Question

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn at the top of the next column, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi .\) (a) Sketch the probability density, \(\psi^{2}(x)\), from \(x=0\) to \(x=2 \pi\) (b) At what value or values of \(x\) will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? \([\) Section \(6.5]\)

Short answer

(a) The probability density function \(\psi^2(x)\) is given by \(\sin^2 x\) over the interval \(x=0\) to \(x=2\pi\). (b) The greatest probability of finding the electron occurs at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). (c) The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a node.

Step-by-step solution

Calculate the probability density function

The probability density function is given by \(\psi^2(x)\). Since \(\psi(x) = \sin x\), we have \[\psi^2(x) = (\sin x)^2 = \sin^2 x\] The wave function \(\psi(x) = \sin x\) is defined on the interval from \(x=0\) to \(x=2\pi\). Therefore, the probability density function \(\psi^2(x) = \sin^2 x\) will also be defined over the same interval.

Analyze the function to find the maximum probability value(s)

To find the values of \(x\) where the probability of finding the electron is the greatest, we need to find the maximum value(s) of \(\psi^2(x)\). Recall that \(\sin^2 x\) varies between 0 and 1. Thus, the maximum value of \(\psi^2(x) = \sin^2 x\) occurs when \(\sin^2 x = 1\), or, equivalently, when \(\sin x = \pm 1\). This will happen at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\).

Determine the probability at \(x=\pi\) and identify the term for such a point

We now need to find the probability of finding the electron at \(x=\pi\). Since the probability density function is given by \(\psi^2(x) = \sin^2 x\), we simply need to evaluate it at \(x=\pi\): \[\psi^2(\pi) = \sin^2 \pi = 0\] The probability of finding the electron at \(x=\pi\) is 0. A point where the wave function is zero is called a node. Now, let's recap our answers to the three questions: (a) The probability density function \(\psi^2(x)\) is given by \(\sin^2 x\) over the interval \(x=0\) to \(x=2\pi\). (b) The greatest probability of finding the electron occurs at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). (c) The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a node.

Key concepts

Probability Density Function

In the world of quantum mechanics, the probability density function is a key concept that helps us understand where we might find a particle, such as an electron, within a given system. It is denoted as \(\psi^2(x)\) when we already know the wave function, which is symbolically represented by \(\psi(x)\). In simple terms, this function gives us the probability that a particle will be located at a specific point in space. When you square the wave function \(\psi(x)\), you create a function that maps out the likelihood of finding the particle at each point along the \(x\)-axis.

For example, if you sketch \(\psi^2(x)\) for our fictitious one-dimensional system where \(\psi(x) = \sin x\), you'll end up with a graph that oscillates between zero and one, since the square of the sine function is always non-negative. The graph reaches its peak where the wave function itself has the maximum amplitude, indicating the highest probability areas. Hence, by analyzing \(\sin^2 x\), we can infer that the electron has a higher chance of being found at the peaks of this function.

Wave Function

The wave function, \(\psi(x)\), is a fundamental concept in quantum mechanics that represents the quantum state of a particle or system. It's like a mathematical description of the particle's position and momentum. But unlike throwing a ball and predicting where it will land, in quantum mechanics, we can only speak in probabilities.

The wave function tells us about the probability amplitude for finding a particle in a particular state. Yet, it isn't probability itself – it's only when we square the wave function that we get our probability density function, which yields actual probabilities. The wave function's characteristics, such as its amplitude and phase, vary across different points in space, and it is typically complex-valued. That means it describes not just the likelihood of finding a particle in a place but also incorporates the phase information, which is integral to understanding interference patterns and the dual wave-particle nature of particles.

Quantum Mechanics Nodes

In quantum mechanics, a 'node' is a significant concept, highlighting points where the wave function \(\psi(x)\) and accordingly the probability density function \(\psi^2(x)\) is zero. It symbolizes a location where there is zero probability of finding the particle. This is different from our everyday experiences where things seem to be either here or there. But in the quantum realm, particles have wave-like properties, causing such curious phenomena.

Returning to our example, when \(\psi(x) = \sin x\) and we evaluate it at \(x=\pi\), \(\psi(\pi)\) is zero, creating a node at this position. Nodes are essential as they help in defining the shape and form of quantum systems by partitioning regions of space where the probability of finding a particle is non-zero. Understanding where these nodes are located aids in the visualization and comprehension of a particle's behavior in its quantum state.