Problem 60 - Fundamentals of Physics

Question

An electron is confined to a narrow evacuated tube of length \(3.0 \mathrm{~m} ;\) the tube functions as a one-dimensional infinite potential well. (a) What is the energy difference between the electron's ground state and its first excited state? (b) At what quantum number \(n\) would the energy difference between adjacent energy levels be \(1.0 \mathrm{eV}-\) which is measurable, unlike the result of (a)? At that quantum number, (c) what multiple of the electron's rest energy would give the electron's total energy and (d) would the electron be relativistic?

Step-by-Step Solution

Answer

(a) \(9.06 \times 10^{-6} \text{eV},\) (b) \(n\approx 6 \times 10^{42},\) (c) much larger multiple, (d) yes, relativistic.

Step 1: Understanding the Problem

In this problem, we have an electron in a one-dimensional infinite potential well of length 3.0 m. We need to calculate the energy difference between the ground state and the first excited state (part a), find a quantum number with a specific energy difference between adjacent levels (part b), and calculate total energy and relativistic status for that quantum number (parts c and d).

Step 2: Calculate Energy Levels

The energy levels for an electron in a one-dimensional infinite potential well are given by \[ E_n = \frac{n^2 h^2}{8mL^2} \]where \( n \) is the quantum number, \( h \) is Planck's constant (6.626 \times 10^{-34} \text{J s}), \( m \) is the mass of the electron (9.109 \times 10^{-31} \text{kg}), and \( L \) is the length of the well (3 m).

Step 3: Energy Difference in First Two States

Calculate the energy of the ground state (\( n = 1 \)) and first excited state (\( n = 2 \)).\[ E_1 = \frac{1^2 \times (6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (3)^2} \approx 3.023 \times 10^{-6} \, \text{eV} \]\[ E_2 = \frac{2^2 \times (6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (3)^2} \approx 1.209 \times 10^{-5} \, \text{eV} \]Thus, the energy difference \( \Delta E = E_2 - E_1 \approx 9.06 \times 10^{-6} \, \text{eV} \).

Step 4: Finding Quantum Number for 1 eV Difference

We need to find a quantum number \( n \) such that the difference between adjacent levels \( \Delta E = E_{n+1} - E_n = 1 \, \text{eV} \). From the expression:\[ E_{n+1} - E_n = \frac{(n+1)^2 - n^2}{8mL^2} h^2 = \frac{2nh^2 + h^2}{8mL^2} \approx 1 \, \text{eV} \]Solving for \( n \), we find \( n \approx 6 \times 10^{42} \), an enormous number resulting from computational approximation challenge without real feasibility.

Step 5: Total Energy as a Multiple of Rest Energy

Energy in nth state for large \( n \) is approximately:\[ E_n = \frac{n^2 h^2}{8mL^2} \gg mc^2 \]Calculate the ratio:\[ \frac{E_n}{mc^2} = \frac{n^2 h^2}{8mL^2 mc^2} \]With \( n \approx 6 \times 10^{42} \), for practical approach \( E_n \) becomes much larger than rest energy \( mc^2 \).

Step 6: Determine the Relativistic Status

An electron would be relativistic if its speed approaches a significant fraction of the speed of light. Use the energy-momentum relation:\[ E^2 = (pc)^2 + (mc^2)^2 \approx pc^2 \text{ if } E \gg mc^2 \]Thus, for found \( n \), total energy calculation exceeds rest energy, making electron relativistic.

Key Concepts

Infinite Potential Well

The infinite potential well is a fundamental concept in quantum mechanics. Imagine it as a box where a particle, like an electron, can move freely inside but cannot escape its boundaries. This is because the potential energy outside the box is infinitely large, effectively trapping the particle inside. The one-dimensional infinite potential well, as used in the exercise, simplifies many calculations. In this setup:

• The well has a specific length, say 3 meters, as mentioned in the exercise.
• Only certain energy levels are possible for the particle, which we call quantized energy levels.
• The particle behaves differently compared to classical physics, highlighting the unique properties of quantum systems.

This phenomenon allows us to understand how electrons behave in confined spaces, like in atoms or in solid-state physics, where electrons can be trapped by potential wells created by atomic nuclei or crystal lattices.

Energy Levels

Energy levels are the specific energies that a particle trapped in an infinite potential well can have. These levels do not overlap, and the electron can only exist at these discrete levels corresponding to different quantum numbers.Energy levels depend on several factors:

• The length of the well, which in this case is 3.0 m.
• Planck's constant, a fundamental constant in quantum mechanics.
• The mass of the electron involved.

For a one-dimensional infinite potential well, the energy levels are given by the formula:\[ E_n = \frac{n^2 h^2}{8mL^2} \]where:

• \( n \) is the quantum number (an integer starting from 1).
• \( h \) is Planck's constant.
• \( m \) is the mass of the electron.
• \( L \) is the length of the well.

In the exercise, calculating the energy difference between the ground state (\( n=1 \)) and first excited state (\( n=2 \)) helps illustrate these quantized levels.

Planck's Constant

Planck's constant, denoted as \( h \), plays a pivotal role in quantum mechanics. It is a fundamental constant that appears in many equations, including those describing energy levels in an infinite potential well.Key properties:

• Planck's constant has a value of approximately \( 6.626 \times 10^{-34} \text{ J s} \).
• It is used to calculate the smallest possible units of energy, showing how energy is quantized.
• It signifies the shift from classical to quantum physics, capturing the idea that energy, rather than being continuous, comes in discrete 'packets' or quanta.

In our exercise, Planck's constant is crucial for determining the specific energy levels of the electron trapped in the potential well. The calculation involves precisely this constant to find how energy varies with the quantum number.

Relativistic Effects

Relativistic effects become significant when particles approach extremely high speeds, close to the speed of light. An electron becomes relativistic when its total energy is a large multiple of its rest energy.Considerations in the exercise:

• The rest energy of an electron is given by \( mc^2 \), where \( m \) is the electron's mass and \( c \) is the speed of light.
• When the energy obtained from the potential well becomes much larger than \( mc^2 \), relativistic effects cannot be ignored.

For large quantum numbers \( n \), electrons in the potential well can achieve such high energies that their behavior conforms to the principles of relativity. In the solution provided, for a certain large \( n \), the energy was found to vastly exceed the rest energy of the electron, thus categorizing it as a relativistic particle. Understanding these effects helps ensure accurate predictions and explanations of particle behavior at high energies.