A typical electron in a piece of metallic sodium has energy −E₀ compared to a free electron, where E₀ is the 2.36 eV work function of sodium. At what distance beyond the surface of the metal is the electron’s probability density 10% of its value at the surface?

Verified step by step guidance

1. Step 1: Understand the problem. The question involves the probability density of an electron outside the surface of metallic sodium. The probability density decreases exponentially with distance from the surface, and we are tasked with finding the distance at which the probability density is 10% of its value at the surface.
2. Step 2: Recall the relationship between the probability density and the work function. The probability density of the electron outside the surface is proportional to \( e^{-2kz} \), where \( k \) is the wave number and \( z \) is the distance from the surface. The wave number \( k \) is related to the work function \( E_0 \) by \( k = \sqrt{\frac{2mE_0}{\hbar^2}} \), where \( m \) is the mass of the electron and \( \hbar \) is the reduced Planck's constant.
3. Step 3: Set up the equation for the probability density. The probability density at distance \( z \) is given by \( P(z) = P(0) e^{-2kz} \), where \( P(0) \) is the probability density at the surface. To find the distance \( z \) where \( P(z) = 0.1P(0) \), substitute \( P(z) = 0.1P(0) \) into the equation: \( 0.1P(0) = P(0)e^{-2kz} \).
4. Step 4: Solve for \( z \). Divide both sides of the equation by \( P(0) \) to isolate the exponential term: \( 0.1 = e^{-2kz} \). Take the natural logarithm of both sides to solve for \( z \): \( \ln(0.1) = -2kz \). Rearrange to find \( z \): \( z = -\frac{\ln(0.1)}{2k} \).
5. Step 5: Substitute the expression for \( k \) into the equation for \( z \). Replace \( k \) with \( \sqrt{\frac{2mE_0}{\hbar^2}} \): \( z = -\frac{\ln(0.1)}{2\sqrt{\frac{2mE_0}{\hbar^2}}} \). This is the final expression for the distance \( z \) in terms of the work function \( E_0 \), the electron mass \( m \), and the reduced Planck's constant \( \hbar \).

Key Concepts

Work Function

The work function is the minimum energy required to remove an electron from the surface of a material. In this case, sodium has a work function of 2.36 eV, meaning that an electron must gain at least this amount of energy to escape the metal's surface. Understanding the work function is crucial for analyzing electron behavior in metals and their interaction with external energy sources.

Quantum Mechanics and Probability Density

In quantum mechanics, the probability density describes the likelihood of finding a particle, such as an electron, in a particular region of space. It is derived from the square of the wave function's amplitude. For this problem, calculating the distance at which the probability density of the electron drops to 10% of its surface value involves understanding how quantum states decay in potential wells.

Exponential Decay in Quantum Tunneling

Exponential decay is a key concept in quantum mechanics, particularly in the context of tunneling phenomena. When an electron is near a potential barrier, its probability density decreases exponentially with distance from the barrier. This behavior is essential for determining how far from the surface the electron's probability density reaches a specific fraction of its initial value, such as 10% in this scenario.