Question

To dexcrihe a matter wave, dues the funstion \(A \sin (k x)\) costwn have a well-detined energy? Explain.

Short answer

No, the function \(A \sin(kx)\) does not provide a well-defined energy since it doesn't contain terms that can give information about the potential energy of the quantum state.

Step-by-step solution

Understanding the Wave Function

Note that the given wave function is \(A \sin(kx)\). This represents a sinusoidal wave, which could be a possible wave function of a particle in quantum mechanics. In this wave function 'A' signifies the amplitude of the wave, 'k' the wave number and 'x' the position.

Energy of the Wave Function

Bear in mind that in quantum physics, the total energy E in a wave function is associated with its frequency and wavelength. It's important to note that this given wave function doesn't have an explicit energy term or frequency term. Therefore, from here it's evident that the function \(A \sin(kx)\) does not contain any term that allows us to extract information about the potential energy of the quantum state.

Final Analysis

Based on the above explanations and analysis, it's clear that the wave function \(A \sin(kx)\) alone does not present the information required to define or determine the energy of the quantum state. Hence, the answer to whether this function has a well-defined energy is no. For it to have a well-defined energy, it would need to be an eigenfunction of the energy operator, i.e., it would need to satisfy the Schrödinger equation.

Key concepts

Wave Function

In quantum mechanics, a wave function is crucial as it provides all the necessary information about the quantum state of a particle. Specifically, it describes the probability amplitude for a particle's position, momentum, and other variables. The wave function, often denoted by the Greek letter \( \psi \), is a complex-valued function that gives the probability density of a particle being found in a particular state. For instance, function \( A \sin(kx) \) represents a sinusoidal wave. Here's a breakdown of its components:

• A: The amplitude of the wave, indicating the maximum height of the wave.
• k: The wave number, related to the wavelength through the relation \( k = \frac{2\pi}{\lambda} \).
• x: The position variable, representing the location at which the wave function is evaluated.

This particular wave function doesn't provide a complete picture of the particle's energy characteristics by itself, as it lacks a term for frequency, which is directly linked to energy in quantum mechanics.

Energy in Quantum Mechanics

Energy in quantum mechanics is a critical concept as it defines the dynamic state of a particle. The energy of a quantum system is often related to the wave function's frequency. According to Planck's relation, the energy \( E \) of a particle is connected to its frequency \( u \) by the equation:\[ E = h u \]where \( h \) is the Planck constant. This relation implies that any well-defined energy state must involve a frequency term in its wave function. However, in the case of the wave function \( A \sin(kx) \), there's no explicit frequency term. Thus, it lacks the direct information needed to determine energy.For energy to be defined, the wave function must account for kinetic and potential energies. Typically, this would mean satisfying conditions set by the Schrödinger equation. Without such terms, like potential energy or explicit frequency representation, we cannot ascertain a straightforward energy interpretation for the sinusoidal wave function discussed.

Schrödinger Equation

The Schrödinger Equation is the fundamental equation of quantum mechanics used to calculate the wave function and energy levels of a quantum system. It defines how the quantum state of a physical system changes over time. The equation is typically presented as:\[ i\hbar \frac{\partial}{\partial t} \psi(x, t) = \hat{H} \psi(x, t)\]where:

• \( i \) is the imaginary unit.
• \( \hbar \) is the reduced Planck constant.
• \( \psi(x, t) \) represents the wave function of the system.
• \( \hat{H} \) is the Hamiltonian operator, which corresponds to the total energy (kinetic plus potential) of the system.

For a wave function to have a well-defined energy as an eigenvalue, it must satisfy the time-independent Schrödinger equation:\[\hat{H} \psi = E \psi\]This indicates that \( \psi \), the wave function, is an eigenfunction of the Hamiltonian operator with \( E \) as the eigenvalue, representing energy. For \( A \sin(kx) \) to be linked definitively to an energy state, it must fulfill this eigenfunction-eigenvalue condition, which it currently doesn't hence it fails to describe an energy-state satisfactorily.