Question

For which of the following states will the particle never be found in the exact center of a square infinite potential well? a) the ground state b) the first excited state c) the second excited state d) any of the above e) none of the above

Short answer

a) the ground state b) the first excited state c) the second excited state d) none of the above Answer: b) the first excited state

Step-by-step solution

Evaluate probability density function for ground state

For the ground state, n = 1. Plug \(n = 1\) into the probability density function and evaluate it at the center of the well \(x = L/2\): $$P_1\left(\frac{L}{2}\right) = \frac{2}{L}\sin^2\left(\frac{1 \pi \cdot L/2}{L}\right) = \frac{2}{L}\sin^2\left(\frac{\pi}{2}\right) = \frac{2}{L}$$ Since the probability density for the ground state at the center of the well is nonzero, the particle can be found in the exact center of the well in the ground state.

Evaluate probability density function for the first excited state

For the first excited state, n = 2. Plug \(n = 2\) into the probability density function and evaluate it at the center of the well \(x = L/2\): $$P_2\left(\frac{L}{2}\right) = \frac{2}{L}\sin^2\left(\frac{2 \pi \cdot L/2}{L}\right) = \frac{2}{L}\sin^2(\pi) = 0$$ The probability density for the first excited state at the center of the well is zero. Thus, the particle will never be found in the exact center of the well in the first excited state.

Evaluate probability density function for the second excited state

For the second excited state, n = 3. Plug \(n = 3\) into the probability density function and evaluate it at the center of the well \(x = L/2\): $$P_3\left(\frac{L}{2}\right) = \frac{2}{L}\sin^2\left(\frac{3 \pi \cdot L/2}{L}\right) = \frac{2}{L}\sin^2\left(\frac{3\pi}{2}\right) = \frac{2}{L}$$ Since the probability density for the second excited state at the center of the well is nonzero, the particle can be found in the exact center of the well in the second excited state.

Determine the correct answer

From the above analysis, we find that the particle will never be found in the exact center of the well only for the first excited state (n=2). Hence, the correct answer is: b) the first excited state

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of matter and energy at the atomic and subatomic levels. It provides a mathematical framework for understanding the properties of particles that are very small, such as electrons, photons, and atoms. In quantum mechanics, particles exhibit wave-like properties and particle-like behavior, a concept known as wave-particle duality. Unlike classical physics, which predicts definite outcomes, quantum mechanics deals with probabilities. The fundamental equations of quantum mechanics describe how these probabilities evolve over time and how they result in quantized energy levels, an idea central to understanding phenomena like the infinite potential well problem.

One of the most important principles in 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. This principle leads to the concept of wave functions, which provide all the information about a particle's position and momentum in terms of probabilities rather than definite values.

Probability Density Function

The probability density function (PDF) in quantum mechanics describes the likelihood of finding a particle at a certain position in space. It is derived from the wave function, a complex mathematical function that encodes the state of a particle. The PDF is the square of the wave function's absolute value and is denoted as \( |\Psi(x)|^2 \).

For a particle confined in a box, like the infinite potential well, the PDF helps calculate where the particle is most likely to be found within the well. The PDF varies with the quantum state of the particle, such as ground state or excited states. In the context of the textbook problem, we can use the PDF to figure out the probability of finding a particle at the center of the infinite potential well, which is crucial for understanding why the particle never occupies the center in certain states — this is directly connected to the wave function being zero at that point for particular energy states.

Excited States

In quantum mechanics, excited states refer to the higher energy levels that a particle can occupy above the ground state, which is its lowest energy state. When a particle such as an electron transitions to an excited state, it has absorbed energy; conversely, when it transitions back to the ground state, it releases energy, often in the form of a photon. The concept of excited states is integral to explaining phenomena in quantum systems - for example, they underpin how atoms emit and absorb light.

Identifying Excited States

In the case of an infinite potential well, excited states are characterized by integer multiples (the quantum number \(n\) greater than 1) in the wave function. These states have unique wave functions and corresponding PDFs. As a practical consideration, the behaviors of these functions are responsible for whether or not a particle can be found in certain regions within the well, such as its center.

Particle in a Box

The 'particle in a box' model is a fundamental problem in quantum mechanics that illustrates how particles behave under extreme constraints. This model, which is also known as the infinite potential well, consists of a single particle trapped in a one-dimensional box with infinitely high walls. Within this well, the particle cannot exist outside the boundaries, as that would require infinite energy.

Quantum States in a Box

The energy levels in a ‘particle in a box’ system are quantized, meaning the particle can only occupy certain discrete energy states. These states are determined by solving the Schrödinger equation for the system. Each quantum state has a unique wave function and PDF. The solution to the Schrödinger equation leads to standing waves with nodes where the probability of finding the particle is zero. Consequently, for specific states—like the first excited state in the textbook problem—the particle will not be found at the center of the box, which coincides with a node in that state's wave function.