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Magazine Chapter 39: Problem 57
An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference \(\Delta E\) between its quantum levels \(n\) and \(n+2\) is \(\left(h^{2} / 2 m L^{2}\right)(n+1)\).
Short Answer
Expert verified
The energy difference is \(\Delta E = \frac{h^2}{2mL^2} (n+1)\).
Step by step solution
01
Understanding the Quantum Well
A one-dimensional infinite potential well is a system where the potential energy is zero inside a certain region (from 0 to \(L\)) and infinite outside. An electron trapped in this well can only have specific energy levels, which are determined by quantum mechanics.
02
Energy Level Formula
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 a positive integer (\(n=1, 2, 3, \ldots\)), \(h\) is Planck's constant, \(m\) is the electron's mass, and \(L\) is the length of the well.
03
Identify Energy Levels
To find the energy difference between levels \(n\) and \(n+2\), we first write their energy expressions. For level \(n\), it is \(E_n = \frac{n^2 h^2}{8mL^2}\). For level \(n+2\), it is \(E_{n+2} = \frac{(n+2)^2 h^2}{8mL^2}\).
04
Calculate the Energy Difference
The energy difference between levels \(n\) and \(n+2\) is: \[ \Delta E = E_{n+2} - E_n = \frac{((n+2)^2 - n^2) h^2}{8mL^2} \] Simplifying the expression inside the brackets: \( (n+2)^2 - n^2 = n^2 + 4n + 4 - n^2 = 4n + 4 \).
05
Simplifying the Expression
Substituting the simplified expression into the energy difference: \[ \Delta E = \frac{(4n + 4) h^2}{8mL^2} \] This can be further simplified by factoring the common terms: \[ \Delta E = \frac{h^2}{2mL^2} (n+1) \]
06
Verification
Let's revisit our initial derivations to ensure all calculations are correct. We computed the difference correctly and verified that the given expression accurately represents the energy difference between energy levels \(n\) and \(n+2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Infinite Potential Well
Imagine a tiny box where an electron is kept trapped inside. This is not an ordinary box but a quantum box with special properties. The inside of this box, called the infinite potential well, has zero potential energy, while the outside has infinite potential energy.
This means an electron cannot exist outside this box鈥攊t is "infinitely" difficult to go beyond its walls. In this setup, the electron can only exist within the confines of this region, from 0 to \(L\), the length of the well.
This means an electron cannot exist outside this box鈥攊t is "infinitely" difficult to go beyond its walls. In this setup, the electron can only exist within the confines of this region, from 0 to \(L\), the length of the well.
- Inside the well, the potential energy is zero, allowing the electron to move freely.
- Outside the bounds (at positions less than 0 or greater than \(L\)), the energy "barriers" are infinitely high.
Energy Levels
In a one-dimensional infinite potential well, energy levels are quantized, meaning the electron can only possess specific energy amounts. These energies depend on the quantum number \(n\), which is a positive integer:
\[E_n = \frac{n^2 h^2}{8mL^2}\]Here, \(n = 1, 2, 3, \ldots\), and every \(n\) represents a different quantum state or energy level. The higher the value of \(n\), the higher the energy level, implying that the electron in that state has more energy.
\[E_n = \frac{n^2 h^2}{8mL^2}\]Here, \(n = 1, 2, 3, \ldots\), and every \(n\) represents a different quantum state or energy level. The higher the value of \(n\), the higher the energy level, implying that the electron in that state has more energy.
- The quantized nature implies electrons cannot have energy values between these specified levels.
- The spacing between energy levels increases as you increase \(n\).
Planck's Constant
Planck's constant is a fundamental quantity in quantum mechanics, symbolized by \(h\). It plays a pivotal role in relating the energy of particles to their frequency. Within the context of the infinite potential well expression for energy levels:
\[E_n = \frac{n^2 h^2}{8mL^2}\]\(h\) features prominently, underscoring its centrality in quantum calculations. Its value is approximately \(6.626 \times 10^{-34} Js\). This small number characterizes many quantum effects, showing that they're subtle and often only observable at microscopic scales.
\[E_n = \frac{n^2 h^2}{8mL^2}\]\(h\) features prominently, underscoring its centrality in quantum calculations. Its value is approximately \(6.626 \times 10^{-34} Js\). This small number characterizes many quantum effects, showing that they're subtle and often only observable at microscopic scales.
- Planck's constant can be thought of as a bridge between the macroscopic world and quantum phenomena.
- It indicates that quantum effects emerge at specific scales and with specific energies.
Quantum States
Quantum states refer to the unique energy configurations an electron can occupy inside our infinite potential well. Each quantum state corresponds to a particular quantum number, \(n\), which dictates the energy level of the electron.
The states are essential not just for describing energy but also for understanding the behavior and properties of quantum particles.
The states are essential not just for describing energy but also for understanding the behavior and properties of quantum particles.
- Each state is quantized, ensuring only discrete values are possible.
- The difference in energy between states helps to explain transitions and photon emissions in quantum systems.
