Exercises: Quantum mechanical model

A beam of electrons travels at a constant speed of $2.20 \times 10^{6}\,\mathrm{m\times s^{-1}}$. Data: mass of the electron $m_{\mathrm{e}} = 9,109 \times 10^{-31}\,\mathrm{kg}$; Planck constant $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$.

1. a) Calculate the de Broglie wavelength associated with each electron in the beam.
2. b) Justify whether this wavelength is comparable to typical interatomic distances in a solid (of the order of $10^{-10}\,\mathrm{m}$) and what physical consequence this has for particle diffraction.

The speed of an electron is determined experimentally as $v = 4.50 \times 10^{5}\,\mathrm{m\times s^{-1}}$ with an uncertainty of $0.10\,\%$. Data: $m_{\mathrm{e}} = 9,109 \times 10^{-31}\,\mathrm{kg}$; $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $\pi = 3.1416$.

1. a) Determine the minimum uncertainty in the measurement of its position ($\Delta x$).
2. b) Using the result, explain why the classical concept of a defined trajectory is not applicable to an electron inside the atom.

Within quantum mechanics, the concept of orbit in the Bohr model is replaced by that of atomic orbital.

1. a) Define precisely the term "atomic orbital", stating which physical quantity the square of the wave function ($|\psi|^2$) represents.
2. b) State whether knowing the orbital allows the position of an electron to be predicted with certainty before it is measured, and justify your answer by citing the relevant physical principle.

The following illustration shows a two-dimensional cross-section of the dot density corresponding to an atomic orbital of type $s$, highlighting an inner region of high relative density (A) and a peripheral region of lower density (B), bounded by a contour surface. Observe the diagram

and answer the following questions:

1. a) Explain the physical meaning of the dot density in this diagram and justify why a precise circular or elliptical orbit is not drawn in the quantum model.
2. b) Describe what the outer contour line usually shown in textbooks for this type of orbital represents, and explain whether the algebraic signs ($+$ or $-$) that are sometimes assigned to wave functions represent positive or negative electric charge.

You can do it on paper and send a photo with your answer.

Consider two particles with the same speed of $15.0\,\mathrm{m\times s^{-1}}$: a very slow free neutron and a microscopic dust particle with a mass of $2.00 \times 10^{-10}\,\mathrm{g}$. Data: mass of the neutron $m_{\mathrm{n}} = 1,675 \times 10^{-27}\,\mathrm{kg}$; Planck constant $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$.

1. a) Calculate the de Broglie wavelength associated with each of the two particles.
2. b) Compare the values obtained with observable dimensions (such as visible light or atomic spacing) and explain why wave behaviour is relevant for the neutron but completely negligible for the dust particle.

The uncertainty in the measurement of motion is analysed for two bodies: an electron confined in a theoretical nuclear region of $\Delta x = 1.00 \times 10^{-14}\,\mathrm{m}$ and a tennis ball of mass $58.0\,\mathrm{g}$ located with a precision of $\Delta x = 1.00 \times 10^{-6}\,\mathrm{m}$. Data: $m_{\mathrm{e}} = 9,109 \times 10^{-31}\,\mathrm{kg}$; $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^{8}\,\mathrm{m\times s^{-1}}$; $\pi = 3.1416$.

1. a) First calculate $\Delta p_{\min}=h/(4\pi\Delta x)$ and the value that would result for $\Delta v$ if the non-relativistic approximation $p=mv$ were applied in both cases.
2. b) Explain why the formal value obtained for the electron shows that the non-relativistic approximation is no longer valid, and does not show that an electron can exceed the speed of light. Compare this case with the ball.

In an electron gun, electrons start from rest and are accelerated by an electric potential difference $\Delta V = 150\,\mathrm{V}$. The beam is then passed through a collimator that restricts its transverse coordinate with an uncertainty $\Delta x = 0.200\,\mathrm{nm}$. Data: elementary charge $e = 1,602 \times 10^{-19}\,\mathrm{C}$; mass of the electron $m_{\mathrm{e}} = 9,109 \times 10^{-31}\,\mathrm{kg}$; Planck constant $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $\pi = 3.1416$.

1. a) Calculate the speed reached by the electrons and determine their associated de Broglie wavelength.
2. b) Calculate the minimum uncertainty introduced in the transverse component of their velocity ($\Delta v_x$) and determine the approximate minimum angular spread of the beam $\theta \approx \Delta v_x / v$. Conclude whether it is acceptable to describe the beam using classical rays with straight-line trajectories and no divergence.

A particle impact detector records individual electrons transmitted through a thin polycrystalline metal foil with interplanar spacing $d = 0.125\,\mathrm{nm}$. At the start, when only 20 electrons have arrived, the screen records apparently random point impacts. After the continued passage of $10^{6}$ electrons, the impacts accumulate to form a sharp pattern of concentric rings identical to optical interference patterns. Data: $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $m_{\mathrm{e}} = 9,109 \times 10^{-31}\,\mathrm{kg}$.

1. a) Given that the first-order constructive diffraction rings follow the geometric condition $\lambda = 2d\sin\theta$ and that the first ring appears at an angle $\theta = 30.0^\circ$, determine the speed of the incident electrons.
2. b) Analyse this experiment in light of wave-particle duality and the quantum mechanical concept of probability: clarify whether each electron behaves as a particle or as a wave during its propagation and detection.