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1. Neka je $H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})$ Hamiltonijan kvantnomehaničkog sustava s centralnim potencijalom $V(r) = -\frac{\alpha}{r}e^{-\mu r}$. Nađite asimptotsko ponašanje radijalne funkcije $R_{nl}(r)$ za $r \to \infty$ i odredite uvjet za vezana stanja u terminima kvantnih brojeva $n$ i $l$. <br>
2. Razmotrite nuklearnu reakciju $\ce{^{235}_{92}U + n -> ^{92}_{36}Kr + ^{141}_{56}Ba + 3n}$. Ako je energija veze po nukleonu za $\ce{^{235}U}$ jednaka $7.59\ \text{MeV}$, za $\ce{^{92}Kr}$ $8.61\ \text{MeV}$, a za $\ce{^{141}Ba}$ $8.34\ \text{MeV}$, izračunajte oslobođenu energiju u ovoj fisiji koristeći formulu $Q = \Delta m c^2$ i odredite koliki postotak mase se pretvara u energiju.<br>
3. Riješite Schrödingerovu jednadžbu za harmonički oscilator u tri dimenzije s Hamiltonijanom $H = \sum_{i=1}^3 \left( \frac{p_i^2}{2m} + \frac{1}{2}m\omega_i^2 x_i^2 \right)$ gdje su $\omega_1 \neq \omega_2 \neq \omega_3$. Nađite energijske nivoe $E_{n_1n_2n_3}$ i odgovarajuće valne funkcije $\psi_{n_1n_2n_3}(x_1,x_2,x_3)$.<br>
4. Izračunajte vjerojatnost tuneliranja alfa-čestice kroz Coulombovu barijeru za nuklid $\ce{^{238}_{92}U}$ koristeći WKB aproksimaciju. Pretpostavite da je potencijal $V(r) = \begin{cases} V_0 & r < R \\ \frac{2(Z-2)e^2}{4\pi\epsilon_0 r} & r \geq R \end{cases}$ gdje je $R = 1.2 \times (238)^{1/3}\ \text{fm}$, a kinetička energija alfa-čestice $E_\alpha = 4.27\ \text{MeV}$.
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5. Odredite vlastite vrijednosti i vlastite funkcije operatora ukupnog momenta impulsa $\hat{\mathbf{J}}^2$ i njegove $z$-komponente $\hat{J}_z$ za sustav dvaju nukleona s spinskim kvantnim brojevima $s_1 = s_2 = \frac{1}{2}$ i orbitalnim momentima impulsa $l_1 = 1$, $l_2 = 2$. Koristite Clebsch-Gordanove koeficijente za izračunavanje vektora stanja u bazi ukupnog momenta impulsa $|j,m_j\rangle$.
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<p> (1) Kolika je razlika u vremenu ?</p>
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<p>4) 假设 \( A = \begin{bmatrix} §§V1(2,4,1)§§ & §§V2(3,5,1)§§ \\ §§V3(1,3,1)§§ & §§V4(4,6,1)§§ \end{bmatrix} \) 和 \( B = \begin{bmatrix} §§V5(5,7,1)§§ & §§V6(-1,1,0.5)§§ \\ §§V7(2,4,1)§§ & §§V8(0,2,0.5)§§ \end{bmatrix} \)。计算 \( A + B \) 和 \( A - B \)。</p>
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<p>5) 计算积分：
$$\int_{§§V0(0,10,1)§§}^{§§V1(10,20,1)§§} (§§V2(1,5,1)§§x^2 + §§V3(1,5,1)§§x + §§V4(0,10,1)§§) \, dx$$</p>
<p>6) 假设函数 \( f(x) \) 定义为 \( f(x) = x^2 - \sqrt{§§V1(3,15,3)§§}x + \sqrt{§§V2(1,10,1)§§} \)，它描述了一个抛物线的形状。确定常数 \( c \) 的值。</p>
<p>7) 计算两颗氘核融合时释放的能量：<p>
<img src="https://www.mathkiss.com/uploads/Krava1.jpg.jpg" width="300"/></p>
$$^2_1\text{H} + ^2_1\text{H} \rightarrow ^3_2\text{He} + n$$
<h2> For §§N0§§ \( \frac{§§V0(1,50,1)§§}{§§V1(1,5,1)§§} \) </h2>
<p>(a) Berechne: \( \sqrt{§§V1(4,25,3)§§} \times \sqrt{§§V2(2,15,2)§§} \)</p>
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<p>(b) Vereinfache den Ausdruck: \( \frac{\sqrt{§§V3(9,36,3)§§}}{\sqrt{§§V4(2,10,2)§§}}
+ \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}} \left(1 + \frac{1}{2} \sin^2(3x) + \frac{\alpha}{\beta}\sqrt{\gamma+\delta} \cdot \frac{\sqrt[3]{\theta^2 + \phi^2}}{\psi + \frac{\omega}{\chi}}\right) \,dx
\)</p>
<p>(c) Multipliziere: \( (\sqrt{§§V5(3,20,2)§§} + \sqrt{§§V6(2,12,2)§§}) \times (\sqrt{§§V7(4,18,2)§§} - \sqrt{§§V8(1,9,1)§§}) \)</p>
<p>(d) Dividiere: \( \frac{\sqrt{§§V9(16,64,8)§§}}{\sqrt{§§V10(4,16,2)§§}} \)</p>
<p>(e) Vereinfache den Ausdruck: \(
\begin{equation}
x = a_0 + \frac{1}{\displaystyle a_1
+ \frac{1}{\displaystyle a_2
+ \frac{1}{\displaystyle a_3 + a_4}}}
\end{equation}
\)</p>
<p>(f) Berechne: \( \frac{\sqrt{§§V13(25,100,5)§§} \times \sqrt{§§V14(49,196,7)§§}}{\sqrt{§§V15(9,36,3)§§}} \)</p>
<p>(g) Multipliziere und vereinfache: \( (\sqrt{§§V16(5,25,5)§§} + \sqrt{§§V17(3,15,3)§§})^2 \)</p>
<p>(h) Dividiere: \( \frac{\sqrt{§§V18(81,324,9)§§}}{\sqrt{§§V19(9,36,3)§§}}
- \begin{bmatrix}
\int_{0}^{1} e^{2t}\sin(t)\,dt & \frac{\cos^2(3\theta)}{\sqrt{2}} & 0 & \frac{\pi}{4} & \ln(2) \\
\frac{\sqrt{5}}{\phi} & \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2}\,dx & \frac{1}{\sqrt[3]{\alpha + \beta}} & \frac{\gamma^2}{\delta} & \frac{\sqrt{\pi}}{2} \\
\frac{\theta}{2\pi} & \frac{1}{\sqrt{3}} & \int_{0}^{1} \frac{e^{2t}}{\sqrt{t}}\,dt & \frac{\omega}{\chi} & \frac{\sqrt{2}}{\sqrt{\pi}} \\
\ln(\sqrt{\pi}) & \frac{\sqrt[4]{\pi^3}}{\sqrt{\alpha}} & \frac{\sqrt{\beta}}{\sqrt[3]{\gamma}} & \int_{-\pi}^{\pi} \cos^2(\phi)\,d\phi & \frac{\sqrt[5]{\delta}}{\sqrt[6]{\varepsilon}}
\end{bmatrix}
\)</p>
<p>(i) Berechne: \( \sqrt{§§V20(20,100,10)§§} \div \sqrt{§§V21(2,8,2)§§} \)</p>
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<p>(j) \( \mathcal{L}_{\mathcal{T}}(\vec{\lambda}) = \sum_{\mathbf{x},\mathbf{s}\in\mathcal{T}} \log P(\mathbf{x}|\mathbf{S}) - \sum_{i=1}^m \frac{\lambda_i^2}{2\sigma^2}
\)
</p>
<td class="border border-dark">§§F(GenerateClock1, §§V6(0,11,1,11)§§, 0, 40)§§ §§F(GenerateClock1, §§V6_2§§, 0, 40)§§ <br>___ sati i ___ minuta</td>
<td> </td>
<td class="border border-dark">§§F(GenerateClock1, §§V6_3§§, 0, 40)§§ §§F(GenerateClock1, §§V6_4§§, 0, 40)§§ <br>___ sati i ___ minuta</td>
<td> </td>
<td class="border border-dark">§§F(GenerateClock1, §§V6_5§§, 0, 40)§§ §§F(GenerateClock1, §§V6_6§§, 0, 40)§§ <br>___ sati i ___ minuta</td>
<td> </td>
<td class="border border-dark">§§F(GenerateClock1, §§V6_7§§, 0, 40)§§ §§F(GenerateClock1, §§V6_8§§, 0, 40)§§ <br>___ sati i ___ minuta</td>
<td> </td>
</tr>
</table>