Question

Calculate the expectation values \(\langle x\rangle\) and \(\left\langle x^{2}\right\rangle\) for a particle in the state \(n=5\) moving in a one- dimensional box of length \(2.50 \times 10^{-10} .\) Is \(\left\langle x^{2}\right\rangle=\langle x\rangle^{2} ?\) Explain your answer.

Short answer

The expectation values for a particle in the state \(n=5\) moving in a one-dimensional box of length \(2.50 \times 10^{-10}\) (given in meters) are calculated as follows: - \(\langle x \rangle = 1\) - \(\langle x^2 \rangle \approx 1.0263\) Since \(\langle x^2 \rangle \neq \langle x \rangle^2\), the uncertainty principle is demonstrated in the system, which means there is inherent uncertainty and dispersion associated with the particle's actual position \(x\).

Step-by-step solution

Set up the Parameters

We have the following parameters: - Quantum number \(n=5\) - Box length \(L = 2.50 \times 10^{-10}\) m

Calculate the Wave Function

We will calculate the wave function \(\psi_{n}(x)\) using the formula: \[ \psi_{n}(x) = \sqrt{\frac{2}{L}}sin\left(\frac{n\pi x}{L}\right) \]

Calculate expectation value \(\langle x \rangle\)

Now we will calculate the expectation value \(\langle x\rangle\) using the formula: \[ \langle x\rangle = \int_{0}^{L}x|\psi_{n}(x)|^2dx = \frac{2}{L}\int_{0}^{L}x\sin^2\left(\frac{5\pi x}{L}\right)dx \] Using the property \(\int_{0}^L x\sin^2\left(\frac{n\pi x}{L}\right)dx = \frac{L}{2}\) for a one-dimensional box, we get: \[ \langle x\rangle = \frac{2}{L}\times\frac{L}{2} = 1\] So, the expectation value \(\langle x\rangle\) is \(1\).

Calculate expectation value \(\langle x^2 \rangle\)

Next, we will calculate the expectation value \(\langle x^2\rangle\) using the formula: \[ \langle x^2\rangle = \int_{0}^{L}x^2|\psi_{n}(x)|^2dx = \frac{2}{L}\int_{0}^{L}x^2\sin^2\left(\frac{5\pi x}{L}\right)dx \] Using the property \(\int_{0}^L x^2\sin^2\left(\frac{n\pi x}{L}\right)dx = \frac{L^2}{3}-\frac{L^2}{2n^2\pi^2}\) for a one-dimensional box, we get: \[ \langle x^2\rangle = \frac{2}{L}\times\left(\frac{L^2}{3}-\frac{L^2}{2(5^2)(\pi^2)}\right) = \frac{L}{3}-\frac{L}{50\pi^2} \] Plugging the value of \(L = 2.50 \times 10^{-10}\) m, we get: \[ \langle x^2\rangle = \frac{L}{3}-\frac{L}{50\pi^2} = 1.0263 \] So, the expectation value \(\langle x^2\rangle\) is approximately \(1.0263\).

Determine if \(\langle x^2 \rangle = \langle x \rangle^2\) and Explain

Now, let's compare the values of \(\langle x^2\rangle\) and \(\langle x\rangle^2\): \[ \langle x^2 \rangle = 1.0263\] \[ \langle x \rangle^2 = (1)^2 = 1\] Here, we observe that: \[ \langle x^2 \rangle \neq \langle x \rangle^2\] The inequality is implied by the uncertainty principle. Even if the particle's average position, represented by \(\langle x \rangle\), is known, there is always inherent uncertainty and dispersion associated with its actual position \(x\). This uncertainty is manifested in \(\langle x^2 \rangle \neq \langle x \rangle^2\).

Key concepts

Expectation Values

Expectation values in quantum mechanics offer a statistical insight into the possible outcomes of measurements on a quantum system. Rather than predicting one specific outcome, they represent the average value of a physical quantity over many measurements. This is especially relevant in quantum systems, where the wave function describes a probability distribution rather than a definite state.

• The expectation value of position, \( \langle x \rangle \), indicates the mean position of a particle across numerous measurements.
• The expectation value of an operator, like \( \langle x^2 \rangle \), generally differs from \( (\langle x \rangle)^2 \), illustrating that these values often capture different aspects of the distribution.
• In mathematical terms, these values are calculated by integrating over the probability density function defined by the wave function.

At the core, expectation values reveal the probabilistic nature of quantum mechanics and help in understanding broader statistical behavior, as seen in the difference between \( \langle x^2 \rangle \) and \( (\langle x \rangle)^2 \).

Wave Function

The wave function, often symbolized as \( \psi(x) \), is a fundamental component in quantum mechanics. It holds all the probabilistic information about a quantum system's state. For a particle in one-dimensional space, the wave function is a mathematical function that describes how the particle's position is distributed over space.For a particle in a box, the wave function takes on a specific form that satisfies the boundary conditions imposed by the box:

• Wave functions must be zero at the boundaries of the box.
• The wave function for any state \( n \), like \( \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) \), reflects oscillatory and sinusoidal behavior.
• The square of the wave function \( |\psi(x)|^2 \) represents the probability density for finding a particle at a position \( x \).

The wave function is crucial as it enables the evaluation of expectation values, demonstrating its importance in predictive and interpretative aspects of quantum mechanics.

Particle in a Box

The particle in a box model is a quintessential quantum mechanics problem that aids in understanding quantum behavior in a confined space. It represents a hypothetical particle free to move within an impenetrable box, experiencing infinite potential walls.

• The confined space introduces quantization: energy levels are discrete and labeled by a quantum number \( n \).
• Only certain wave functions that satisfy boundary conditions are possible, restricting energy and momentum to quantized states.
• It serves as a foundational model for understanding more complex systems and behaviors in quantum mechanics.

The model illustrates important principles, such as quantization and probability densities, offering a clear view of how confinement influences quantum phenomena and leading to insights into principles like the uncertainty principle.

Uncertainty Principle

The uncertainty principle, formulated by Werner Heisenberg, is a fundamental theory in quantum mechanics. It asserts that certain pairs of physical properties, like position and momentum, cannot both be precisely known at the same time.

• Mathematically, it is described with the inequality \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) and \( \Delta p \) are uncertainties in position and momentum, and \( \hbar \) is the reduced Planck's constant.
• This principle is inherent to the very nature of quantum systems, not a limitation of measurement technology.
• It reflects the wave-like properties of particles, dictating that a discrete and certain determination of position yields uncertainty in momentum and vice versa.

In the context of the exercise, the difference between \( \langle x^2 \rangle \) and \( (\langle x \rangle)^2 \) highlights the uncertainty and inherent spread in the particle's properties, showcasing the ubiquitous influence of the uncertainty principle in non-commutative spaces.