Question

Find the ground-state energy (in eV) of an electron in a onedimensional box, if the box is of length \(L=0.100 \mathrm{nm}\).

Short answer

Answer: The ground-state energy of an electron in a one-dimensional box with a length of 0.100 nm is approximately 1.505 eV.

Step-by-step solution

Identify the necessary constants and variables

We'll need the following constants to find the ground-state energy: 1. Planck's constant (\(\hbar\)): \(1.0545718 \times 10^{-34} Js\) 2. Electron mass (m): \(9.10938356 \times 10^{-31}kg\) 3. Length of the box (L): \(0.100 nm\) or \(0.100 \times 10^{-9} m\) 4. Energy level (n): 1 (Ground-state) We also need to recall the conversion factor from joules to electron volts (eV): \(1eV = 1.602176634\times 10^{-19}J\)

Calculate the Ground-State Energy in Joules

Using the formula provided, \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\) and plugging in the given values for n, L, \(\hbar\), and m, we get: \(E_1 = \frac{(1)^2\pi^2(1.0545718 \times 10^{-34}Js)^2}{2(9.10938356 \times 10^{-31}kg)((0.100 \times 10^{-9} m)^2)}\) Now, just plug in the numbers to find the energy in joules.

Convert the Ground-State Energy to Electron Volts (eV)

To convert the energy calculated in Step 2 from joules to electron volts, we can use the conversion factor mentioned earlier: \(1eV = 1.602176634\times 10^{-19}J\) Divide the energy value obtained in Joules by the conversion factor: \(E_1(eV) = \frac{E_1(J)}{1.602176634\times 10^{-19}J/eV}\)

Calculate the Result and Provide the Answer

Now, complete the calculations in Step 2 and Step 3 to find the ground-state energy in electron volts and make sure to include the correct units for the answer.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. At its core, quantum mechanics departs from classical mechanics in that it introduces the concept of quantization, where certain properties, such as energy, are not continuous but rather occur in discrete 'quanta'.

One of the most significant features of quantum mechanics is the wave-particle duality, which postulates that every particle or quantum entity exhibits both wave and particle properties. This duality is apparent in the 'particle in a box' model, an idealized system that shows how quantization emerges in a confined space.

Another cornerstone of quantum mechanics is the uncertainty principle, which states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. This principle dictates that the more precisely one property is known, the less precisely the other can be measured. This inherent uncertainty is crucial to understanding why only specific energy levels are allowed for a quantum system, such as an electron in a box.

Particle in a Box

The 'particle in a box' model is a simplified quantum mechanical system that helps us understand how quantization of energy levels arises. In this model, a particle such as an electron is confined to move in a one-dimensional space with impenetrable walls at each end.

The model assumes that the potential energy within the box is zero and infinite outside, prohibiting the particle from escaping. As a consequence, the particle's wavefunction, which describes its quantum state, can only take specific forms that fit neatly within the box, leading to quantized energy levels.

From the 'particle in a box' model, the ground-state energy can be calculated as the lowest allowable energy state of the particle. This model is essential for understanding more complex quantum systems and finds applications in quantum chemistry, solid-state physics, and nanotechnology.

Planck's Constant

Planck's constant, denoted by \(\hbar\) (often the reduced Planck's constant) or \(h\), is a physical constant that plays a significant role in quantum mechanics. It is the proportionality factor between the minimal increment of energy, known as a quantum, and the frequency of the corresponding electromagnetic wave.

With the value of \(\hbar = 1.0545718 \times 10^{-34} Js\), Planck's constant enables calculation of the energy levels accessible to particles confined in quantum systems. The existence of this constant implies that all energy changes in the quantum scale are discrete and quantized. This fundamental constant is utilized in the calculation of energy states such as those in the 'particle in a box' problem.

Energy Level Quantization

Energy level quantization is the concept that energy of a quantum mechanical system can only exist in certain discrete values, rather than any value within a range. This quantization arises from the constraints imposed on a particle's wavefunction, which must satisfy the boundary conditions of the system it occupies.

For the 'particle in a box' problem, energy levels of a particle are quantized and can be calculated using the formula \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\), where \(n\) is the principal quantum number corresponding to different energy levels. In this equation, \(L\) is the length of the box and \(m\) is the mass of the particle, illustrating that the size of the box and the mass of the particle influence the spacing of the energy levels.

When applying this to determine the ground-state energy, we take \(n = 1\), which yields the lowest energy that the electron can possess in the box. This property of quantized energy levels is foundational to various fields, including quantum computing and the design of optical and electronic devices.