Question

Electron and positron beams are collided, and pairs of tau leptons are produced. If the angular distribution of the tau leptons varies as \(\left(1+\cos ^{2} \theta\right),\) what fraction of the tau lepton pairs will be captured in a detector that covers only the angles from \(60^{\circ}\) to \(120^{\circ} ?\)

Short answer

Answer: The fraction of tau lepton pairs detected in the given angular range is \(\frac{1}{N} \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \left(1+\cos^2 \theta\right)d\theta\), where \(N\) is the normalization factor calculated as \(\int_{0}^{2\pi} \left(1+\cos^2 \theta\right)d\theta\). Calculate the integral to find the exact value.

Step-by-step solution

Convert degrees to radians

Since we are working with trigonometric functions, convert the given angles from degrees to radians. To convert degrees to radians, we multiply the degree value by \(\frac{\pi}{180}\). So, \(60^{\circ}=\frac{\pi}{3}\) and \(120^{\circ}= \frac{2\pi}{3}\).

Normalize the probability distribution function

Since the function \((1+\cos^2 \theta)\) represents the probability distribution, we need to ensure it is normalized, i.e., the total probability should be equal to 1. To normalize the function, first, we will integrate it from \(0\) to \(2\pi\). Let's denote the normalized function as \(P(\theta)\). The normalization factor is given by: \(\displaystyle N= \int_{0}^{2\pi} \left(1+\cos^2 \theta\right)d\theta\) Now divide the original function by N to get the normalized distribution function. Then, we will have a normalized function, \(P(\theta) = \frac{1+\cos^2 \theta}{N}\).

Calculate the area under the curve within the range \([\frac{\pi}{3}, \frac{2\pi}{3}]\)

To find the fraction of tau lepton pairs detected between angles \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\), we need to find the area under the curve of the normalized function within this range. \(\text{Area detected} = \displaystyle \int_{\frac{\pi}{3}}^{\frac{2\pi}{3} } P(\theta)d\theta = \frac{1}{N} \int_{\frac{\pi}{3}}^{\frac{2\pi}{3} } \left(1+\cos^2 \theta\right)d\theta\)

Calculate the total area under the curve

Since we have already normalized the function, the total area under the curve of \(P(\theta)\) is equal to 1.

Determine the fraction of tau lepton pairs detected

Divide the area detected (step 3) by the total area under the curve (step 4) to determine the fraction of tau lepton pairs detected. \(\text{Fraction of detected pairs} = \frac{\text{Area detected}}{\text{Total area}} = \frac{1}{N} \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \left(1+\cos^2 \theta\right)d\theta\) Calculate the integral, and you will get the fraction of tau lepton pairs detected in the given angular range.

Key concepts

Normalization of Probability Distribution

Understanding the normalization of a probability distribution is essential in physics, particularly in quantum mechanics and statistical mechanics. Normalization ensures that the total probability for all possible outcomes equals one, which aligns with the fundamental principle of probability.

Let's take the tau leptons angular distribution problem as an example. The angular distribution function given is \(1+\cos^2 \theta\). However, this function is not immediately useful as a probability distribution because it doesn't account for the fact that the total probability should be equal to one. To correct this, we need to normalize the function.

Normalization involves integrating the function over the entire range of possible outcomes, which, in this case, is the complete range of angles from 0 to \(2\pi\) radians. After integrating, we obtain a normalization factor, which we use to scale our original function. The result is a probability density function that can be used to find probabilities of certain outcomes within a given range - such as the angle range covered by a detector.

To normalize, we first calculate the integral of the original function to find the normalization factor \(N\). The normalized probability distribution \(P(\theta)\) becomes \(\frac{1+\cos^2 \theta}{N}\), ensuring that when \(P(\theta)\) is integrated over the entire range, the total probability is equal to one.

Integration in Physics

Integration plays a pivotal role in physics; it allows us to calculate quantities like area, volume, and other physical properties that depend on summing over continuous variables. In the case of the tau leptons problem, we deal with the concept of integration to find the probability of certain events occurring within a specific range.

We begin by integrating the normalized probability distribution function \(P(\theta)\) over the range of interest, which is from \(\frac{\pi}{3}\) to \(\frac{2\pi}{3}\) radians. This process calculates the 'area under the curve' which represents the probability of detecting tau lepton pairs within that angular range.

The calculation of this area, using definite integration, gives us the desired fraction of events that our detector would capture. This example illustrates the power of integration in translating a theoretical probability distribution into tangible, physical predictions about the behavior of particles in an experiment.

Trigonometric Functions in Physics

Trigonometric functions such as sine, cosine, and tangent are essential tools in physics, used to describe oscillations, waves, and rotations. The tau lepton problem utilizes the squared cosine function, representing an angular distribution that is dependent on the angle \(\theta\).

In physics, these functions allow us to model the periodic nature of phenomena like the movement of a pendulum or the distribution of particles in space. The squaring of the cosine function in the expression \(1 + \cos^2 \theta\) suggests that the probability of detecting tau leptons isn't uniform across all angles. Instead, it's more likely to detect them at certain angles due to the distribution's dependence on \(\cos^2 \theta\).

By integrating this trigonometric function over the given range, we get insight into the physical system in question. For example, the squared cosine factor increases the detected fraction around the angles where the cosine function is at its maximum, which is at 0 and \(\pi\) radians. Trigonometry provides us with the language needed to quantify and understand these patterns in the physical world.