Question

Although quantum systems are frequently characterized by their stationary states, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass \(m\) in a one dimensional potential well with infinite walls (a box) of width \(a\) is actually in a state with the wave function $$\Psi(x, t)=\frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t)+\Psi_{2}(x, t)\right]$$ where \(\Psi_{1}\) denotes the stationary state with quantum number \(n=1\) and \(\Psi_{2}\) denotes the state with \(n=2 .\) Calculate the probability density distribution for the position \(x\) of the particle in this state.

Short answer

Answer: The probability density distribution for the position \(x\) of the particle is given by the expression: $$P(x) = \frac{1}{a}\left[\sin^2\left(\frac{\pi x}{a}\right)+\sin^2\left(\frac{2\pi x}{a}\right)+2\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{2\pi x}{a}\right)\right]$$

Step-by-step solution

Write the expressions for the wave functions \(\Psi_1(x, t)\) and \(\Psi_2(x, t)\)

Using the general expression for the wave function of a stationary state in a potential well, we can write the wave functions for states with quantum numbers \(n = 1\) and \(n = 2\) as: $$\Psi_1(x, t) = \sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)e^{-iE_1t/\hbar}$$ $$\Psi_2(x, t) = \sqrt{\frac{2}{a}}\sin\left(\frac{2\pi x}{a}\right)e^{-iE_2t/\hbar}$$ To find the probability density distribution, we need to consider the combined wavefunction, which is given by: $$\Psi(x, t)=\frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t)+\Psi_{2}(x, t)\right]$$

Substitute the expressions for \(\Psi_1(x, t)\) and \(\Psi_2(x, t)\) into \(\Psi(x, t)\) equation

Substituting the expressions for \(\Psi_1(x, t)\) and \(\Psi_2(x, t)\) into the equation for \(\Psi(x, t)\), we get: $$\Psi(x, t)=\frac{1}{\sqrt{2}}\left[\sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)e^{-iE_1t/\hbar}+\sqrt{\frac{2}{a}}\sin\left(\frac{2\pi x}{a}\right)e^{-iE_2t/\hbar}\right]$$

Calculate the probability density distribution

The probability density distribution is given by the square of the magnitude of the wave function: $$P(x) = |\Psi(x,t)|^2$$ Let's compute \(P(x)\) for the wave function: $$P(x) = \left|\frac{1}{\sqrt{2}}\left[\sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)e^{-iE_1t/\hbar}+\sqrt{\frac{2}{a}}\sin\left(\frac{2\pi x}{a}\right)e^{-iE_2t/\hbar}\right]\right|^2$$ First, we'll drop the time-dependent terms, as they don't affect the probability density distribution for position \(x\), which simplifies to: $$P(x) = \left|\frac{1}{\sqrt{2}}\left[\sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)+\sqrt{\frac{2}{a}}\sin\left(\frac{2\pi x}{a}\right)\right]\right|^2$$ Then, we'll square the magnitude of the wave function: $$P(x) = \frac{1}{2}\left[\frac{2}{a}\sin^2\left(\frac{\pi x}{a}\right)+\frac{2}{a}\sin^2\left(\frac{2\pi x}{a}\right)+2\sqrt{\frac{2}{a}}\sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{2\pi x}{a}\right)\right]$$ Simplifying, we arrive at the final expression for the probability density distribution: $$P(x) = \frac{1}{a}\left[\sin^2\left(\frac{\pi x}{a}\right)+\sin^2\left(\frac{2\pi x}{a}\right)+2\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{2\pi x}{a}\right)\right]$$ This expression gives the probability density distribution for the position \(x\) of the particle in the given wave function, which is a linear combination of the stationary states with quantum numbers \(n=1\) and \(n=2\).

Key concepts

Quantum Particle

When we speak of a quantum particle, we refer to an entity such as an electron or a photon that behaves according to the strange and counterintuitive rules of quantum mechanics. These particles can display wave-like properties, an aspect showcased by the famous double-slit experiment. In a classical sense, we'd expect particles to have well-defined positions and velocities at any given time. However, quantum particles are described probabilistically.

A crucial aspect of quantum particles is that their behavior is mathematically represented by a wave function. This function encodes information such as the probability of finding the particle in a particular position when measured. Unlike classical particles, quantum particles can exist in superpositions of states, a fact that is key in understanding complex quantum phenomena like entanglement and quantum computing.

Stationary States

Stationary states are a foundational concept in quantum mechanics. These special quantum states are stable over time—meaning that a system in a stationary state has a wave function that experiences only a phase change due to time evolution. In simpler terms, the probability distribution associated with finding the particle in space does not change as time passes.

In the context of the textbook exercise, stationary states are represented with discreet energy levels within a one-dimensional potential well. When a quantum particle like an electron is trapped in such a well, it can only occupy specific energy levels (quantized energy states). The textbook solutions show how wave functions for different stationary states are combined to describe the particle's overall state, which influences where the particle may be found within the confining walls of the potential well.

Wave Function

The wave function is the heart of quantum mechanics. It is a fundamental mathematical function that provides the complete description of the quantum state of a particle or a system of particles. For a single quantum particle, the wave function typically depends on position and time and is represented symbolically by \(\Psi(x, t)\).

Physically, the absolute square of the wave function represents the probability density distribution. It tells us the likelihood of finding the particle at a particular position and time. Understanding the wave function and its evolution is crucial for calculating and predicting quantum phenomena. The exercise you're looking at demonstrates the process of combining wave functions to describe a more complex state, which is common practice in quantum mechanics when dealing with superpositions of states.

One Dimensional Potential Well

A one-dimensional potential well, often referred to as an 'infinite square well', is a fundamental model in quantum mechanics. It is a hypothetical scenario where a quantum particle is trapped in a box with infinitely high walls. This model is used to illustrate some of the most essential concepts of quantum mechanics, such as quantized energy levels and wave function confinement.

In our context, the width of the well \(a\) is a vital parameter that determines the quantization of energy. As per textbook solution approaches, the formulation of wave functions \(\Psi_1(x, t)\) and \(\Psi_2(x, t)\) for the first and second energy levels, respectively, is informed by the width of the well. These functions reflect the probability distributions for these stationary states and are bound by the constraints of the well's physical dimensions.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at the molecular, atomic, nuclear, and even smaller microscopic levels. It is renowned for challenging classical notions of physics, introducing concepts like quantization, wave-particle duality, and entanglement. Unlike classical physics, which is deterministic, quantum mechanics is fundamentally probabilistic.

The exercise at hand is a classical example of quantum mechanics in action. It demonstrates how particles behave within the confines of quantum constraints (the one-dimensional potential well) and how their physical properties, such as position and momentum, are not deterministic but described by a probability distribution. Through these concepts, quantum mechanics not only enhances our understanding of the universe at fundamental levels but also opens up possibilities for technologies like quantum computing and quantum cryptography.