Question

When a hydrogen atom undergoes a transition from the \(n\) = 2 to the \(n\) = 1 level, a photon with \(\lambda\) = 122 nm is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n\) = 2 to \(n\) = 1 transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (\(Hint\): Compare the spacing between adjacent energy levels as a function of n.)

Short answer

(a) Calculate \( L \) using photon energy. (b) Compare \( E_1 \) to \( -13.6 \text{ eV} \). (c) 1D box poorly models hydrogen atom levels.

Step-by-step solution

Calculate Energy of Emitted Photon

Use the equation for the energy of a photon: \[ E = \frac{hc}{\lambda} \]where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \), \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength \( 122 \times 10^{-9} \text{ m} \). Substitute these values to find \( E \).

Energy Levels of a 1D Box

In a one-dimensional box, the energy levels are given by:\[ E_n = \frac{n^2 h^2}{8mL^2} \]where \( n \) is the quantum number, \( m \) is the electron mass \( 9.109 \times 10^{-31} \text{ kg} \), and \( L \) is the width of the box. Set up the equation for the transition energy from \( n = 2 \) to \( n = 1 \):\[ E_2 - E_1 = E \]

Solve for Width L of the Box

Substitute the expressions for \( E_2 \) and \( E_1 \) into the transition energy equation and solve for \( L \):\[ \frac{4h^2}{8mL^2} - \frac{h^2}{8mL^2} = E \]Rearrange to solve for \( L \):\[ L = \sqrt{\frac{3h^2}{8mE}} \]Substitute known values for \( h \), \( m \), and \( E \) and calculate \( L \).

Calculate Ground-State Energy

The ground-state energy \( E_1 \) for the calculated \( L \) is:\[ E_1 = \frac{h^2}{8mL^2} \]Substitute the calculated value of \( L \) from Step 3, along with the constants \( h \) and \( m \), to find \( E_1 \). Compare this with the known ground-state energy of a hydrogen atom \(-13.6 \text{ eV}\).

Evaluate Suitability of the Model

Compare the energy spacing in the one-dimensional box model with the actual hydrogen atom. The energy levels in a hydrogen atom go as \( 1/n^2 \), while in a box they are linear in \( n^2 \). This results in very different energy level spacing, especially as \( n \) increases.

Key concepts

Photon Emission

In the realm of quantum mechanics, understanding photon emission requires us to delve into the behavior of atoms and the energy they emit. When an atom's electron transitions between energy levels, it can release energy in the form of a photon. This process is instrumental in phenomena such as fluorescence and laser action.

In the exercise, a hydrogen atom transitions from the second energy level \(n = 2\) to the first energy level \(n = 1\). This loss of energy results in the emission of a photon with a specific wavelength, \(\lambda = 122 \text{ nm}\). The energy of this photon can be calculated using Planck's relation, \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light.

This transition exemplifies the discrete nature of energy in quantum systems. Each photon carries away the exact energy difference between the two levels, emphasizing the quantization of electronic states.

Hydrogen Atom

The hydrogen atom is the simplest atom, composed of a single proton and electron. Its simplicity is precisely what makes it an ideal subject for studying fundamental quantum concepts. The energy levels in a hydrogen atom are determined by the Rydberg formula, which suggests that the energy is proportional to \(1/n^2\), where \(n\) is a principal quantum number.

When a hydrogen atom transitions between levels, it emits or absorbs energy as photons. This is because the electron jumps from a higher to a lower energy level, or vice versa. The quantum model of the atom successfully explains this by visualizing electrons as existing in distinct energy states or 'shells'.

Understanding this behavior in hydrogen atoms helps us grasp more complex atoms. For example, the patterns observed in hydrogen’s spectrum have applications ranging from astrophysics to quantum computing.

One-Dimensional Box Model

The one-dimensional box model is a simplified framework used in quantum mechanics to illustrate how particles behave in constrained systems. Imagine an electron trapped in a perfectly rigid box with impenetrable walls. This model, though hypothetical, helps research the energy quantization principle.

In this model, the electron's energy levels are given by the equation \(E_n = \frac{n^2 h^2}{8mL^2}\), where \(n\) denotes the principal quantum number, \(h\) is Planck's constant, \(m\) is the electron's mass, and \(L\) is the box's width. The energy levels depend squarely on \(n\), which shows a linear relation as opposed to the reciprocal relation seen in hydrogen atoms.

This model predicts distinct, quantized states for the electron, similar to what is observed experimentally. While it provides insights, the model oversimplifies the actual complex atomic interactions, limiting its application to more realistic scenarios.

Energy Levels

Energy levels are core to understanding how quantum systems behave. In quantum mechanics, energy is quantized, meaning electrons in an atom can't have just any energy level, but specific, discrete levels. These levels are determined by the system in which the particle exists.

In the hydrogen atom, energy levels follow a \(1/n^2\) dependence, which results in decreasing gaps between levels as \(n\) increases. This results in unique spectral lines for each element, which can be observed in hydrogen's emission spectrum.

In contrast, the one-dimensional box model predicts energy levels dependent on \(n^2\), and hence, these levels do not converge as they do in hydrogen atoms. This unique discrepancy highlights the practical limitations of certain models in representing real-world quantum phenomena accurately. The difference in energy spacing, particularly for higher quantum numbers, illustrates why this model can be misleading if applied to hydrogen atoms.