Problem 39 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An electron approaches a potential barrier \(18 \mathrm{eV}\) high and \(0.55 \mathrm{nm}\) wide. If the electron has a \(1.0 \%\) probability of tunneling through the barrier, what is the electron's energy?

Step-by-Step Solution

Answer

The energy of the electron is approximately 5.41 eV.

Step 1: Identify the Given Values

The problem provides us with several pieces of information: a potential barrier height (\( V_0 = 18 \, \text{eV} \)), a width of the barrier (\( L = 0.55 \, \text{nm} \)), and the tunneling probability (\( P = 0.01 \) or 1.0%). We need to find the energy of the electron (\( E \)).

Step 2: Apply the Tunneling Probability Formula

The probability of tunneling through a barrier is given by the formula \[ P = e^{-2aL} \] where \( a = \frac{\sqrt{2m(V_0 - E)}}{\hbar} \) and \( L \) is the width of the barrier. \( m \) represents the mass of the electron and \( \hbar \) is the reduced Planck's constant.

Step 3: Simplify the Equation

Substitute for \( a \) in the equation: \[ P = e^{-2 \frac{\sqrt{2m(V_0 - E)}}{\hbar} L} \] Take the natural logarithm on both sides to solve for \( \sqrt{2m(V_0 - E)} \). This results in \[ -2 \frac{\sqrt{2m(V_0 - E)}}{\hbar} L = \ln P \] Rearranging gives: \[ \sqrt{2m(V_0 - E)} = -\frac{\hbar \ln P}{2L} \]

Step 4: Calculate Constants and Further Simplify

The values of the constants are \( m \approx 9.11 \times 10^{-31} \text{kg} \), \( \hbar \approx \frac{6.63 \times 10^{-34} \text{Js}}{2\pi} \). Substitute these constants and simplify the equation to find expressions for \( V_0 - E \).

Step 5: Solve for Electron's Energy

With the rearranged equation from previous steps, calculate \( V_0 - E \) by squaring the expression obtained for \( \sqrt{2m(V_0 - E)} \). Finally, rearrange to find \[ E = V_0 - \frac{(\hbar \ln P)^2}{8mL^2} \] Ensure all substituting for\( P\), \( m\), \( \hbar\), and \( L\).

Step 6: Compute the Numerical Value

Calculate the numerical result using the previous formula to find the value of \( E \) with the specific constants provided. This value represents the energy of the electron matching the tunneling probability.

Key Concepts

Potential Barrier

A potential barrier is a critical concept in quantum tunneling, acting as an obstacle that must be overcome by an electron in order to pass from one side to another. Imagine a hill that an electron must climb to continue on its path. This barrier is defined by two key characteristics: its height and width. The height of the potential barrier represents the energy level that the electron must surpass. In this case, the barrier's height is given as 18 eV (electron volts), a unit of energy commonly used in atomic and particle physics.

The width of the barrier, which is 0.55 nm (nanometers), reflects how thick this barrier is. Both the height and width are crucial because they determine how likely an electron is to "tunnel through" this barrier, a term that describes the peculiar quantum mechanical phenomenon where particles pass through barriers that would normally be insurmountable in classical physics.

• Height of the barrier impacts the required energy for overcoming it.
• Width of the barrier affects the distance over which tunneling occurs.

Combining these two determines the difficulty of tunneling, making the study of potential barriers essential for understanding electron energy in quantum systems.

Electron Energy

The energy of an electron is vital when examining the potential for tunneling through a barrier. In quantum mechanics, electron energy influences whether a particle can pass through a potential barrier like the one described. If an electron's energy is lower than the height of the barrier, classical physics suggests it shouldn't pass through. However, quantum mechanics allows for some interesting possibilities.

In the given problem, we need to find the electron's energy based on a 1% probability of tunneling, using the value of the potential barrier height (18 eV). The goal is to determine what energy level gives our electron not just a fighting chance, but a quantifiable probability, such as 1%, of getting through the barrier.

When calculating this, various formulas and constants ( - like the electron mass and Planck's constant) will come into play, providing a way to find the balance where the energy of the electron allows this improbable feat:

• Classically, insufficient energy cannot cross the barrier.
• Quantum mechanics introduces a probability aspect to tunneling.

Finding this energy is a mind-opening exploration into the weird and wonderful world of quantum particles.

Planck's Constant

Planck's constant is a fundamental constant in physics denoting the scale at which quantum mechanical effects become significant. In our exercise involving quantum tunneling, Planck's constant plays an indispensable role. It is denoted by the symbol \( h \), and in equations concerning quantum mechanics, its reduced form, \( \hbar = \frac{h}{2\pi} \), is often used.

This constant allows us to calculate important aspects of quantum behavior, such as the tunneling probability and the corresponding energy levels of particles like electrons. In the tunneling probability formula, it dictates the pre-factor that helps find how likely an electron can "tunnel" through a potential barrier.

• \( h \approx 6.63 \times 10^{-34} \text{ Js} \)
• Used in calculating the electron wave characteristics.

Understanding Planck's constant helps us grasp the size and scope of quantum activities. It's a bridge between the classical interpretations of physics and the quantum world, explaining why certain calculations yield bizarre but true phenomena as that of quantum tunneling.