Quiz: Quantum Description of the Atom — 10 questions

1. What energy does an electron have in the second principal energy level of a hydrogenic species?

A. $-6.8\ \mathrm{eV}$

B. $-27.2\ \mathrm{eV}$

C. $-3.4\ \mathrm{eV}$

D. $-13.6\ \mathrm{eV}$

Answer: (C)

Explanation

Substituting $n=2$ into $E=-\frac{13.6\ \mathrm{eV}}{n^2}$ gives $E=-\frac{13.6}{4}\ \mathrm{eV}=-3.4\ \mathrm{eV}$. The value $-13.6\ \mathrm{eV}$ corresponds to the ground state with $n=1$, not the second level.

2. Why do atoms produce line spectra rather than spectra containing every possible wavelength?

A. They produce radiation with a fixed wavelength determined by atomic mass.

B. They emit light continuously as electrons move through every possible orbit.

C. They absorb wavelengths according to the total number of electrons present.

D. They exchange energy in defined amounts associated with allowed transitions.

Answer: (D)

Explanation

Line spectra arise because atoms exchange specific amounts of energy, so transitions produce defined wavelengths. A continuous spectrum instead contains a broad uninterrupted range of wavelengths rather than discrete spectral lines.

3. An electron moves non-relativistically with its speed doubled while its mass remains constant; how does its de Broglie wavelength change?

A. It becomes twice as large.

B. It becomes half as large.

C. It becomes four times as large.

D. It remains unchanged.

Answer: (B)

Explanation

Because $\lambda=\frac{h}{p}$ and $p=mv$, doubling the speed doubles the momentum and reduces the wavelength by a factor of two. The wavelength would increase when momentum decreases, not when speed increases.

4. What observation in the Davisson–Germer experiment demonstrated the wave behavior of electrons?

A. Electrons accelerated through nickel lost all their kinetic energy.

B. Electrons scattered from nickel formed a diffraction pattern.

C. Electrons absorbed by nickel followed circular atomic orbits.

D. Electrons emitted from nickel produced a continuous spectrum.

Answer: (B)

Explanation

The experiment showed that electrons scattered by a nickel crystal formed a diffraction pattern, a characteristic wave phenomenon. A continuous spectrum does not provide the specific diffraction evidence that established electron wave behavior.

5. What does the formula $E=-\frac{13.6\,\text{eV}}{n^2}$ represent in the context of atomic physics?

A. The probability of photon emission during electron transitions.

B. The energy levels of a hydrogenic atom depending on the principal quantum number $n$.

C. The classical kinetic energy of an electron in a hydrogen atom.

D. The wavelength of light emitted by hydrogen atoms.

Answer: (B)

Explanation

This formula describes the quantized energy levels of a hydrogenic atom, where $n$ is the principal quantum number. It shows that energy depends inversely on the square of $n$, which is fundamental to the Bohr model.

6. What is the formula that describes the energy levels of an electron in a hydrogenic species?

A. $E=13.6\ \mathrm{eV} \times n^2$

B. $E=-\frac{13.6\ \mathrm{eV}}{\sqrt{n}}$

C. $E=\frac{13.6\ \mathrm{eV}}{n}$

D. $E=-\frac{13.6\ \mathrm{eV}}{n^2}$

Answer: (D)

Explanation

The formula $E=-\frac{13.6\ \mathrm{eV}}{n^2}$ describes the quantized energy levels of an electron in a hydrogen atom, where $n$ is the principal quantum number. The other options do not correctly represent the energy levels in this context.

7. What is the primary purpose of the wave function in quantum mechanics?

A. It predicts the particle's behavior based solely on its energy levels.

B. It determines the classical trajectory of a particle in space.

C. It completely describes the quantum state of a particle and allows the calculation of physical observables.

D. It provides the exact position and momentum of a particle simultaneously.

Answer: (C)

Explanation

The wave function fully characterizes a quantum system's state and enables the calculation of measurable quantities. Unlike classical descriptions, it does not specify exact position and momentum simultaneously, reflecting the uncertainty principle.

8. When was the Heisenberg uncertainty principle formally introduced into quantum mechanics?

A. 1927, during the development of matrix mechanics

B. 1932, with Schrödinger's wave equation

C. 1913, following the discovery of the quantum hypothesis by Planck

D. 1924, in the PhD thesis of Werner Heisenberg

Answer: (D)

Explanation

The Heisenberg uncertainty principle was formally introduced in 1924 by Werner Heisenberg in his PhD thesis. The other dates correspond to different milestones: 1927 relates to the formulation of matrix mechanics, 1913 to Planck's quantum hypothesis, and 1932 to Schrödinger's wave equation.

9. How do operators in quantum mechanics differ from classical physical quantities?

A. Operators are physical objects that can be observed directly, while classical quantities are abstract mathematical constructs.

B. Operators are mathematical entities that act on wave functions to produce measurable quantities, whereas classical quantities are directly measurable physical properties.

C. Operators always commute with each other, unlike classical quantities which do not commute.

D. Operators are only used for position and momentum, while classical quantities include all physical properties.

Answer: (B)

Explanation

Operators in quantum mechanics are mathematical tools that act on wave functions to extract physical observables, unlike classical quantities which are directly measurable properties. The key difference is that operators can be non-commutative, which is not a feature of classical variables.

10. What is the effect of applying the Hamiltonian operator to an energy eigenstate in the Schrödinger equation?

A. It transforms the wave function into a different state, representing a change in energy.

B. It measures the probability density of finding a particle at a certain position.

C. It determines the velocity of the particle within the potential well.

D. It returns the same wave function multiplied by the energy value, indicating a quantized energy level.

Answer: (D)

Explanation

Applying the Hamiltonian operator to an energy eigenstate yields the same wave function multiplied by its energy, demonstrating that the state has a definite energy. This is a fundamental aspect of quantum mechanics, showing the cause-effect relationship between the Hamiltonian and energy measurement.