Question

Is it possible for the expectation value of the position of an electron to occur at a position where the electron's probability function, \(\Pi(x)\), is zero? If it is possible, give a specific example.

Short answer

Short answer: Yes, the expectation value of an electron's position can occur at a position where the probability function is zero. A specific example is an even probability function, \(\Pi(x) = A(1-x^2)\), in the range \(-1 \le x \le 1\), where the expectation value of position \(\langle{x}\rangle = 0\) occurs at \(x = 0\), even though the probability function is also zero at that position.

Step-by-step solution

Review expectation value formula and probability function

The expectation value, usually denoted as \(\langle{x}\rangle\), of a random variable \(x\) can be found using the formula: $$\langle{x}\rangle = \int_{-\infty}^{\infty} x\Pi(x)dx$$ where \(x\) represents the position of an electron in this case and \(\Pi(x)\) is the probability function that gives the probability of finding the electron at a particular position. The probability function is a non-negative function that integrates to 1 over its domain.

Analyze the condition of \(\Pi(x) = 0\) at some position

Notice that if \(\Pi(x) = 0\) at some position, let's say \(x = x_0\), it means that the probability of finding the electron exactly at that position is zero. However, the expectation value \(\langle{x}\rangle\) is an average of all positions in the domain weighted by their probabilities. So, even if the probability function \(\Pi(x)\) becomes zero at a certain position, it doesn't necessarily mean that the expectation value also becomes zero at that position, but it suggests that the contributions from other positions persist.

Provide a specific example

Consider an even probability function, given by \(\Pi(x) = A(1-x^2)\) in the range \(-1 \le x \le 1\), and zero otherwise, where \(A\) is a normalization constant. In this case, the expectation value of the position \(\langle{x}\rangle\) is given by: $$\langle{x}\rangle = \int_{-1}^{1} xA(1-x^2)dx = A\left(\frac{x^2}{2}-\frac{x^4}{4}\right)\Bigg|_{-1}^1 = 0$$ The probability function \(\Pi(x)\) is zero at \(x = -1, 0, 1\), and yet the expectation value \(\langle{x}\rangle = 0\) occurs precisely at \(x=0\), where the probability function is also zero. In conclusion, it is possible for the expectation value of the position of an electron to occur at a position where the electron's probability function is zero, as demonstrated by the given example.

Key concepts

Probability Function

Understanding the concept of a probability function is essential in quantum mechanics. This mathematical tool describes the likelihood of different outcomes. In simpler terms, it tells us how probable it is to find a particle, like an electron, at various locations in space. Imagine tossing a fair die; the probability function would assign an equal chance to landing on any of the six sides.
In quantum mechanics, the probability function, often expressed as \( \Pi(x) \), is not about dice but about the position of particles. This function is especially crucial when it comes to electrons since they don't have a defined position until measured. Instead of being at one specific spot, they exist in a range of places with varying probabilities. Importantly, the probability function must meet two conditions: it can never be negative, as probabilities are never less than zero, and when you add up all the probabilities over all possible positions, the total must be one, a concept called normalization.

Electron Position Probability

The likelihood of finding an electron at any given point is quantified by electron position probability. Unlike objects we see in our everyday lives, an electron doesn't have a fixed route or position until it's observed. Understanding electron behavior is like predicting where a particular drop of spray paint will land on a wall; it's largely unpredictable and governed by the rules of quantum mechanics.
This unpredictability is represented mathematically by the electron's probability function, which we've mentioned before as \( \Pi(x) \). This probability function has high values where the electron is more likely to be found, and it reduces to zero at points where finding the electron is impossible. However, it's important to note that while the probability can be zero at some points, this does not directly influence the overall expected position of the electron—which is what brings us to expectation values.

Expectation Value Formula

Now, onto the expectation value formula. In quantum mechanics, this formula isn't about hoping for an outcome; it's a precise calculation. It's akin to a weighted average of all possible outcomes. The expectation value formula encapsulates the idea that every possible outcome has a certain weight, which is the probability of that outcome. As seen in our textbook solution, the formula is mathematically represented as
\[ \langle{x}\rangle = \int_{-\infty}^{\infty} x\Pi(x)dx \]
When applying this formula to electron position, each position \( x \) is weighted by how probable that position is—given by the function \( \Pi(x) \). An essential insight is that the expectation value may not coincide with any particular position's probability; it's the 'center of mass' of all position probabilities. The provided exercise underlines this, showing that even a point with zero probability can be the expectation value, underscoring the fascinatingly counterintuitive nature of quantum physics.

Quantum Mechanics Principles

Finally, quantum mechanics principles form the bedrock upon which all these concepts stand. Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Unlike classical mechanics, where objects are considered to have definite positions and speeds, quantum mechanics introduces a range of probabilities and uncertainties.
Key principles include the uncertainty principle, wave-particle duality, and quantization of energy. These principles imply that particles can act as both waves and particles, their behavior cannot be predicted precisely, and phenomena like the position of an electron can be described only in terms of probability. Our textbook problem mirrors one of these principles by revealing that an electron’s expected position need not be where we're most likely to find it—defying classical intuition and affirming the probabilistic nature of the quantum world.