Question

An electron is trapped in a one-dimensional infinite potential well that is \(L=300\). pm wide. What is the probability that one can detect the electron in the first excited state in an interval between \(x=0.500 L\) and \(x=0.750 L ?\)

Short answer

Answer: The probability of detecting the electron in the first excited state in the interval between x=0.5L and x=0.75L is 0.25 or 25%.

Step-by-step solution

Write down the wave function for the first excited state

For the first excited state, which corresponds to the second energy level, the quantum number n=2. The wave function can be written as ψ(x) = √(2/L) * sin(2 * π * x/L)

Define the limits for the given interval

The interval in which we want to determine the probability of detecting the electron is between 0.5L and 0.75L. Therefore, the limits of the interval are: Lower limit: x1 = 0.5L = 150 pm Upper limit: x2 = 0.75L = 225 pm

Determine the probability integral formula

The probability of detecting the electron in the given interval is given by the integration of the square of the wave function over that interval: probability = ∫(ψ²(x)dx) from x1 to x2

Substitute the limits and wave function into the probability integral

Using the wave function for the first excited state and the limits of the interval, we can rewrite the probability integral as: probability = ∫([2/L * sin²(2 * π * x/L)]dx) from x=150 pm to x=225 pm

Calculate the probability integral

To find the probability, we need to calculate: probability = ∫([2/300 * sin²(2 * π * x/300)]dx) from x=150 pm to x=225 pm After evaluating the integral, we get: probability = 0.25

Interpret the result

The probability of detecting the electron in the first excited state in the interval between x=0.5L and x=0.75L is 0.25, or 25%. This means that there is a 25% chance of finding the electron in that specific interval in the first excited state.

Key concepts

Wave Function

In the fascinating world of quantum mechanics, the wave function is a fundamental concept. It's a mathematical description that encapsulates the quantum state of a particle, like an electron in an atom. You can think of the wave function, often represented by the symbol \( \psi \), as providing all the information we could possibly know about a particle's behavior and position.

For our trapped electron, the wave function varies with position and the energy state of the electron. It's important to note that we can't use the wave function to predict the exact position of an electron at a given time. Instead, when we square the wave function, \( |\psi(x)|^2 \), it gives us the probability density, meaning how likely it is to find the electron at a particular point in space. This is why the wave function is instrumental in quantum mechanics—it bridges what we know with what we can predict about particles at the quantum level.

First Excited State

Quantum mechanics tells us that particles like electrons in an atom or a well can only exist in certain energy states. The first excited state is like the second rung on a ladder of energy levels—a particle in this state has more energy than it does in the ground state but less than in higher excited states.

When we talk about an electron in the first excited state of an infinite potential well, we're describing its second-lowest energy condition. This state has a unique wave function associated with it, which determines how the electron behaves. When the electron is in this first excited state, it's like a wave inside the well with nodes and antinodes—the points where the wave is always zero and the points of maximum amplitude, respectively. Mathematically, this state has a defined wave function that we use to calculate various probabilities, including where the electron is likely to be found within the well.

Probability Integral

Calculating the probability of finding an electron in a specific region involves what's called the probability integral. It's an integral over the squared magnitude of the wave function, \( \int |\psi(x)|^2 dx \), which yields the likelihood or probability of locating the particle within a particular interval.

This mathematical tool is like a magnifying glass for probabilities. When you're asked about the electron's whereabouts within an interval, you set the limits of this interval into the integral. The wave function is squared because we're interested in the probability density, not the wave function itself. The final value of this integral, which ranges between 0 and 1, then tells us the probability as a percentage of finding the electron within the specified limits. It's an essential technique for answering many questions in quantum mechanics, providing a clear and calculable way of predicting outcomes.

Infinite Potential Well

The concept of an infinite potential well is a model used to simplify the somewhat complex reality of quantum particles. It's a hypothetical scenario where a particle, such as an electron, is confined to a specific region in space with impenetrable boundaries. These walls are said to have 'infinite potential,' meaning the electron doesn't have enough energy to escape—like a marble in a bowl that's perfectly smooth and infinitely tall.

In this scenario, the electron can only occupy specific energy levels, quite similar to standing waves on a string with fixed ends. Between these walls, the electron exhibits wave-like properties described by the wave function. The infinite potential well is a cornerstone in teaching quantum mechanics, as it provides a clear example of how quantum confinement can lead to quantized energy states and illustrates the principles of wave-particle duality and quantization in a relatively simple system.