The wave function of a quantum particle of mass m is ψ(x)=A...

The wave function of a quantum particle of mass $m$ is $$ \psi(x)=A \cos (k x)+B \sin (k x) $$ where $A, B,$ and $k$ are constants. (a) Assuming the particle is free $(U=0),$ show that $\psi(x)$ is a solution of the Schrödinger equation (Eq. 40.15 ). (b) Find the corresponding energy $E$ of the particle.

The wave function of a quantum particle of mass $m$ is $$
\psi(x)=A \cos (k x)+B \sin (k x) $$ where $A, B,$ and $k$ are constants. (a) Assuming the particle is free $(U=0),$ show that $\psi(x)$ is a solution of the Schrödinger equation (Eq. 40.15 ). (b) Find the corresponding energy $E$ of the particle.

Key Concepts

Energy Eigenvalue

The energy eigenvalue is the specific value of energy corresponding to a wave function that satisfies the Schrödinger equation. In the context of a free particle, substituting the sinusoidal wave function into the Schrödinger equation reveals a relationship between the energy, the mass of the particle, and the wave number k. This concept is central to understanding quantization in quantum mechanics.

Free Particle

A free particle in quantum mechanics is a particle that is not subject to any external forces or potential fields (i.e., U=0). For such particles, the Schrödinger equation simplifies, and the solutions typically involve sinusoidal functions. This scenario provides fundamental insight into the behavior of particles in the absence of interactions and lays the groundwork for understanding more complex systems.

Time-Independent Schrödinger Equation

This is a central equation in quantum mechanics that describes how the quantum state (wave function) of a system with a given energy evolves spatially when the potential is time-independent. It is formulated as an eigenvalue problem where the Hamiltonian operator acting on the wave function yields an energy eigenvalue multiplied by the wave function. For a free particle with zero potential energy, this takes on a specific form that supports sinusoidal solutions.

Wave Function

In quantum mechanics, the wave function represents the state of a system and contains all the probabilistic information about a particle's position, momentum, and other physical properties. It is a fundamental concept that encapsulates the behavior of quantum systems and allows for the calculation of measurable quantities through operators acting on it.

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