Problem 27 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Determine the lowest four energy levels and wave functions for an electron trapped in an infinitely deep potential well of width 2.0 \(\mathrm{nm} .\)

Step-by-Step Solution

Answer

The lowest four energy levels are calculated and their wave functions are expressed in sine terms, considering the well width of 2 nm.

Step 1: Understand the Potential Well

An infinitely deep potential well imposes boundary conditions where the potential energy inside the well is zero, and it is infinite outside. This means the wave function must be zero at the boundaries of the well, at \( x = 0 \) and \( x = L \) where \( L = 2 \text{ nm} \).

Step 2: Apply the Schrödinger Equation

Use the time-independent Schrödinger equation for a particle in an infinite potential well: \( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi \). Since the potential inside the well is zero, this simplifies to \( \frac{d^2 \psi}{dx^2} + k^2 \psi = 0 \) where \( k^2 = \frac{2mE}{\hbar^2} \).

Step 3: Determine the Form of the Wave Function

The general solution to the differential equation is \( \psi(x) = A \sin(kx) + B \cos(kx) \). The boundary conditions \( \psi(0) = 0 \) and \( \psi(L) = 0 \) imply \( B = 0 \) and \( \sin(kL) = 0 \).

Step 4: Solve for Allowed Wave Numbers

The condition \( \sin(kL) = 0 \) implies \( kL = n\pi \), where \( n \) is a positive integer. Therefore, \( k = \frac{n\pi}{L} \). For \( L = 2 \text{ nm} \), \( k = \frac{n\pi}{2 \times 10^{-9}} \).

Step 5: Calculate Energy Levels

The energy levels are given by \( E_n = \frac{\hbar^2 k^2}{2m} = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \). Using \( \hbar = 1.055 \times 10^{-34} \text{ J}\cdot\text{s} \) and \( m = 9.11 \times 10^{-31} \text{ kg} \), we find \( E_1, E_2, E_3, E_4 \) by substituting \( n = 1, 2, 3, 4 \).

Step 6: Calculate Specific Energies

Substitute values to find specific energies. For example, \( E_1 = \frac{\pi^2 (1.055 \times 10^{-34})^2}{2 \times 9.11 \times 10^{-31} \times (2 \times 10^{-9})^2} \approx 6.024 \times 10^{-20} \text{ J} \). This process is repeated for \( E_2, E_3, \) and \( E_4 \) with \( n = 2, 3, 4 \).

Step 7: Write the Wave Functions

For each energy level \( n \), the wave function \( \psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right) \). Use normalization conditions to find \( A \), typically resulting in \( A = \sqrt{\frac{2}{L}} \). Therefore, \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \).

Key Concepts

Infinitely Deep Potential Well

The concept of an infinitely deep potential well is pivotal in quantum mechanics for understanding how particles behave when constrained within a particular region. Imagine this well as a box with walls so high that a particle can never escape. The potential energy inside this well is zero, meaning a particle can move freely within, but becomes infinite at the boundaries. As a result, the wave function, which describes the probability of finding a particle, must be zero at these boundaries. In the case of a well with width 2 nm, this condition holds at both ends, specifically when the position is at 0 nm and 2 nm.

Wave Function

In quantum mechanics, the wave function is a fundamental concept that captures all the information about a quantum system. When dealing with particles, like electrons in an infinitely deep potential well, the wave function determines the probability of finding the particle at various positions.
The wave function for a particle in such a well is typically given by:

• For boundary conditions \( \psi(0) = 0 \) and \( \psi(L) = 0 \, \text{where } L = 2 \text{ nm}\)
• This leads to solutions of the form \( \psi(x) = A \sin(kx) + B \cos(kx) \) where \( B = 0 \).

Thus, the wave function becomes entirely sinusoidal, dictated by the constants set by the particle’s energy and the specific boundary conditions.

Energy Levels

The energy levels in an infinitely deep potential well are quantized, meaning they can only take on discrete values. Quantum mechanics dictates that these levels are proportional to the square of the integers known as quantum numbers, represented as \( n \, = 1, 2, 3, ... \).
The energies are given by:

• \( E_n = \frac{\hbar^2 k^2}{2m} \)
• Substituting for \( k \, \text{the expression simplifies to } E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \)
• Calculating for the first few quantum numbers (\( n = 1, 2, 3, \,\text{and } 4\)) yields specific energy values with increasing magnitude.

These calculations show that energy levels are not evenly spaced and become increasingly larger as you move to higher quantum numbers.

Schrödinger Equation

The Schrödinger Equation is the cornerstone of quantum mechanics, governing how the quantum state of a physical system changes over time. For an infinitely deep potential well, we work with the time-independent Schrödinger Equation:

• \( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E\psi \)
• Inside the well, this simplifies to \( \frac{d^2 \psi}{dx^2} + k^2 \psi = 0 \), where \( k^2 = \frac{2mE}{\hbar^2} \)

Solving this differential equation involves finding the wave function that satisfies boundary conditions at \( x = 0 \) and \( x = L \, \text{thus leading to quantized values of } k \).The Schrödinger Equation not only predicts the allowed energy levels but also the corresponding wave functions, providing a complete picture of the quantum system within the potential well.