Question

Exercises \(90-92\) refer to a particle described by the wave function $$ \psi(x)=\sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} $$ Calculate the uncertainty in the particle's position.

Short answer

The detailed calculations depend on the specific value of \(a\). After performing the integral calculations, the uncertainty can be found by taking square root of the difference between \(\langle x^2 \rangle\) and \(\langle x \rangle^2\).

Step-by-step solution

Calculate the expectation value of x

The formula for the expectation value of \(x\) in quantum mechanics is given by: \[ \langle x \rangle = \int_{-\infty}^{+\infty} \psi^* x \psi \, dx \]. Given the wave function is real, we can use this to calculate \[\langle x \rangle = \int_{-\infty}^{+\infty} x |\psi(x)|^2 \, dx \]

Calculate the expectation value of \(x^2\)

Next, calculate the expectation value of \(x^2\) using the following equation: \[\langle x^2 \rangle = \int_{-\infty}^{+\infty} x^2 |\psi(x)|^2 \, dx \]

Calculate the square of the expectation value of \(x\)

The square of the expectation value of position is given by: \[\langle x \rangle^2 \]

Calculate the Uncertainty

Finally, the uncertainty in position, \(\Delta x\), is given by the following equation: \[ \Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2} \]

Key concepts

Wave Function

The wave function, symbolized as \( \( \( \psi(x) \) \)\), is a fundamental concept in quantum mechanics that describes the quantum state of a particle or system. Instead of providing a single, definite trajectory for a particle, the wave function encapsulates all the possible positions a particle can be found. Mathematically, it is a complex-valued probability amplitude, and the probability of finding a particle at a specific place can be determined by taking the square of the modulus of the wave function, \(|\psi(x)|^2\). In essence, the wave function contains information about the likelihood of a particle's properties, such as position and momentum, and evolves over time according to the Schrödinger equation.

For the given particle described by the wave function \( \( \( \psi(x)=\sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} \) \)\), it reflects a scenario where the particle's position is more spread out as opposed to being at a specific point. Analyzing this wave function allows physicists to make predictions about the behavior of the particle, such as where it is likely to be found.

Expectation Value

In quantum mechanics, the expectation value is a statistical mean of a certain quantity, which means it provides the average outcome if an experiment is repeated many times. More specifically, given an observable corresponding to a physical quantity (like position, momentum, or energy), the expectation value is the average of that observable's measurement outcomes from a quantum system in a given state. The calculation of expectation values is derived from the wave function by integrating the product of the observable's operator, the wave function, and the complex conjugate of the wave function across all space.

For position, the expectation value of \(x\), represented as \( \( \( \langle x \rangle \) \)\), provides a sort of 'center of mass' or the average location of the particle's probability distribution. It is calculated as: \[ \langle x \rangle = \int_{-\infty}^{+\infty} \psi^* x \psi \, dx \]. In this case, since the wave function is real and positive, it simplifies to: \[ \langle x \rangle = \int_{-\infty}^{+\infty} x |\psi(x)|^2 \, dx \]. Similarly, \( \( \( \langle x^2 \rangle \) \)\) gives the average value of the square of the particle's position.

Uncertainty in Position

A keystone of quantum mechanics is the Heisenberg uncertainty principle which states that certain pairs of physical properties, like position and momentum, cannot be simultaneously known to arbitrary precision. The 'uncertainty' in position, denoted as \( \( \( \Delta x \) \)\), is a quantitative measure of how much the position of a particle is spread out. It represents the standard deviation of the position from its mean value and reflects the fuzziness or indeterminacy at the quantum level.

The uncertainty is calculated using the expectation values: \[ \Delta x = \sqrt{ \langle x^2 \rangle - (\langle x \rangle)^2} \], where \( \( \( \langle x^2 \rangle \) \)\) is the expectation value of the square of the position and \( \( \( \langle x \rangle \) \)\) is the expectation value of the position. This equation is a direct application of the definition of standard deviation in statistics. A large uncertainty in position means the particle is more delocalized, while a small uncertainty means its position is more precisely defined. Through the exercise provided, one can understand that even with a definite wave function such as \( \( \( \psi(x)=\sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} \) \)\), the principle of uncertainty still applies and the exact position of the particle remains inherently uncertain.