the parameters given in this problem statement, that is $T$, $\lambda $, $L$, $A$ and $\omega _0$. For certain frequencies, the solution has a diverging amplitude (i.e. growing beyond any limits). Find their values and the respective wavelenghts.
Let's have a point source of light and a planar glass panel with a refractive index $n = 1,50$. In the foot of the perpendicular from the source to the panel there are wavefronts with a radius of curvature $R = 5,00 \mathrm{m}$ inside the glass. What is the real distance between the source and the panel?