Quiz 1, with solutions

Write your UNI on the top left of each page. Read each question carefully. Partial credit will be awarded if logical steps are shown. This is a take-home quiz. You may look up relevant information you need to complete the questions. Good luck!

Excited particle in a well

Imagine a particle in a 1D infinite square well. The particle goes from its ground state ($n=1$) to its 4th excited state ($n=4$).

(a) What wavelength of light is needed to excite the particle from $n=1$ to $n=4$?

Solution. Recall

$$E = \frac{n^2h^2}{8mL^2}$$

and $E = h\nu$.

(b) Approximately, what is the probability of the particle being in the region $x=0$ to $x=L/4$ in its $n=4$ excited state? (Hint, you do not need to use an integral if you do not want to.)

Solution. Recall the $n=4$ state has 3 nodes. The integral of a sinusoidal wave with 3 nodes from $x=0$ to $x=L/4$ is $1/4$ by inspection.

(c) Say the particle is replaced by another particle with twice its mass. How will this change the amount of energy needed to excite the bigger particle from $n=1$ to $n=4$? How will this change the probability of the bigger particle being in $x=0$ to $x=L/4$ for the $n=4$ state?

Solution. The energy will decrease by half, while the probability stays the same.

Spring bond

Two particles with the same mass have a bond that stretches around 3000 wavenumbers. If one of the particles is replaced with its isotope twice its mass, calculate the new stretching frequency of the bond.

Solution. Recall

$$\nu \propto \sqrt{\frac{k}{\mu}} \implies \frac{\nu_1}{\nu_2} \propto \sqrt{\frac{\mu_2}{\mu_1}}$$

For two particles of the same mass, $\mu = \frac{m^2}{2m} = \frac{m}{2}$, and for particles where $m_1 = 2m_2$, $\mu_2 = \frac{2m}{3}$. Therefore

$$\nu_2 = \sqrt{\tfrac{3}{4}}\,\nu_1$$

Multiple choice

1. An excited hydrogen atom emits light with a wavelength of 397.2 nm to reach the energy level for which $n=2$. In which principal quantum level did the electron begin?

• 7 ✓
• 6
• 5
• 4

2. Which of the following sets of quantum numbers is allowed?

• $n=3$, $l=3$, $m_l=0$, $m_s=-1/2$
• $n=4$, $l=3$, $m_l=2$, $m_s=-1/2$ ✓
• $n=2$, $l=1$, $m_l=-1$, $m_s=-1$
• $n=3$, $l=1$, $m_l=2$, $m_s=-1/2$

3. Arrange the following atoms in order of increasing size. (All four orderings are correct.)

• S Se Te ✓
• O P Fe ✓
• Be Na Rb ✓
• F Si Ba ✓

4. Indicate the bond order of $\text{O}_2$, $\text{O}_2^+$, benzene.

• 2, 1.5, 1.5
• 2, 2.5, 3
• 2, 2.5, 1.5 ✓
• 2, 1.5, 3

5. In the rotational spectrum of DCl, the rotational line spacings compared to those of HCl are approximately

• halved ✓
• doubled
• tripled
• unchanged