Question

Although quantum systems are frequently characterized by their stationary states or energy eigenstates, 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 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 formula: $$ P(x,t) = \frac{1}{a}\left(\sin^2\frac{\pi x}{a} + \sin^2\frac{2\pi x}{a} + \sin\frac{\pi x}{a}\sin\frac{2\pi x}{a}\right).$$

Step-by-step solution

Identify the Wavefunctions Psi1 and Psi2

To find the stationary states (energy eigenstates) wavefunctions for a particle in an infinite potential well, we use the following form of the wavefunction for stationary states: $$ \Psi_n(x) = \sqrt{\frac{2}{a}}\sin\frac{n\pi x}{a}, $$ where "n" is the quantum number and "a" is the width of the infinite box. We're given \(\Psi_1\) and \(\Psi_2\), which denotes the stationary states with quantum numbers n=1 and n=2 respectively: $$\Psi_1(x) = \sqrt{\frac{2}{a}}\sin\frac{\pi x}{a} \quad\text{and}\quad \Psi_2(x) = \sqrt{\frac{2}{a}}\sin\frac{2\pi x}{a}.$$

Calculate the Wavefunction Psi

Now, substituting the expressions for \(\Psi_1\) and \(\Psi_2\) into the given wavefunction: $$ \Psi(x,t) = \frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t) + \Psi_{2}(x, t)\right], $$ we get: $$ \Psi(x,t) = \frac{1}{\sqrt{2}}\left[\sqrt{\frac{2}{a}}\sin\frac{\pi x}{a} + \sqrt{\frac{2}{a}}\sin\frac{2\pi x}{a}\right]. $$

Compute the Probability Density Distribution

To compute the probability density distribution of the position, we need to calculate the square of the modulus of the wavefunction: $$P(x,t) = |\Psi(x,t)|^2.$$ In our case, $$P(x,t) = \left|\frac{1}{\sqrt{2}}\left[\sqrt{\frac{2}{a}}\sin\frac{\pi x}{a} + \sqrt{\frac{2}{a}}\sin\frac{2\pi x}{a}\right]\right|^2$$ so we have: $$ P(x,t) = \frac{1}{2}\left(\frac{2}{a}\sin^2\frac{\pi x}{a} + \frac{2}{a}\sin^2\frac{2\pi x}{a} + 2\sqrt{\frac{2}{a}}\sin\frac{\pi x}{a}\sin\frac{2\pi x}{a}\right).$$ Simplifying the expression, we obtain the probability density distribution for the position x of the particle: $$ P(x,t) = \frac{1}{a}\left(\sin^2\frac{\pi x}{a} + \sin^2\frac{2\pi x}{a} + \sin\frac{\pi x}{a}\sin\frac{2\pi x}{a}\right).$$

Key concepts

Quantum Mechanics

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. It introduces a significant departure from classical mechanics and gives rise to phenomena that cannot be explained by classical physics, such as wave-particle duality and quantization of energy.

The quantum world is governed by probability, with particles not having definitive positions or velocities until they are measured. Instead, their behavior is described statistically in terms of probability distributions. Quantum mechanics employs complex mathematical structures to make predictions about the behavior of particles, and these predictions have been confirmed with high levels of accuracy through numerous experiments.

Wave Function

In quantum mechanics, the wave function is a mathematical expression used to describe the quantum state of a system. Represented by the symbol \( \Psi \), the wave function contains all the information about a particle's position, momentum, and other physical properties.

A key feature of the wave function is that it is generally complex-valued, meaning it contains both a magnitude and a phase component. To derive physically meaningful information, such as the probability of finding a particle in a certain location, one must calculate the square of the wave function's magnitude, which gives the probability density. The act of squaring this complex function to find real values that can be measured is a central concept in quantum mechanics.

Stationary States

Stationary states, or energy eigenstates, are quantum states of a system where the system's energy remains constant with time. These states are beautifully exemplified by the particle in a box (infinite potential well) scenario, where discrete energy levels are allowed.

A quantum particle in a stationary state is described by a wave function that has a time-independent probability density. That means if you were to map out the probability of finding the particle at various positions, that map would not change over time. Stationary states are associated with specific energy levels, and when a particle transitions between these states, it emits or absorbs energy in discrete quantities called quanta.

Infinite Potential Well

An infinite potential well, often referred to as a quantum box, is a theoretical construct used to demonstrate many principles of quantum mechanics, including quantization and the concept of stationary states. It is a potential well with infinitely high walls, where a particle is confined to a finite space.

In such a well, the particle cannot exist outside a certain region, as it would require an infinite amount of energy to escape. Thus, it is confined to a one-dimensional space between the well's boundaries. The wave functions that describe the particle's states within this well take on distinct forms, characterized by standing waves that correspond to different energy levels. The wave functions for these states are sinusoidal, with nodes at the walls of the well, and vary based on quantum numbers, which denote the energy level or the state of the particle.