Problem 3 - Fundamentals of Physics

Question

The ground-state energy of an electron trapped in a onedimensional infinite potential well is \(2.6 \mathrm{eV}\). What will this quantity be if the width of the potential well is doubled?

Step-by-Step Solution

Answer

0.65 eV

Step 1: Understanding the Relationship

The energy levels of an electron in a one-dimensional infinite potential well are given by the formula: \[E_n = \frac{n^2 h^2}{8mL^2}\], where:- \(E_n\) is the energy of the level, - \(n\) is the principal quantum number (\(n = 1\) for ground state), - \(h\) is Planck's constant,- \(m\) is the mass of the electron,- \(L\) is the width of the potential well.Since the problem involves the ground state, we use \(n = 1\).

Step 2: Identify the Current Scenario

Given that the ground state energy \(E_1\) is 2.6 eV for the current width \(L\), the equation becomes:\[E_1 = \frac{h^2}{8mL^2} = 2.6 \text{ eV}\].

Step 3: Adjusting for the New Width

If the width of the potential well is doubled, then the new width is \(2L\). The new ground state energy \(E'_1\) can be calculated with:\[E'_1 = \frac{h^2}{8m(2L)^2} = \frac{h^2}{32mL^2}\].

Step 4: Relating Old and New Energies

The new energy \(E'_1\) can be expressed in terms of the original energy \(E_1\) by noting:\[E'_1 = \frac{1}{4} \cdot \frac{h^2}{8mL^2} = \frac{1}{4} \cdot E_1\].

Step 5: Substitute and Calculate

Substituting \(E_1 = 2.6 \text{ eV}\) into the expression for \(E'_1\), we get:\[E'_1 = \frac{1}{4} \times 2.6 \text{ eV} = 0.65 \text{ eV}\].

Key Concepts

Infinite Potential Well

An infinite potential well is a core concept in quantum mechanics. It's essentially a model that helps us understand the behavior of particles, like electrons, at very small scales. The 'well' part of the name refers to the concept of a particle being trapped in a specific region of space, unable to escape due to infinitely high potential energy barriers at the edges.
The 'infinite' aspect means that these barriers are insurmountable, making the particle permanently confined to this area.
However, unlike in classical mechanics where a particle can have any energy, in an infinite potential well, the energy is quantized. This means the particle can only occupy specific energy levels. The formula for these energy levels, given in the solution, illustrates how certain factors like the particle's mass and the width of the well affect these levels.

• The well is one-dimensional, simplifying calculations but still demonstrating key quantum principles.
• Energy levels depend inversely on the square of the well's width.

Doubling the width of the well, as explored in the exercise, shows how energy levels decrease, demonstrating the sensitivity of quantum systems to changes in confinement structures.

Ground-State Energy

The ground-state energy is the lowest possible energy that a quantum mechanical system, like an electron in an infinite potential well, can have. It's important because it sets the base level from which other energy states are calculated.
In this context, the ground-state is when the principal quantum number ( =1) is at its minimum.
For the electron in the well, this means it occupies the lowest allowed energy level according to the formula provided in the steps.
The provided value of 2.6 eV for the ground-state energy gives us a starting point to find out how changes in the system, like doubling the well's width, influence this energy:

• Energy decreases when the well's width increases because they are inversely related.
• From the calculations in the step-by-step solution, doubling the width results in the energy being quartered, highlighting the inverse square dependence.

This demonstrates a key principle of quantum mechanics: small changes in the system can lead to significant changes in energy.

Planck's Constant

Planck's constant is one of the fundamental constants in physics, crucial for the development of quantum mechanics. It's represented by the letter \( h \) and holds the value of approximately \( 6.63 \times 10^{-34} \, ext{Js} \). This tiny number is incredibly important because it quantifies the scale at which quantum effects become significant.

Planck's constant appears in equations that describe the quantization of energy, like the formula for energy levels in the infinite potential well. Planck's constant helps us understand:

• How energy levels are quantized — only certain energies are allowed.
• The relationship between energy and frequency via its role in formulas like \( E = hf \), where \( f \) is frequency.

In our exercise, Planck's constant plays a crucial role in determining the electron's energy levels within the well. By using this constant, we can calculate the specific allowed energies, and observe how changes in the system affect these energies, showcasing the profound implications Planck's constant has within the quantum realm.