Question

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

Short answer

Answer: The ground state energy of an electron in the given one-dimensional quantum box is approximately 3.000 eV.

Step-by-step solution

Formula for the energy levels of an electron in a quantum box

The formula for the energy levels (E) of an electron in a one-dimensional box of length L is given by: \[ E_n = \frac{h^2 n^2}{8 m_e L^2} \] where E_n - the energy level of the electron n - quantum number (n = 1, 2, 3, ...) h - Planck's constant (\(h = 6.626 \times 10^{-34} \mathrm{Js}\)) m_e - mass of an electron (\(m_e = 9.109 \times 10^{-31} \mathrm{kg}\)) L - length of the box Ground state energy corresponds to n = 1.

Convert the length of the box to meters

The length of the box, L = 0.100 nm is given in nanometers. We need to convert it to meters (m) to match the SI units used in Planck's constant and the mass of the electron. To convert nanometers to meters, we can use the following conversion factor: 1 nm = \(10^{-9}\) m Hence, L = 0.100 nm × \(10^{-9}\) m/nm = \(1.000 \times 10^{-10}\) m

Calculate the ground state energy in joules

Now that we have L in meters, we can use the energy level formula to calculate the ground state energy E_1, with n = 1: \[ E_1 = \frac{h^2 (1)^2}{8 m_e (1.000 \times 10^{-10})^2} \] Plug in the values for h and m_e, and solve for E_1: \[ E_1 = \frac{(6.626 \times 10^{-34})^2}{8 (9.109 \times 10^{-31})(1.000 \times 10^{-10})^2} \mathrm{J} \] \[ E_1 \approx 4.802 \times 10^{-19} \text{ J} \]

Convert the ground state energy into electron volts (eV)

The energy E_1 is given in Joules (J), but the problem asks for the energy in electron volts (eV). To convert Joules to electron volts, we can use the following conversion factor: 1 eV = \(1.602 \times 10^{-19}\) J So the ground state energy in electron volts (eV) is: \[ E_1 = \frac{4.802 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 3.000 \text{ eV} \] Thus, the ground state energy of an electron in this one-dimensional quantum box is approximately 3.000 eV.

Key concepts

Quantum Physics

Quantum physics is the branch of physics that studies the behavior of matter and energy at the quantum scale—the smallest scale of energy levels in an atom. Here, traditional rules of classical physics do not apply, and instead, particles exhibit dual characteristics, behaving as both waves and particles.

Understanding the ground state energy of an electron in a quantum box is a fundamental problem in quantum physics. An electron within a quantum box is an example of quantum confinement where its energy levels become quantized due to spatial constraints. This concept of quantization is fundamental to understanding how quantum systems differ from classical systems, which could theoretically have continuous energy values.

Planck's Constant

Planck's constant is a crucial value in quantum mechanics, symbolized by the letter 'h'. It represents the smallest possible physical action that can take place in nature, as well as the scale at which quantum effects become significant.

As seen in the exercise, Planck's constant plays a key role in the equation for determining the energy levels within a quantum box. It relates the energy carried by photons to their frequency and serves as the proportional constant in the equation \( E = h u \), where \( u \) is the frequency of the photon. The value of Planck's constant is \( h = 6.626 \times 10^{-34} \mathrm{Js}.\)

Quantum Number

A quantum number is an integer that appears in the solution to a quantum mechanical problem, which like an electron in a box, often serves as an index to the energy levels of a system. In our example, the quantum number 'n' signifies the state of the electron; with \( n = 1 \) representing the ground state, or lowest energy level.

The energy levels \( E_n \) depend on the square of the quantum number, indicating that energy levels become more widely spaced as 'n' increases. So the ground state (where \( n = 1 \) ) has the least possible energy, which is what we were solving for in our exercise.

Electron Energy Levels

Electron energy levels or electronic states are the possible energies that an electron can have, typically within an atom or a condensed matter system. In an atom, these are discrete levels corresponding to electron orbitals. However, in the context of our quantum box problem, the energy levels are determined by the confinement dimensions, the Planck’s constant, and the electron's mass.

Energy levels become key in understanding phenomena such as absorption and emission spectra, and chemical bonding. By calculating the ground state energy—in this case, approximately 3.000 eV—one can deduce the minimum energy required for an electron to occupy the smallest or 'ground' quantum box.