Question

Determine the two lowest energies of a wave function of an electron in a box of width \(2.0 \cdot 10^{-9} \mathrm{~m} .\)

Short answer

Answer: The two lowest energy levels of the electron in the box are: - Lowest energy level (\(n=1\)): \(E_1 \approx 3.768 \cdot 10^{-19} \mathrm{~J}\) - Second lowest energy level (\(n=2\)): \(E_2 \approx 1.507 \cdot 10^{-18} \mathrm{~J}\)

Step-by-step solution

Write down the given parameters.

The width of the box is given as \(L=2.0 \cdot 10^{-9} \mathrm{~m}\). The mass of the electron is \(m = 9.109 \times 10^{-31} \mathrm{~kg}\), and the Planck constant is \(h = 6.626 \times 10^{-34} \mathrm{~J \cdot s}\).

Use the energy formula for n=1.

To find the lowest energy level, we need to calculate the energy for \(n=1\). The formula for the energy level is: $$ E_{n} = \frac{n^2h^2}{8mL^2}. $$ Now, plug in the values for \(n=1\), \(h\), \(m\), and \(L\): $$ E_{1}=\frac{(1)^{2}(6.626\cdot10^{-34}\mathrm{~J\cdot s})^{2}}{8(9.109\cdot10^{-31}\mathrm{~kg})(2.0\cdot10^{-9}\mathrm{~m})^{2}}. $$

Calculate the energy for n=1.

Compute the energy for \(n=1\) as follows: $$ E_{1}=\frac{(6.626\cdot10^{-34})^{2}}{8(9.109\cdot10^{-31})(4.0\cdot10^{-18})}\mathrm{~J} \approx 3.768 \cdot 10^{-19} \mathrm{~J}. $$ The lowest energy level of the electron is approximately \(3.768 \cdot 10^{-19} \mathrm{~J}\).

Use the energy formula for n=2.

To find the second lowest energy level, calculate the energy for \(n=2\). Use the same energy level formula as in step 2: $$ E_{n} = \frac{n^2h^2}{8mL^2}. $$ Plug in the values for \(n=2\), \(h\), \(m\), and \(L\): $$ E_{2}=\frac{(2)^{2}(6.626\cdot10^{-34}\mathrm{~J\cdot s})^{2}}{8(9.109\cdot10^{-31}\mathrm{~kg})(2.0\cdot10^{-9}\mathrm{~m})^{2}}. $$

Calculate the energy for n=2.

Compute the energy for \(n=2\) as follows: $$ E_{2}=\frac{(4)(6.626\cdot10^{-34})^{2}}{8(9.109\cdot10^{-31})(4.0\cdot10^{-18})}\mathrm{~J} \approx 1.507 \cdot 10^{-18} \mathrm{~J}. $$ The second lowest energy level of the electron is approximately \(1.507 \cdot 10^{-18} \mathrm{~J}\).

Report the two lowest energies.

The two lowest energy levels of the electron in the box are: - Lowest energy level (\(n=1\)): \(E_1 \approx 3.768 \cdot 10^{-19} \mathrm{~J}\) - Second lowest energy level (\(n=2\)): \(E_2 \approx 1.507 \cdot 10^{-18} \mathrm{~J}\)

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental branch of physics that deals with the behavior of particles at atomic and subatomic levels. At this scale, particles such as electrons do not follow the classical laws of motion. Instead, they exhibit behaviors that can only be described probabilistically. Unlike classical objects, particles can exist in multiple states and locations simultaneously, a concept known as superposition.

A key idea in quantum mechanics is that particles like electrons have properties of both particles and waves. This duality is essential in explaining phenomena such as interference and diffraction. Quantum mechanics also introduces the idea of quantization, meaning certain properties, like energy, can only take on discrete values called quantized states. This is vastly different from how energy is understood in the classical physics realm.

• Wave-Particle Duality: Particles exhibit both wave-like and particle-like characteristics.
• Superposition: Particles can be in multiple states simultaneously.
• Quantization: Energy and other properties exist in discrete states rather than being continuous.

Energy Levels

In a quantum system, energy levels are discrete quantities that an electron can possess. For an electron in a box, these levels are determined by the size of the box and the properties of the electron itself. The concept of energy levels is crucial for understanding the behavior of electrons confined in small spaces, like atoms or molecules.

When an electron moves between these energy levels, it absorbs or emits precise amounts of energy, corresponding to the difference between the levels. These are known as quanta. The formula used to calculate the energy levels for an electron in a box is:\[ E_n = \frac{n^2h^2}{8mL^2} \]Here, \(n\) represents the quantum number, which indicates the level or state of energy, \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(L\) is the length of the box. Only specific energy levels are allowed, and these correspond to specific values of the quantum number \(n\).

• Discrete Values: Electrons can only exist in specific energy levels.
• Quantum Transitions: Electrons change energy levels via emission or absorption of quanta.
• Dependence on Physical Parameters: Energy levels depend on electron mass, Planck's constant, and box size.

Electron

Electrons are fundamental particles with negative charge, which play a crucial role in electricity, magnetism, and chemistry. In the "particle in a box" model, the electron is considered as a particle constrained to move in a closed region of space with infinitely high walls. This model is a simplistic way to understand how electrons behave in confined spaces like atoms or quantum wells.

In the quantum world, electrons do not orbit the nucleus like planets around a sun. Instead, they exist in regions of probability called orbitals. An electron's energy and position are not precisely defined, leading to the idea of uncertainty at this scale.

• Negative Charge: Electrons have a fundamental negative charge.
• Confined Movement: In the particle in a box model, the electron's movement is limited by the box walls.
• Probability Regions: Electrons exist in probability clouds rather than distinct paths.

Wave Function

The wave function in quantum mechanics is a mathematical description of the quantum state of a particle, such as an electron. It contains all the information about the particle, including its energy, position, and momentum, and is usually denoted by the Greek letter \( \psi \).

The wave function is key to calculating probabilities. By taking the square of the wave function's magnitude, \(|\psi|^2\), we obtain the probability density of finding the particle at a specific location at a specific time. For an electron in a box, the wave function takes on a sinusoidal form, reflecting the standing wave pattern.

• Probability Tool: The wave function helps find the probability of a particle's location.
• Wave-Like Nature: For electrons in a box, the wave function resembles a standing wave.
• Unique Characteristics: Each energy level corresponds to a unique wave function.

The properties of wave functions illustrate the wave-like nature of particles in quantum mechanics, leading to phenomena such as interference and diffractions in confined spaces.