Problem 6 - University Physics with Modern Physics

Question

Compute \(|\Psi|^{2}\) for \(\Psi=\psi \sin \omega t,\) where \(\psi\) is time independent and \(\omega\) is a real constant. Is this a wave function for a stationary state? Why or why not?

Step-by-Step Solution

Answer

\(|\Psi|^2 = \psi^2 \sin^2 \omega t\). It is not stationary due to time dependence.

Step 1: Understand the Problem

We are given a function \(\Psi = \psi \sin \omega t\), where \(\psi\) is time independent and \(\omega\) is a real constant. We need to compute \(|\Psi|^2\) and determine if this represents a wave function of a stationary state.

Step 2: Express the Magnitude

To find \(|\Psi|^2\), we need to understand that \(|\Psi|\) denotes the magnitude of \(\Psi\). Since \(\Psi\) is a product of a real time-independent function \(\psi\) and \(\sin \omega t\), which is also real, the magnitude is a straightforward multiplication of the two values.

Step 3: Calculate \(|\Psi|^2\)

Calculate the squared magnitude:\[ |\Psi|^2 = |\psi \sin \omega t|^2 = (\psi \sin \omega t) \cdot (\psi \sin \omega t) = \psi^2 \sin^2 \omega t \]This shows that \(|\Psi|^2 = \psi^2 \sin^2 \omega t\).

Step 4: Analyze the Stationarity

For a wave function to be stationary, \(|\Psi|^2\) must be time-independent. However, since \(\sin^2 \omega t\) is a function of time, \(|\Psi|^2\) is clearly not constant over time. Hence, the wave function \(\Psi = \psi \sin \omega t\) is not that of a stationary state.

Key Concepts

Stationary States

Stationary states are a fascinating concept in quantum mechanics. These states refer to wave functions whose probability density, represented by \(|\Psi|^2\), does not change over time. In simpler terms, if you have a stationary wave function, the likelihood of finding a particle in a specific region remains constant. This leads to stability in the observed properties of a quantum system.

When we talk about stationary states, the wave function \(\Psi(x, t)\) typically takes the form of a product of a spatial part \(\psi(x)\) and a time-dependent phase factor \(e^{-i\omega t}\). Notably, the magnitude squared \(|\Psi|^2\) involves just the spatial function \(\psi(x)\), because the time-dependent factors cancel out during this calculation.

In the problem, however, the presence of the \(\sin \omega t\) term in the wave function suggests a time-dependent nature for \(|\Psi|^2\). Since it oscillates with time, it indicates the wave function is not stationary. Therefore, analyzing the time dependence is key in determining the nature of a state.

Wave Function Properties

Wave functions are central to understanding quantum mechanics. They describe the quantum state of a system and contain all the information about a system's behavior. Here are some critical properties:

• Normalization: Wave functions must be normalized, meaning the total probability over all space must equal one. Mathematically, this requirement is expressed as \(\int |\Psi(x)|^2 dx = 1\).
• Continuity: A valid wave function should be continuous and smooth. This ensures that probabilities remain well-defined across all points.
• Linearity: Wave functions can be added together to form new solutions. This property is crucial for understanding phenomena such as interference and superposition.

In this context, the wave function \(\Psi = \psi \sin \omega t\) suggests a physical scenario where \(\psi\) represents the spatial distribution of the particle, while \(\sin \omega t\) introduces periodic time variation. The problem highlights that \(|\Psi|^2 = \psi^2 \sin^2 \omega t\) indicates non-stationarity, showing us how wave function properties influence quantum states.

Time-Dependent Functions

Time-dependent functions are critical in describing how physical systems evolve with time in quantum mechanics. Time-dependence typically appears in the form of oscillations or exponential decay, often represented through sine, cosine, or complex exponential functions.

In many quantum problems, the time part of a wave function is extracted separately for clarity. For example, a common form includes a phase factor such as \(e^{-i \omega t}\), where \(\omega\) is the angular frequency. This form provides a direct link to the energy of the state via the relation to Planck's constant.

In our exercise, the function \(\Psi = \psi \sin \omega t\) shows how time-dependent factors explicitly determine a state's non-stationary nature. Here, \(\sin \omega t\) introduces periodic time fluctuations, significantly impacting \(|\Psi|^2\) and ensuring it changes over time. Understanding these time-dependent behaviors is pivotal for interpreting dynamical processes in quantum systems.