Question

Determine the lowest 3 energies of the wave function of a proton in a box of width $1.0\cdot {10}^{-10}\mathrm{m}$.

Short answer

Answer: The lowest three energy levels for the proton in the box are approximately $6.04\cdot {10}^{-25}\mathrm{J}$, $2.42\cdot {10}^{-24}\mathrm{J}$, and $5.44\cdot {10}^{-24}\mathrm{J}$.

Step-by-step solution

Recall the energy level formula for a particle in a box

For a particle in a one-dimensional box of width L, the allowed energy levels can be determined by the formula: $E_n={\displaystyle \frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}}$, where $\hslash$ is the reduced Planck's constant, $m$ is the mass of the particle (proton), and $n$ is an integer (quantum number). In this case, the box has a width of $L=1.0\cdot {10}^{-10}\mathrm{m}$, and the mass of the proton is approximately $m=1.67\cdot {10}^{-27}\mathrm{kg}$. The reduced Planck's constant, $\hslash$, is approximately $1.05\cdot {10}^{-34}\mathrm{J}\cdot \mathrm{s}$.

Determine the lowest three energy levels for the proton

Using the energy level formula, we can calculate the energies for the lowest three levels by plugging in the values of $n=1,2,3$ in the formula. For $n=1$, $E_1={\displaystyle \frac{1^2{\pi }^2{\hslash }^2}{2m{L}^2}}$ For $n=2$, $E_2={\displaystyle \frac{2^2{\pi }^2{\hslash }^2}{2m{L}^2}}$ For $n=3$, $E_3={\displaystyle \frac{3^2{\pi }^2{\hslash }^2}{2m{L}^2}}$

Calculate the energy levels

Now plug in the values of $\hslash$, $m$, and $L$ into the energy level formula to find the energies: $E_1={\displaystyle \frac{(1.0{)}^2(\pi {)}^2(1.05\cdot {10}^{-34}{)}^2}{2(1.67\cdot {10}^{-27})(1.0\cdot {10}^{-10}{)}^2}}\approx 6.04\cdot {10}^{-25}\mathrm{J}$ $E_2={\displaystyle \frac{(2.0{)}^2(\pi {)}^2(1.05\cdot {10}^{-34}{)}^2}{2(1.67\cdot {10}^{-27})(1.0\cdot {10}^{-10}{)}^2}}\approx 2.42\cdot {10}^{-24}\mathrm{J}$ $E_3={\displaystyle \frac{(3.0{)}^2(\pi {)}^2(1.05\cdot {10}^{-34}{)}^2}{2(1.67\cdot {10}^{-27})(1.0\cdot {10}^{-10}{)}^2}}\approx 5.44\cdot {10}^{-24}\mathrm{J}$ The lowest three energy levels for the proton in the box are approximately $6.04\cdot {10}^{-25}\mathrm{J}$, $2.42\cdot {10}^{-24}\mathrm{J}$, and $5.44\cdot {10}^{-24}\mathrm{J}$.

Key concepts

Particle in a Box

The concept of a 'particle in a box' is a fundamental model in quantum mechanics that describes a particle which is confined to move freely within a one-dimensional region with zero potential energy inside and infinite potential energy at the boundaries. Essentially, this model provides a simplified scenario to study quantum behavior and is also known as the 'infinite potential well'.

The key takeaway from the 'particle in a box' model is that only certain discrete energy levels are allowed for the particle. These levels are dependent upon the width of the box, as seen in the exercise where the width was given as $1.0\cdot {10}^{-10}\mathrm{m}$. This width constrains the motion of the particle, such as a proton, and determines its possible energy states.

This model is not just a theoretical construct but has practical implications in fields like quantum dots in semiconductors and spectroscopy where the behavior of electrons in confined spaces is analogous to a particle in a one-dimensional box.

Energy Levels

The energy levels of a quantum system refer to the discrete energies that an electron or any quantum particle can possess. Unlike classical particles, which can have a continuous range of energy values, quantum particles can only occupy certain levels, a bit like the distinct rungs of a ladder.

In the context of our exercise, the energy levels for a proton within a one-dimensional box are determined by the size of the box and the mass of the proton. For each energy level, there is a designated quantum number, which is a positive integer that signifies the state of the particle. As the quantum number increases, the energy level becomes higher, indicating that the particle has more energy. It's worth noting that each energy state is uniquely defined and no two particles can occupy the same state simultaneously in accordance with the Pauli exclusion principle, relevant in systems with multiple fermions like electrons.

Planck's Constant

One of the pivotal constants in quantum mechanics is Planck's constant, denoted as $h$, which represents the quantization of energy. However, in the equation used for the 'particle in a box' model, we see the reduced Planck's constant, $\hslash$, which is equal to $h/(2\pi )$.

Planck's constant has a value of approximately $6.626\times {10}^{-34}$ joule-seconds, and it links the amount of energy a photon has with the frequency of its electromagnetic wave. The presence of Planck's constant in quantum equations, including the energy level formula, is a reminder that the energy of quantum systems is quantized. In our exercise, the reduced Planck's constant is crucial for calculating the discrete energy levels of a proton trapped in a one-dimensional box. Its small numerical value reflects the fact that quantum effects become significant at very small scales.

Quantum Numbers

Quantum numbers are integers that serve as the 'DNA' of quantum states, uniquely identifying the properties of a particle in a quantum system. In the 'particle in a box' scenario, the principal quantum number, denoted as $n$, dictates the energy level of the particle.

The principal quantum number starts at 1 and increases to infinity. Each number corresponds to a particular energy level: the higher the quantum number, the higher the energy. In our exercise, the quantum numbers 1, 2, and 3 represent the first, second, and third energy states respectively for the proton in the box. These quantum numbers are essential for determining the allowed energies, orbital shapes, and angular momentum of particles in quantum systems, playing a vital role in the structure and behavior of atoms and molecules.